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b(of)h(them)g(emplo)n(y)f(a)g(similar)g(set)h(of)g(ideas;)g(among)f (these,)i(the)515 922 y(use)24 b(of)f(the)i(higher)e(order)g(analogues) f(of)i(the)g(standard)f(homological)f(op)r(erator)g(\(already)515 1021 y(suggested)34 b(in)h([33)o(]\),)j(and)d(the)g(Lie)g(algebraic)e (prop)r(erties)i(of)g(the)g(set)g(of)g(v)n(ector)f(\014elds)515 1121 y(in)g(normal)e(form)h(with)i(resp)r(ect)e(to)g(a)h(giv)n(en)e (linear)h(part)g(\(already)g(suggested)f(in)i([9]\).)515 1220 y(Eac)n(h)22 b(approac)n(h)g(has)i(also,)g(of)f(course,)h(its)g(o) n(wn)f(features,)h(and)g(leads)g(to)f(similar)h(but)g(not)515 1320 y(iden)n(tical)j(results.)681 1420 y(In)43 b([20)o(,)f(21)o(])h(I) g(prop)r(osed)e(an)i(approac)n(h)d(based)i(on)h(iteration)f(of)g(the)h (standard)515 1519 y(P)n(oincar)n(\023)-39 b(e)39 b(normalization)i (pro)r(cedure)h(\(the)h(reduced)g(normal)e(forms)i(th)n(us)f(obtained) 515 1619 y(w)n(ere)34 b(therefore)g(called)h(P)n(oincar)n(\023)-39 b(e)32 b(renormalized)h(forms,)k(or)d(PRF)h(for)f(short\);)39 b(this)c(is)515 1719 y(completely)26 b(algorithmic)f(and)h(requires)f (only)h(to)g(solv)n(e)f(linear)h(equations)f(at)h(eac)n(h)g(step.)681 1818 y(The)35 b(PRF)g(algorithm)e(uses)i(only)f(the)h(grading)f(b)n(y)g (homogeneit)n(y)g(degree)g(of)h(the)515 1918 y(algebra)29 b Fm(G)858 1930 y Fl(A)944 1918 y Fr(of)j(p)r(olynomial)e(v)n(ector)g (\014elds)i(in)f(normal)g(form)g(with)h(resp)r(ect)f(to)g(a)g(giv)n(en) 515 2017 y(linear)j(part)h Fk(A)p Fr(;)40 b(it)c(is)f(th)n(us)h(quite)g (general,)g(but)g(fails)f(to)g(tak)n(e)g(adv)-5 b(an)n(tage)34 b(of)h(the)h(Lie)515 2117 y(algebraic)d(structure)i(of)g Fm(G)1388 2129 y Fl(A)1442 2117 y Fr(.)60 b(Not)35 b(surprisingly)-7 b(,)35 b(taking)g(this)g(Lie)g(algebra)f(structure)515 2217 y(in)n(to)29 b(accoun)n(t)f(can)h(lead)g(to)g(a)g(considerable)f (simpli\014cation)h(of)h(the)f(computations)g(and)515 2316 y(of)e(the)h(resulting)f(simpli\014ed)h(normal)f(form.)681 2416 y(The)18 b(main)h(purp)r(ose)f(of)g(this)h(note)f(is)h(indeed)g (to)f(conjugate)g(the)h(PRF)f(approac)n(h)f(with)515 2516 y(Lie)26 b(algebraic)e(considerations,)h(so)g(to)h(obtain)g(a)g (pro)r(cedure)f(whic)n(h)h(tak)n(es)f(adv)-5 b(an)n(tage)25 b(of)515 2615 y(the)36 b(Lie)f(algebra)f(structure)h(of)g Fm(G)41 b Fr(and)35 b(k)n(eeps)g(the)h(computational)f(simplicit)n(y)g (of)h(the)515 2715 y(PRF)29 b(approac)n(h.)41 b(Rather)28 b(than)i(aiming)f(at)g(the)h(greater)e(generalit)n(y)-7 b(,)28 b(w)n(e)h(will)h(fo)r(cus)f(on)515 2814 y(a)e(structure)g(whic)n (h)h(is)f(non-generic,)f(but)i(relativ)n(ely)f(common)g(in)h (applications.)681 3014 y(The)e(discussion)g(giv)n(en)g(in)g(this)h (note)g(should)f(also)f(clarify)h(some)g(p)r(oin)n(ts)g(related)g(to) 515 3113 y(standard)h(PRF)g(approac)n(h)f(\(as)i(w)n(e)f(discuss)h(in)g (the)g(lines)g(b)r(elo)n(w\);)g(it)g(should)g(also)f(mak)n(e)515 3213 y(clear)j(ho)n(w)h(m)n(uc)n(h)g(the)g(PRF)g(approac)n(h)f(is)h (related)f(to)i(Bro)r(er's)d(ideas)i(and)g(to)g(previous)515 3313 y(w)n(ork)21 b(along)g(the)i(same)f(lines.)35 b(Actually)-7 b(,)24 b(the)f(set)g(of)f(ideas)g(emplo)n(y)n(ed)g(here)g(is)g(essen)n (tially)515 3412 y(the)39 b(same)g(\(higher)f(order)g(homological)f(op) r(erators,)j(Lie)f(algebras)e(\014ltrations\))h(as)h(in)515 3512 y(those)f(w)n(orks;)43 b(they)d(are)e(blended)h(here)f(in)i(a)e (form)h(whic)n(h)g(is)g(suitable)g(for)f(concrete)515 3611 y(computational)27 b(implemen)n(tation.)681 3711 y(It)c(should)g(b)r(e)g(stressed)f(that)h(concrete)f(computations)g(in) h([20,)g(21)o(])g(w)n(ere)f(p)r(erformed)515 3811 y(b)n(y)g(using)g (this)g(the)h(Lie)f(algebraic)f(structure,)i(although)e(this)i(p)r(oin) n(t)f(w)n(as)g(not)g(su\016cien)n(tly)515 3910 y(stressed)31 b(there;)j(this)e(could)g(ha)n(v)n(e)e(caused)i(some)f(confusion,)i (whic)n(h)f(the)g(presen)n(t)f(note)515 4010 y(should)d(hop)r(efully)h (dissipate.)39 b(T)-7 b(o)28 b(mak)n(e)f(things)h(w)n(orse)f(and)h(add) h(o)r(ccasions)d(for)i(confu-)515 4110 y(sion,)33 b(in)g([20)o(,)f(21)o (])h(the)g(term)f(PRF)h(w)n(as)e(on)h(the)h(one)f(hand)h(precisely)e (de\014ned,)j(but)f(on)515 4209 y(the)d(other)g(end)h(also)e(used)h(to) g(indicate)h(generically)d(reduced)i(normal)g(forms)f(obtained)515 4309 y(b)n(y)k(use)f(of)i(a)e(sequence)h(of)g(P)n(oincar)n(\023)-39 b(e)30 b(transformations,)i(suc)n(h)h(as)g(those)f(based)h(on)g(the)515 4408 y(Lie)27 b(algebra)f(structure.)681 4508 y(This)31 b(can)h(cause)f(confusion)g(due)h(to)f(the)i(follo)n(wing)d(fact:)45 b(if)33 b(w)n(e)e(consider)g(the)h(se-)515 4608 y(quence)h(of)h(Lie-P)n (oincar)n(\023)-39 b(e)30 b(transformations)h(prescrib)r(ed)i(b)n(y)h (the)g(PRF)f(pro)r(cedure,)h(w)n(e)515 4707 y(obtain)j(some)g(reduced)h (normal)f(form)g Fk(F)1872 4719 y Fj(1)1948 4707 y Fr(\(whic)n(h)h(is)f (of)h(course)f(a)g(PRF\).)h(If)g(the)h(se-)515 4807 y(quence)26 b(of)g(Lie-P)n(oincar)n(\023)-39 b(e)23 b(transformations)h(is)i(not)g (tak)n(en)g(in)h(this)f(order)f(but)i(according)515 4907 y(to)f(a)g(di\013eren)n(t)g(sc)n(heme,)g(e.g.)36 b(to)26 b(tak)n(e)f(adv)-5 b(an)n(tage)25 b(of)h(the)h(Lie)f(algebra)e (structure)i(on)g Fm(G)3325 4919 y Fl(A)515 5006 y Fr(\(as)g(in)h (considering)f(the)h(ab)r(o)n(v)n(e)f(men)n(tioned)h(example)f(in)h ([20)o(,)g(21)o(]\),)h(w)n(e)e(obtain)h(another)1926 5255 y(2)p eop %%Page: 3 3 3 2 bop 515 523 a Fr(reduced)21 b(normal)g(form)g Fk(F)1337 535 y Fj(2)1375 523 y Fr(.)35 b(In)22 b(general)e Fk(F)1864 535 y Fj(2)1925 523 y Fm(6)p Fr(=)j Fk(F)2066 535 y Fj(1)2103 523 y Fr(,)h(but)e(moreo)n(v)n(er)d(this)j Fk(F)2857 535 y Fj(2)2917 523 y Fr(can)f(fail)h(to)g(b)r(e)515 623 y(a)27 b(PRF)h(according)f(to)h(the)g(prop)r(er)g(de\014nition;)g (this)h(happ)r(ened)f(for)g(the)g(main)g(example)515 722 y(in)22 b([20)o(,)h(21)o(])f(and)g(will)h(also)e(b)r(e)h(the)h (case)e(for)h(the)h(example)e(B)h(considered)g(in)g(detail)g(b)r(elo)n (w.)681 922 y(The)36 b(pap)r(er)g(is)h(organized)d(as)i(follo)n(ws:)54 b(in)36 b(section)g(1)g(w)n(e)g(recall)g(the)h(main)f(facts)515 1021 y(ab)r(out)f(P)n(oincar)n(\023)-39 b(e-Dulac)31 b(\(standard\))j(normal)g(forms)g(and)h(\014x)g(notation;)j(in)d (section)f(2)515 1121 y(w)n(e)39 b(discuss)f(t)n(w)n(o)h(further)g (reduction)g(sc)n(heme,)j(i.e.)72 b(the)40 b(\\generic")d(PRF)i(pro)r (cedure)515 1220 y(and)32 b(a)g(mo)r(di\014cation)g(of)g(it)h(whic)n(h) f(mak)n(es)g(use)g(of)g(the)h(Lie)f(algebraic)e(structure)i(of)h Fm(G)3325 1232 y Fl(A)515 1320 y Fr(and)25 b(applies)g(when)h(this)g (structure)f(has)g(fa)n(v)n(ourable)f(prop)r(erties)h(\(this)h(will)g (b)r(e)g(the)g(LRF)515 1420 y(approac)n(h\);)34 b(in)f(section)g(3)f(w) n(e)h(brie\015y)f(consider)g(a)h(three-dimensional)f(example,)i(with) 515 1519 y(linear)f(part)h(corresp)r(onding)e(to)i(a)f(cen)n(ter-fo)r (cus,)i(sho)n(wing)e(the)h(simplicit)n(y)g(of)g(compu-)515 1619 y(tations)f(required)g(b)n(y)g(the)h(LRF)g(approac)n(h;)h(in)e (section)h(4)f(w)n(e)g(consider)g(a)g(case)g(where)515 1719 y(the)h(structure)g(of)g Fm(G)1177 1731 y Fl(A)1266 1719 y Fr(is)g(not)g(the)h(optimal)f(one)g(for)f(use)h(of)g(the)h(LRF)f (approac)n(h,)g(but)515 1818 y(still)h(allo)n(ws)e(for)h(its)h(use.)58 b(Finally)-7 b(,)37 b(in)e(section)f(5)g(w)n(e)h(analize)f(in)g(full)i (detail)f(a)f(simple)515 1918 y(t)n(w)n(o-dimensional)25 b(example,)i(giving)f(a)h(completely)g(explicit)h(description)e(of)h (the)h(renor-)515 2017 y(malizing)e(transformations)f(and)i (renormalized)e(forms)i(up)g(to)g(order)f(six,)h(and)f(compare)515 2117 y(the)i(results)f(obtained)g(with)h(the)g(PRF)g(and)f(with)h(the)g (LRF)g(approac)n(h.)681 2217 y(W)-7 b(e)28 b(also)f(pro)n(vide)g(t)n(w) n(o)g(app)r(endices:)38 b(in)28 b(the)h(\014rst)f(one)f(w)n(e)h (discuss)g(Bruno's)f(treat-)515 2316 y(men)n(t)c(of)g(PRF)f(and)h (remark)e(that)i(his)g(de\014nition)g(is)g(not)g(equiv)-5 b(alen)n(t)22 b(to)h(the)g(one)g(giv)n(en)f(in)515 2416 y([20)o(,)h(21)o(],)g(so)f(that)h(the)g(example)g(he)f(considers)g(in)h ([12)o(,)f(13)o(])h({)f(and)h(whic)n(h)g(falls)f(in)h(the)g(case)515 2516 y(considered)d(in)i(section)f(5)g({)g(do)r(es)g(not)h(apply)f(to)g (PRFs;)i(in)f(app)r(endix)g(B)f(w)n(e)g(brie\015y)g(recall)515 2615 y(the)j(main)f(ideas)g(put)i(forw)n(ard)d(b)n(y)h(Bro)r(er)f(and)i (Baider,)f(and)h(describ)r(e)f(the)h(Bro)r(er-Baider)515 2715 y(reduction)j(pro)r(cedure)g(in)g(the)h(language)e(emplo)n(y)n(ed) h(in)h(the)g(presen)n(t)f(pap)r(er.)515 2947 y Fi(Ac)m(kno)m(wledgemen) m(t)515 3100 y Fr(This)k(w)n(ork)e(w)n(as)h(started)h(in)g(the)g(Ph)n (ysics)f(Departmen)n(t)h(of)g(Univ)n(ersit\023)-42 b(a)30 b(di)h(Roma,)g(and)515 3200 y(completed)22 b(in)g(the)g(Mathematics)g (Departmen)n(t)g(of)g(Univ)n(ersit\023)-42 b(a)21 b(di)h(Milano.)35 b(I)22 b(w)n(ould)f(lik)n(e)515 3300 y(to)26 b(thank)g(A.)g(Degasp)r (eris)f(and)h(P)-7 b(.)25 b(San)n(tini)h(\(Roma\))g(and)g(D.)h(Bam)n (busi)e(and)h(L.)g(Galgani)515 3399 y(\(Milano\))31 b(for)f(their)g (kindest)h(hospitalit)n(y)-7 b(.)46 b(The)31 b(supp)r(ort)g(of)f(\\F)-7 b(ondazione)30 b(CARIPLO)515 3499 y(p)r(er)20 b(la)g(ricerca)f(scien)n (ti\014ca")h(under)g(the)h(pro)5 b(ject)20 b Fh(T)-6 b(e)l(oria)24 b(del)t(le)h(p)l(erturb)l(azioni)f(p)l(er)g(sistemi)515 3599 y(c)l(on)29 b(simmetria)g Fr(is)f(gratefully)e(ac)n(kno)n (wledged.)515 3873 y Fq(1)134 b(Standard)45 b(normal)h(forms)515 4055 y Fr(Let)37 b(us)f(\014rst)h(collect)f(some)g(basic)h(form)n(ulas) e(ab)r(out)i(\(standard\))f(normal)g(forms.)64 b(W)-7 b(e)515 4155 y(w)n(ork)29 b(in)i Fs(R)893 4118 y Fl(n)969 4155 y Fr(with)h(basis)e Fm(f)p Fs(e)1455 4167 y Fj(1)1491 4155 y Fk(;)14 b(:::;)g Fs(e)1678 4167 y Fl(n)1723 4155 y Fm(g)31 b Fr(and)g(co)r(ordinates)e(\()p Fk(x)2484 4124 y Fj(1)2522 4155 y Fk(;)14 b(:::;)g(x)2712 4124 y Fl(n)2758 4155 y Fr(\);)33 b(and)e(consider)e(a)515 4254 y(v)n(ector)d(\014eld)h Fk(X)34 b Fr(in)28 b Fs(R)1215 4224 y Fl(n)1287 4254 y Fr(ha)n(ving)e(a)h(zero)g(in)g(the)h(origin.)36 b(This)27 b(is)g(written)h(in)f(co)r(ordinates)515 4354 y(as)k Fk(X)37 b Fr(=)30 b Fk(f)872 4324 y Fl(i)899 4354 y Fr(\()p Fk(x)p Fr(\))p Fk(@)1054 4366 y Fl(i)1115 4354 y Fr(\(here)h(and)h(b)r(elo)n(w,)h Fk(@)1805 4366 y Fl(i)1863 4354 y Fr(=)d Fk(@)5 b(=@)g(x)2145 4324 y Fl(i)2172 4354 y Fr(\);)34 b(w)n(e)e(expand)g(the)g(v)n(ector)e(function)515 4453 y Fk(f)9 b Fr(\()p Fk(x)p Fr(\))33 b(in)g(homogeneous)e(terms)i (as)f Fk(f)9 b Fr(\()p Fk(x)p Fr(\))32 b(=)1957 4391 y Fg(P)2045 4412 y Ff(1)2045 4478 y Fl(k)q Fj(=0)2183 4453 y Fk(f)2224 4465 y Fl(k)2265 4453 y Fr(\()p Fk(x)p Fr(\),)j(with)e Fk(f)2669 4465 y Fl(k)2710 4453 y Fr(\()p Fk(ax)p Fr(\))f(=)g Fk(a)3038 4423 y Fl(k)q Fj(+1)3162 4453 y Fk(f)3203 4465 y Fl(k)3244 4453 y Fr(\()p Fk(x)p Fr(\).)515 4553 y(When)c(considering)e(co)r(ordinate)g(expressions,)f (w)n(e)i(will)h(write)f Fm(V)2596 4565 y Fl(k)2664 4553 y Fr(for)g(the)h(set)f(of)g(v)n(ector)515 4653 y(\014elds)i(with)h (comp)r(onen)n(ts)f(homogeno)r(eus)f(of)i(order)e(\()p Fk(k)23 b Fr(+)c(1\))29 b(in)h(the)g Fk(x)p Fr(.)43 b(It)30 b(is)f(clear)g(that)515 4752 y(under)e(comm)n(utator)g(w)n(e)g(ha)n(v)n (e)f([)p Fm(V)1606 4764 y Fl(k)1647 4752 y Fk(;)14 b Fm(V)1735 4764 y Fl(m)1798 4752 y Fr(])23 b Fm(\032)g(V)1983 4764 y Fl(k)q Fj(+)p Fl(m)2133 4752 y Fr(.)681 4852 y(W)-7 b(e)34 b(also)e(consider)h(the)g(linearization)g Fk(Ax)h Fr(of)f Fk(f)42 b Fr(in)34 b(the)g(origin,)g(giv)n(en)f(b)n(y)g Fk(A)3212 4822 y Fl(i)3223 4874 y(j)3291 4852 y Fr(:=)515 4962 y(\()p Fk(@)5 b(f)646 4932 y Fl(i)673 4962 y Fk(=@)g(x)811 4932 y Fl(j)846 4962 y Fr(\)\(0\).)73 b(W)-7 b(e)39 b(write)h(in)g (general)e Fk(f)9 b Fr(\()p Fk(x)p Fr(\))43 b(=)g Fk(Ax)27 b Fr(+)f Fk(F)12 b Fr(\()p Fk(x)p Fr(\),)43 b(where)c Fk(F)12 b Fr(\()p Fk(x)p Fr(\))41 b(collects)1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fr(nonlinear)26 b(terms)i(only)-7 b(.)681 623 y(W)g(e)31 b(denote)h(b)n(y)f Fk(\033)h Fr(:=)d Fm(f)p Fk(\025)1504 635 y Fj(1)1541 623 y Fk(;)14 b(:::;)g(\025)1732 635 y Fl(n)1778 623 y Fm(g)31 b Fr(the)g(sp)r(ectrum)h(of)f Fk(A)p Fr(.)48 b(If)32 b(the)f Fk(\025)2876 635 y Fl(i)2936 623 y Fr(satisfy)g(some)515 722 y(relation)1533 764 y Fl(n)1494 789 y Fg(X)1500 966 y Fl(i)p Fj(=1)1628 868 y Fk(\026)1678 880 y Fl(i)1705 868 y Fk(\025)1753 880 y Fl(i)1804 868 y Fr(:=)23 b(\()p Fk(\026)c Fm(\001)f Fk(\025)p Fr(\))52 b(=)e Fk(\025)2352 880 y Fl(\013)3273 868 y Fr(\(1\))515 1094 y(where)30 b Fk(\013)g Fr(=)f(1)p Fk(;)14 b(:::;)g(n)p Fr(,)32 b Fk(\026)1275 1106 y Fl(i)1334 1094 y Fr(are)e(non-negativ)n(e)f(in)n(tegers,)i(and)h Fm(j)p Fk(\026)p Fm(j)d Fr(:=)2711 1032 y Fg(P)2798 1119 y Fl(i)2840 1094 y Fk(\026)2890 1106 y Fl(i)2947 1094 y Fm(\025)f Fr(2,)k(w)n(e)f(sa)n(y)515 1194 y(that)d Fk(A)g Fr(is)f Fh(r)l(esonant)p Fr(.)681 1294 y(A)j(v)n(ector)f(of)h (the)h(form)f Fs(v)f Fr(=)e(\()p Fk(x)1717 1254 y Fl(\026)1757 1262 y Fe(1)1717 1316 y Fj(1)1795 1294 y Fk(:::x)1911 1263 y Fl(\026)1951 1271 y Fd(n)1911 1314 y Fl(n)1996 1294 y Fr(\))p Fs(e)2072 1306 y Fl(\013)2120 1294 y Fr(,)k(with)g Fk(\026)2416 1306 y Fl(i)2474 1294 y Fr(and)f Fk(\013)g Fr(as)g(in)g(\(1\),)h(is)f(called)515 1393 y(a)25 b Fh(r)l(esonant)i (monomial)j(ve)l(ctor)p Fr(;)c(the)g(linear)f(span)g(of)g(resonan)n(t)f (monomial)h(v)n(ector)f(is)i(the)515 1493 y(linear)i(space)h(of)g Fh(r)l(esonant)i(ve)l(ctors)p Fr(.)42 b(When)30 b(some)e(am)n(biguit)n (y)g(could)h(arise,)g(w)n(e)g(sp)r(ecify)515 1592 y(these)e(are)g Fh(r)l(esonant)i(with)i Fk(A)p Fr(.)681 1692 y(The)23 b(P)n(oincar)n(\023)-39 b(e-Dulac)20 b(theorem)i(a\016rms)h(that)g(it)g (is)g(p)r(ossible)g(to)g(\014nd)h(a)f(sequence)f(of)515 1792 y(near-iden)n(tit)n(y)j(c)n(hanges)h(of)h(co)r(ordinates)e(suc)n (h)i(that)g(in)g(the)g(new)g(co)r(ordinates)f(\(denoted)515 1891 y(again)j(b)n(y)g Fk(x)p Fr(\))i(the)g(v)n(ector)e(\014eld)h Fk(X)37 b Fr(is)30 b(expressed)f(as)g Fk(X)34 b Fr(=)26 b Fk(g)2454 1861 y Fl(i)2482 1891 y Fr(\()p Fk(x)p Fr(\))p Fk(@)2637 1903 y Fl(i)2665 1891 y Fr(,)31 b(with)g Fk(g)s Fr(\()p Fk(x)p Fr(\))d(=)f Fk(Ax)20 b Fr(+)515 1991 y Fk(G)p Fr(\()p Fk(x)p Fr(\),)29 b(and)e(with)h(the)g(nonlinear)f(term)h Fk(G)p Fr(\()p Fk(x)p Fr(\))h(b)r(eing)e(the)h(sum)g(of)g(resonan)n(t)e (terms)h(only)-7 b(.)515 2091 y(The)39 b(c)n(hange)f(of)h(co)r (ordinates)f(de\014ned)h(b)n(y)g(the)h(whole)f(sequence)f(is)h(in)h (general)d(only)515 2190 y(formal,)25 b(but)h(it)f(is)g(analytic)g (\(in)h(some)f(op)r(en)g(neigh)n(b)r(ourho)r(o)r(d)f(of)h(the)h (origin\))e(if)i(zero)e(lies)515 2290 y(out)j(of)h(the)g(con)n(v)n(ex)e (h)n(ull)i(of)f Fk(\033)k Fr(in)d(the)g(complex)f(plane)h(\(P)n(oincar) n(\023)-39 b(e)24 b(condition\).)681 2389 y(It)e(is)f(actually)g(also)f (p)r(ossible)h(to)g(express)f(the)i(c)n(hanges)e(of)h(co)r(ordinates)f (as)h(the)h(time-)515 2489 y(one)27 b(action)f(of)i(v)n(ector)e (\014elds)h Fk(H)1540 2501 y Fl(k)1604 2489 y Fr(=)c Fk(h)1740 2459 y Fl(i)1740 2513 y(k)1780 2489 y Fk(@)1824 2501 y Fl(i)1879 2489 y Fr(\(this)28 b(has)f(sev)n(eral)f(theoretical)g (and)h(practical)515 2589 y(adv)-5 b(an)n(tages)17 b([8)o(,)i(17)o(,)g (30)o(,)f(37)o(]\);)23 b(in)18 b(this)h(case)f(w)n(e)g(sp)r(eak)g(of)h Fh(Lie-Poinc)l(ar)n(\023)-40 b(e)23 b(tr)l(ansformations)p Fr(.)681 2688 y(When)31 b(a)f(nonlinear)f(v)n(ector)h(\014eld)h Fk(X)j Fr(=)27 b([)p Fk(A)2081 2658 y Fl(i)2093 2710 y(j)2128 2688 y Fk(x)2175 2658 y Fl(j)2231 2688 y Fr(+)20 b Fk(G)2381 2658 y Fl(i)2409 2688 y Fr(\()p Fk(x)p Fr(\)])p Fk(@)2587 2700 y Fl(i)2646 2688 y Fr(in)31 b(giv)n(en)e(co)r(ordinates) 515 2798 y(satisfy)g(the)i(condition)e(that)h(the)h(nonlinear)e(part)g Fk(G)2230 2768 y Fl(i)2258 2798 y Fr(\()p Fk(x)p Fr(\))i(is)f(a)f (resonan)n(t)f(v)n(ector)h({)h(with)515 2898 y(resp)r(ect)c(to)g(the)h (linear)f(part)g Fk(A)h Fr({)f(w)n(e)g(sa)n(y)f(that)i Fk(X)33 b Fr(is)26 b(resonan)n(t)f(\(in)i(the)g Fk(x)g Fr(co)r(ordinates\).)681 2997 y(Notice)32 b(that)g(the)g(resonance)e (condition)i(in)n(v)n(olv)n(es)e(only)h(the)i(eigen)n(v)-5 b(alues)30 b(of)i Fk(A)p Fr(;)j(if)515 3097 y(w)n(e)24 b(decomp)r(ose)g Fk(A)h Fr(in)n(to)g(a)f(semisimple)h(and)f(a)h(nilp)r (oten)n(t)g(part)f(\(m)n(utually)h(comm)n(uting\),)515 3197 y Fk(A)e Fr(=)g Fk(A)750 3209 y Fl(s)804 3197 y Fr(+)18 b Fk(A)949 3209 y Fl(n)994 3197 y Fr(,)28 b(then)g(only)f Fk(A)1478 3209 y Fl(s)1542 3197 y Fr(en)n(ters)g(in)h(the)g (de\014nition)g(of)f(resonan)n(t)f(v)n(ectors.)681 3296 y(If)32 b Fk(A)g Fr(is)f(not)h(semisimple,)h(it)f(is)g(actually)f(p)r (ossible)g(to)g(re\014ne)h(ulteriorly)e(the)i(form)515 3396 y(to)25 b(whic)n(h)g Fk(G)p Fr(\()p Fk(x)p Fr(\))h(can)f(b)r(e)h (tak)n(en.)35 b(Denoting)25 b Fk(X)1994 3408 y Fl(A)2071 3396 y Fr(=)e Fk(A)2221 3366 y Fl(i)2232 3418 y(j)2267 3396 y Fk(x)2314 3366 y Fl(j)2350 3396 y Fk(@)2394 3408 y Fl(i)2447 3396 y Fr(the)i(v)n(ector)f(\014eld)h(asso)r(ciated)515 3496 y(to)30 b(the)h(linear)f(part)g(of)h Fk(X)37 b Fr(\(notice)31 b(this)g(is)f(in)n(v)-5 b(arian)n(t)30 b(under)g(near-iden)n(tit)n(y)g (c)n(hanges)f(of)515 3595 y(co)r(ordinates\),)h(w)n(e)h(denote)g(b)n(y) f Fm(L)1587 3607 y Fj(0)1656 3595 y Fr(the)h(op)r(erator)e(of)i(comm)n (utation)f(with)h Fk(X)3007 3607 y Fl(A)3061 3595 y Fr(;)i(that)e(is,) 515 3695 y Fm(L)572 3707 y Fj(0)609 3695 y Fr(\()p Fk(Y)19 b Fr(\))24 b(=)e([)p Fk(X)943 3707 y Fl(A)997 3695 y Fk(;)14 b(Y)19 b Fr(].)37 b(Ob)n(viously)-7 b(,)26 b Fm(L)1647 3707 y Fj(0)1708 3695 y Fr(:)d Fm(V)1805 3707 y Fl(k)1868 3695 y Fm(!)h(V)2026 3707 y Fl(k)2066 3695 y Fr(.)681 3794 y(W)-7 b(e)35 b(can)f(then)h(force)f Fk(G)1459 3806 y Fl(k)1536 3794 y Fr(=)g Fk(G)23 b Fm(\\)h(V)1853 3806 y Fl(k)1929 3794 y Fr(to)34 b(b)r(e)h(in)g(a)g(complemen)n(tary)e (space)h(to)h(the)515 3894 y(range)f(of)j Fm(L)910 3906 y Fj(0)983 3894 y Fr(in)g(eac)n(h)e Fm(V)1335 3906 y Fl(k)1376 3894 y Fr(.)62 b(If)37 b(w)n(e)e(in)n(tro)r(duce)h(a)g (scalar)e(pro)r(duct)2667 3864 y Fj(1)2741 3894 y Fr(in)i Fm(V)2897 3906 y Fl(k)2974 3894 y Fr(\(and)g(hence)515 3994 y(in)f Fm(V)43 b Fr(=)36 b Fm(V)865 4006 y Fj(0)926 3994 y Fm([)24 b(V)1056 4006 y Fj(1)1117 3994 y Fm([)g Fk(:::)p Fr(\),)38 b(then)e(w)n(e)f(can)g(force)g Fk(G)h Fm(2)h Fr(Ker)o(\()p Fm(L)2470 3958 y Fj(+)2470 4016 y(0)2526 3994 y Fr(\).)60 b(When)36 b(the)g(condition)515 4093 y Fk(X)584 4105 y Fl(G)662 4093 y Fm(2)24 b Fr(Ker)o(\()p Fm(L)964 4058 y Fj(+)964 4115 y(0)1019 4093 y Fr(\))j(is)g (satis\014ed,)f(w)n(e)g(sa)n(y)f(that)i Fk(X)33 b Fr(is)26 b(in)h(P)n(oincar)n(\023)-39 b(e-Dulac)23 b(normal)i(form)i(\(in)515 4193 y(the)h(giv)n(en)e(co)r(ordinates\).)681 4293 y(Notice)33 b(that)h(in)g(general)e([)p Fk(X)1620 4305 y Fl(A)1674 4293 y Fk(;)14 b(X)1780 4308 y Fl(A)1830 4292 y Fe(+)1881 4293 y Fr(])33 b Fm(6)p Fr(=)f(0,)j(and)e(th)n(us)h(the)g(adjoin)n(t)f (of)h(the)g(linear)515 4392 y(part)22 b(of)g(a)g(v)n(ector)f(\014eld)i (in)g(normal)e(form)h(do)r(es)g(not)h(comm)n(ute)f(with)h(the)g(full)g (v)n(ector)e(\014eld,)515 4492 y(but)31 b(only)g(with)g(the)h (nonlinear)e(part.)46 b(On)31 b(the)g(other)g(side,)g(if)h([)p Fk(X)2659 4504 y Fl(A)2713 4492 y Fk(;)14 b(X)2819 4507 y Fl(A)2869 4491 y Fe(+)2919 4492 y Fr(])29 b(=)f(0)j(\(whic)n(h)515 4591 y(is)g(equiv)-5 b(alen)n(t)31 b(to)g([)p Fk(A;)14 b(A)1289 4561 y Fj(+)1345 4591 y Fr(])29 b(=)g(0,)j(i.e.)48 b(to)31 b Fk(A)h Fr(b)r(eing)f(a)g(normal)f(matrix\),)i(then)g Fk(X)3118 4607 y Fl(A)3168 4591 y Fe(+)39 b Fr(is)31 b(a)515 4691 y(symmetry)c(of)g Fk(X)7 b Fr(.)p 515 4744 1146 4 v 607 4804 a Fc(1)642 4827 y Fo(Cho)r(osing)18 b(the)g(Bargmann)f(scalar)g(pro)r(duct)h([19])f(will)f(guaran)n(tee)j (that)g Fu(L)2616 4798 y Fc(+)2616 4850 y(0)2666 4827 y Fo(\()p Fb(Y)d Fo(\))k(=)g([)p Fb(X)2950 4846 y Fa(A)2996 4832 y Fe(+)3046 4827 y Fb(;)12 b(X)5 b Fo(],)18 b(where)515 4906 y Fb(A)568 4883 y Fc(+)642 4906 y Fo(is)23 b(the)h(adjoin)n(t)h (of)e Fb(A)p Fo(.)31 b(Another)24 b(p)r(opular)g(c)n(hoice)h (\(standard)g(scalar)e(pro)r(duct\))i(is)e(describ)r(ed)h(in)g([2)o(].) 1926 5255 y Fr(4)p eop %%Page: 5 5 5 4 bop 681 523 a Fr(Here)30 b(w)n(e)g(will)h Fh(assume)f Fr(that)h Fk(A)i Fh(is)g(semisimple)f Fk(A)c Fr(=)g Fk(A)2496 535 y Fl(s)2562 523 y Fr(and)i(has)g(b)r(een)h(tak)n(en)f(to)515 623 y Fh(Jor)l(dan)g(normal)g(form)1248 593 y Fj(2)1286 623 y Fr(,)e(so)f(that)1537 769 y Fk(A)51 b Fr(=)g(diag\()p Fk(\025)1999 781 y Fj(1)2037 769 y Fk(;)14 b(:::;)g(\025)2228 781 y Fl(n)2273 769 y Fr(\))28 b Fk(:)917 b Fr(\(2\))515 915 y(Th)n(us,)23 b(in)f(the)g(presen)n(t)g(discussion,)g(v)n(ector)f (\014elds)h(in)g(normal)f(form)h(and)g(resonan)n(t)e(v)n(ector)515 1015 y(\014elds)26 b(will)g(b)r(e)g(the)h(same.)35 b(W)-7 b(e)27 b(also)d(ha)n(v)n(e)h(that)h Fm(L)2104 1027 y Fj(0)2165 1015 y Fr(=)d Fm(L)2310 979 y Fj(+)2310 1037 y(0)2391 1015 y Fr(\(pro)n(vided)i(w)n(e)h(ha)n(v)n(e)e(c)n(hosen)h(a) 515 1115 y(suitable,)i(e.g.)37 b(the)28 b(Bargmann,)e(scalar)g(pro)r (duct\).)681 1214 y(Resonan)n(t)g(v)n(ector)h(\014elds)g(are)g(th)n(us) h(iden)n(ti\014ed)g(b)n(y)f(the)h(condition)1711 1360 y([)p Fk(X)1803 1372 y Fj(0)1840 1360 y Fk(;)14 b(X)7 b Fr(])50 b(=)g(0)1090 b(\(3\))515 1507 y(where)37 b Fk(X)834 1519 y Fj(0)910 1507 y Fr(is)g(the)i(linear)e(part)h(of)g Fk(X)7 b Fr(.)68 b(In)38 b(order)f(to)h(emphasize)f(that)i(w)n(e)e(are) h(actu-)515 1606 y(ally)27 b(dealing)g(with)h(co)r(ordinate)f (expressions)f(and)i(not)f(with)i(geometrical)d(ob)5 b(jects,)27 b(it)h(is)515 1706 y(ma)n(yb)r(e)g(b)r(etter)h(to)f(deal)g (with)h(comp)r(onen)n(ts)f(of)g Fk(X)35 b Fr(in)28 b(the)h Fk(x)g Fr(co)r(ordinates.)38 b(It)28 b(is)h(easy)e(to)515 1806 y(see)e(that)g(if)h Fk(X)967 1818 y Fl(f)1033 1806 y Fr(=)d Fk(f)1171 1775 y Fl(i)1198 1806 y Fr(\()p Fk(x)p Fr(\))p Fk(@)1353 1818 y Fl(i)1407 1806 y Fr(and)i Fk(X)1635 1818 y Fl(g)1697 1806 y Fr(=)d Fk(g)1827 1775 y Fl(i)1854 1806 y Fr(\()p Fk(x)p Fr(\))p Fk(@)2009 1818 y Fl(i)2038 1806 y Fr(,)k(then)g([)p Fk(X)2366 1818 y Fl(f)2409 1806 y Fk(;)14 b(X)2515 1818 y Fl(g)2553 1806 y Fr(])23 b(=)g Fk(X)2756 1818 y Fl(h)2822 1806 y Fr(=)f Fk(h)2957 1775 y Fl(i)2985 1806 y Fr(\()p Fk(x)p Fr(\))p Fk(@)3140 1818 y Fl(i)3169 1806 y Fr(,)k(with)854 1952 y Fk(h)p Fr(\()p Fk(x)p Fr(\))e(=)f Fm(f)p Fk(f)9 b Fr(\()p Fk(x)p Fr(\))p Fk(;)14 b(g)s Fr(\()p Fk(x)p Fr(\))p Fm(g)51 b Fr(:=)f(\()p Fk(f)1832 1918 y Fl(j)1867 1952 y Fr(\()p Fk(x)p Fr(\))19 b Fm(\001)g Fk(@)2083 1964 y Fl(j)2118 1952 y Fr(\))p Fk(g)s Fr(\()p Fk(x)p Fr(\))g Fm(\000)f Fr(\()p Fk(g)2481 1918 y Fl(j)2516 1952 y Fr(\()p Fk(x)p Fr(\))i Fm(\001)e Fk(@)2732 1964 y Fl(j)2767 1952 y Fr(\))p Fk(f)2849 1918 y Fl(i)2877 1952 y Fr(\()p Fk(x)p Fr(\))28 b Fk(:)234 b Fr(\(4\))515 2098 y(The)23 b(brac)n(k)n(et)g Fm(f)p Fk(:;)14 b(:)p Fm(g)22 b Fr(is)i(just)g(expressing)e(the)i(comm)n (utator)f(in)h(terms)f(of)h(the)g(comp)r(onen)n(ts)515 2198 y(of)j(v)n(ector)g(\014elds)g(in)h(giv)n(en)f(co)r(ordinates.)35 b(Relation)28 b(\(3\))f(reads)g(then)1485 2344 y Fm(f)p Fr([\()p Fk(D)r(f)9 b Fr(\)\(0\)])p Fk(x;)14 b(f)9 b Fr(\()p Fk(x)p Fr(\))p Fm(g)51 b Fr(=)f(0)27 b Fk(:)841 b Fr(\(3)3324 2314 y Ff(0)3347 2344 y Fr(\))681 2490 y(The)30 b(p)r(oin)n(t)h(is)g(that)g(once)f Fk(f)9 b Fr(\()p Fk(x)p Fr(\))31 b(is)g(in)g(normal)e(form,)i(w)n(e)g(can)f (still)h(consider)e(near-)515 2590 y(iden)n(tit)n(y)35 b(Lie-P)n(oincar)n(\023)-39 b(e)32 b(c)n(hanges)i(of)h(co)r(ordinates)f (taking)g(it)i(in)n(to)f(a)g(di\013eren)n(t)g(normal)515 2690 y(form)723 2668 y Fg(b)707 2690 y Fk(f)9 b Fr(\()p Fk(x)p Fr(\);)26 b(in)f(order)d(to)i(b)r(e)h(guaran)n(teed)d(that)2047 2668 y Fg(b)2031 2690 y Fk(f)9 b Fr(\()p Fk(x)p Fr(\))25 b(is)f(still)h(in)f(NF,)g(i.e.)36 b(\(3'\))25 b(is)e(satis\014ed)515 2801 y(b)n(y)649 2779 y Fg(b)633 2801 y Fk(f)9 b Fr(,)31 b(the)g(generator)d Fk(h)j Fr(m)n(ust)g(b)r(e)g(c)n(hosen)e(to)i(b)r(e) g(itself)g(resonan)n(t)e(with)i Fk(A)d Fr(=)g(\()p Fk(D)r(f)9 b Fr(\)\(0\).)515 2900 y(Indeed,)22 b(the)e(set)g Fm(G)1108 2912 y Fl(A)1182 2900 y Fr(of)g(v)n(ector)f(\014elds)h(comm)n(uting)g (with)g Fk(X)2390 2912 y Fl(A)2464 2900 y Fr(is)g(ob)n(viously)e(a)i (Lie)g(algebra.)681 3000 y(W)-7 b(e)28 b(recall)f(that)h(if)h Fk(h)p Fr(\()p Fk(x)p Fr(\))24 b(=)g Fk(x)19 b Fr(+)1724 2938 y Fg(P)1812 2958 y Ff(1)1812 3025 y Fl(k)q Fj(=1)1951 3000 y Fk(h)1999 3012 y Fl(k)2039 3000 y Fr(\()p Fk(x)p Fr(\),)29 b(the)g(c)n(hanges)d(of)i(co)r(ordinates)f(giv)n(en)515 3114 y(b)n(y)e(the)g(time-one)g(\015o)n(w)f(of)h Fk(X)1438 3126 y Fl(h)1506 3114 y Fr(maps)g Fk(X)31 b Fr(in)n(to)2009 3093 y Fg(b)1987 3114 y Fk(X)f Fr(=)22 b Fk(e)2212 3084 y Fl(H)2275 3114 y Fk(X)7 b(e)2390 3084 y Ff(\000)p Fl(H)2504 3114 y Fr(.)36 b(This)25 b(can)g(b)r(e)g(explicitely)515 3214 y(computed)33 b(b)n(y)f(the)h(classical)e(Bak)n(er-Campb)r (ell-Haussdor\013)e(form)n(ula;)35 b(with)e(\002\()p Fk(X)7 b Fr(\))30 b(:=)515 3313 y([)p Fk(H)r(;)14 b(X)7 b Fr(],)27 b(w)n(e)g(ha)n(v)n(e)1563 3458 y Fg(b)1541 3479 y Fk(X)58 b Fr(=)1810 3376 y Ff(1)1783 3401 y Fg(X)1785 3576 y Fl(s)p Fj(=0)1903 3479 y Fr(\(1)p Fk(=s)p Fr(!\)\002)2178 3445 y Fl(s)2212 3479 y Fr(\()p Fk(X)7 b Fr(\))921 b(\(5\))515 3679 y(whic)n(h)27 b(in)h(terms)f(of)h(homogeneous)e(comp)r(onen)n(ts)h (reads)1471 3891 y Fg(b)1456 3913 y Fk(f)1497 3925 y Fl(m)1610 3913 y Fr(=)1725 3802 y Fj([)p Fl(m=k)q Fj(])1749 3834 y Fg(X)1751 4009 y Fl(s)p Fj(=0)1954 3856 y Fr(1)p 1944 3893 62 4 v 1944 3969 a Fk(s)p Fr(!)2044 3913 y Fm(H)2115 3878 y Fl(s)2150 3913 y Fr(\()p Fk(f)2223 3925 y Fl(m)p Ff(\000)p Fl(sk)2406 3913 y Fr(\))835 b(\(6\))515 4135 y(where)27 b([)p Fk(m=k)s Fr(])g(denotes)g(the)h(in)n(teger)f (part)g(of)g(\()p Fk(m=k)s Fr(\))h(and)f Fm(H)q Fr(\()p Fk(f)9 b Fr(\))23 b(=)g Fm(f)p Fk(h;)14 b(f)9 b Fm(g)p Fr(.)681 4235 y(It)36 b(is)f(con)n(v)n(enien)n(t,)i(for)e(further)g (discussion,)i(to)e(de\014ne)h(the)g(higher)f(order)f(homo-)515 4334 y(logical)27 b(op)r(erators)h(\(already)f(considered)h(b)n(y)h(T) -7 b(ak)n(ens)28 b([33)o(]\))i(as)e Fm(L)2627 4346 y Fl(k)2694 4334 y Fr(:=)d Fm(f)p Fk(f)2890 4346 y Fl(k)2930 4334 y Fk(;)14 b(:)p Fm(g)p Fr(.)41 b(W)-7 b(e)29 b(also)515 4434 y(denote,)c(for)e(later)g(discussion,)i(b)n(y)e Fk(G)h Fr(=)e Fk(C)6 b Fr(\()p Fk(A)p Fr(\))26 b(the)e(cen)n(tralizer)f (of)h Fk(A)g Fr(in)g(the)h(Lie)f(algebra)515 4534 y(of)i Fk(n)p Fr(-dimensional)f(matrices)h(\(with)h(Lie)f(op)r(eration)f(the)i (matrix)f(comm)n(utator\);)g(a)g(basis)515 4633 y(for)21 b(this)h(will)f(b)r(e)h(giv)n(en)f(b)n(y)g(matrices)g Fm(f)p Fk(K)1809 4645 y Fj(1)1845 4633 y Fk(;)14 b(:::;)g(K)2059 4645 y Fl(d)2097 4633 y Fm(g)22 b Fr(\(if)g Fk(A)h Fm(6)p Fr(=)g(0,)f(w)n(e)f(can)h(c)n(ho)r(ose)e Fk(K)3114 4645 y Fj(1)3174 4633 y Fr(=)i Fk(A)p Fr(\);)515 4733 y(notice)27 b(that)h Fk(d)23 b Fm(\024)g Fk(n)p Fr(.)p 515 4771 1146 4 v 607 4825 a Fc(2)642 4848 y Fo(The)j(theory)i(of)d(normal)g(forms)f (is)i(w)n(ell)g(dev)n(elop)r(ed)h(without)h(these)f(assumptions,)f(but) h(reduction)515 4927 y(encoun)n(ters)i(a)f(n)n(um)n(b)r(er)f(of)g (substan)n(tial)h(obstacles;)j(see)d(e.g.)43 b([32,)27 b(36)q(])g(for)g(normal)f(forms)f(in)j(the)g(case)515 5006 y(where)c Fb(A)772 5014 y Fa(n)834 5006 y Fu(6)p Fo(=)19 b(0)24 b(and/or)g Fb(A)g Fo(is)f(not)h(in)f(Jordan)i(normal)d (form.)1926 5255 y Fr(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fq(2)134 b(Tw)l(o)45 b(non-equiv)-7 b(alen)l(t)46 b(further)f(reduction)716 672 y(sc)l(hemes:)61 b(PRF)45 b(and)g(LRF)515 854 y Fr(Ha)n(ving)26 b(de\014ned)i(normal)e (forms)h(and)h(obtained)f(the)g(form)n(ulas)g(for)g(the)g(c)n(hange)g (of)g(v)-5 b(ari-)515 954 y(ables,)34 b(w)n(e)f(can)h(w)n(onder)e(if)i (one)f(can)h(c)n(hange)e(v)-5 b(ariables)32 b(th)n(us)i(transforming)e (a)h(normal)515 1054 y(form)d(in)n(to)h(another)f(one,)h(more)f(con)n (v)n(enien)n(t)g(in)h(some)f(resp)r(ect)g(\(in)i(particular,)e(ha)n (ving)515 1153 y(a)h(smaller)g(n)n(um)n(b)r(er)h(of)g(nonlinear)f (terms,)i(or)e(a)h(smaller)f(n)n(um)n(b)r(er)g(of)h(lo)n(w)g(order)e (ones\).)515 1253 y(The)j(answ)n(er)e(is)i(ob)n(viously)e(y)n(es,)j(as) e(already)f(noted)i(b)n(y)g(Dulac)g([18)o(],)h(in)f(the)h(form)e(and) 515 1352 y(within)c(the)g(limits)g(implied)g(b)n(y)g(\(5\),)f(\(6\))h (ab)r(o)n(v)n(e.)681 1452 y(In)35 b(recen)n(t)g(w)n(orks)f({)h(as)g (men)n(tioned)g(in)h(the)f(In)n(tro)r(duction)g({)g(I)h(ha)n(v)n(e)e (prop)r(osed)g(a)515 1552 y(general)28 b(pro)r(cedure)h(\(and)h (algorithm,)f(requiring)g(to)g(solv)n(e)g(only)g(linear)g(equations\))h (for)515 1651 y(suc)n(h)h(a)f(reduction)h([20)o(,)g(21)o(].)48 b(Since)31 b(this)h(approac)n(h)d(represen)n(ts)h(a)g(direct)h (extension)g(of)515 1751 y(P)n(oincar)n(\023)-39 b(e)29 b(pro)r(cedure,)k(the)g(normal)f(form)g(\(in)h(general)f(not)g (unique\))h(obtained)g(in)g(this)515 1851 y(w)n(a)n(y)25 b(has)g(b)r(een)i(called)f(\\P)n(oincar)n(\023)-39 b(e)22 b(renormalized)j(form")g(\(PRF\),)i(and)f(w)n(e)g(refer)f(to)h(it)h(as) 515 1950 y(the)h(PRF)f(approac)n(h.)681 2050 y(On)c(the)h(other)e (hand,)i(in)g(some)f(cases)f(it)h(is)h(also)e(p)r(ossible)h(to)g(pro)r (ceed)g(in)g(a)g(di\013eren)n(t)515 2149 y(w)n(a)n(y:)35 b(one)28 b(can)f(use)g(the)h(Lie)g(algebraic)e(structure)h(of)g(the)h (set)g(of)f(resonan)n(t)f(v)n(ectors.)36 b(W)-7 b(e)515 2249 y(will)21 b(call)g(the)g(reduced)g(normal)f(forms)g(obtained)h(in) g(this)h(w)n(a)n(y)-7 b(,)21 b(\\Lie)f(renormalized)g(form")515 2349 y(\(LRF\),)28 b(and)g(w)n(e)f(refer)g(to)g(the)h(pro)r(cedure)f (as)g(the)h(LRF)g(approac)n(h.)681 2448 y(It)37 b(should)g(b)r(e)h(men) n(tioned)f(that)g(the)h(relev)-5 b(ance)36 b(of)h(Lie)g(algebraic)e (structures)i(in)515 2548 y(normal)g(forms)h(theory)g(w)n(as)f (stressed)h(\(in)g(his)h(thesis\))g(b)n(y)f(Bro)r(er,)i(who)e(ga)n(v)n (e)e(a)i(v)n(ery)515 2648 y(general)26 b(and)h(p)r(o)n(w)n(erful)g (reduction)h(pro)r(cedure)e(\(see)i(app)r(endix)f(B\).)681 2747 y(Our)h(pro)r(cedure)g(will)i(b)r(e)f(less)g(general)f(and)h(p)r (o)n(w)n(erful,)f(and)h(apply)g(only)g(in)g(fav)n(ou-)515 2847 y(rable)g(cases)h(\(see)g(b)r(elo)n(w\);)i(but)f(it)g(is)g (simpler)f(and)g({)h(when)f(applicable)g({)g(its)h(practical)515 2946 y(implemen)n(tation)j(in)g(completely)g(explicit)h(computations)e (is)h(elemen)n(tary)-7 b(,)35 b(as)e(w)n(e)h(also)515 3046 y(sho)n(w)26 b(b)n(y)i(example.)515 3279 y Fi(2.1)112 b(The)38 b(PRF)f(approac)m(h)515 3432 y Fr(I)29 b(will)h(no)n(w)f (recall)g(the)h(basic)f(asp)r(ects)g(of)h(the)g(PRF)g(approac)n(h,)e (dev)n(elop)r(ed)h(in)h([20)o(,)g(21)o(];)515 3531 y(the)e(reader)e(is) h(referred)g(to)g(these)h(w)n(orks)e(and)h(to)h([16)o(])f(for)h (further)f(detail.)681 3631 y(Let)37 b Fk(W)50 b Fr(b)r(e)38 b(the)f(v)n(ector)f(\014eld)i(\(sa)n(y)f(already)e(in)j(standard)f(NF,) h(to)f(a)n(v)n(oid)f(trivial)515 3731 y(steps\))d(under)g (consideration.)52 b(Let)34 b(us)f(write)g(it)h(as)e Fk(W)45 b Fr(=)32 b Fk(X)2529 3743 y Fj(0)2588 3731 y Fr(+)2675 3668 y Fg(P)2762 3689 y Ff(1)2762 3756 y Fl(k)q Fj(=1)2901 3731 y Fk(W)2979 3743 y Fl(k)3020 3731 y Fr(,)j(with)f Fk(X)3342 3743 y Fj(0)515 3830 y Fr(linear)d(and)i Fk(W)995 3842 y Fl(k)1068 3830 y Fr(homogeneous)e(of)h(degree)g(\()p Fk(k)24 b Fr(+)e(1\).)51 b(F)-7 b(or)31 b(the)i(sak)n(e)e(of)i (simplicit)n(y)3215 3800 y Fj(3)3284 3830 y Fr(w)n(e)515 3930 y(will)28 b(assume)e Fk(X)1027 3942 y Fj(0)1088 3930 y Fm(6)p Fr(=)c(0.)681 4030 y(T)-7 b(ak)n(e)31 b(the)i(\014rst)e (nonzero)g Fk(W)1601 4042 y Fl(k)1643 4030 y Fr(,)i(sa)n(y)e Fk(W)1925 4042 y Fl(p)1959 4050 y Fe(0)1996 4030 y Fr(;)k(op)r(erating) c(with)i(transformations)d(gen-)515 4148 y(erated)k(b)n(y)h Fk(h)947 4105 y Fj(\(0\))947 4173 y Fl(k)1072 4148 y Fm(2)h(W)1252 4105 y Fj(\(0\))1245 4173 y Fl(k)1377 4148 y Fr(:=)g(Ker)o(\()p Fm(L)1724 4160 y Fj(0)1761 4148 y Fr(\))24 b Fm(\\)g(V)1947 4160 y Fl(k)2023 4148 y Fr(\(successiv)n (ely)34 b(for)h Fk(k)k Fr(=)c(1)p Fk(;)14 b Fr(2)p Fk(;)g(:::)p Fr(\))35 b(w)n(e)f(can)515 4268 y(eliminate)39 b(all)f(terms)g(in)h([)p Fk(W)1463 4280 y Fl(p)1497 4288 y Fe(0)1535 4268 y Fk(;)14 b Fm(W)1661 4225 y Fj(\(0\))1654 4293 y Fl(k)1749 4268 y Fr(],)42 b(i.e.)70 b(in)39 b(the)h(range)d(of)h Fm(M)2718 4280 y Fl(p)2752 4288 y Fe(0)2789 4268 y Fr(,)k(de\014ned)d(as)f(the) 515 4368 y(restriction)26 b(of)i(the)g(op)r(erator)e Fm(L)1542 4380 y Fl(p)1576 4388 y Fe(0)1640 4368 y Fr(to)i(Ker)o(\()p Fm(L)1965 4380 y Fj(0)2003 4368 y Fr(\).)p 515 4421 1146 4 v 607 4475 a Fc(3)642 4498 y Fo(Notice)c(that)h(the)g(PRF)e(can)i(as) f(w)n(ell)f(deal)h(with)g(cases)g(where)g(the)h(linear)e(part)h(v)l (anishes:)32 b(no)n(w)24 b(the)515 4577 y(standard)29 b(homological)f(op)r(erator)g(is)g(only)g(the)i(\014rst)e(in)g(a)g(c)n (hain)h(of)f(op)r(erators,)h(and)g(w)n(e)g(can)f(use)h(the)515 4656 y(other)g(ones)h(for)e(reduction.)48 b(In)29 b(practice,)i (computations)f(will)d(b)r(e)i(v)n(ery)h(di\016cult)f(for)f Fb(A)g Fo(=)g(0,)i(unless)515 4735 y(some)23 b(other)i(constrain)n(t)g (\(e.g.)33 b(symmetry)22 b(prop)r(erties\))i(reduces)h(the)g(set)g(of)e (allo)n(w)n(ed)i(nonlinear)f(v)n(ector)515 4814 y(\014elds;)f(see)h ([23])f(for)g(PRF)h(analysis)f(of)h(symmetric)d(systems)i(with)h Fb(A)19 b Fo(=)h(0.)1926 5255 y Fr(6)p eop %%Page: 7 7 7 6 bop 681 523 a Fr(Let)35 b(no)n(w)f Fk(W)1095 535 y Fl(p)1129 543 y Fe(1)1201 523 y Fr(b)r(e)h(the)g(\014rst)f(nonzero)f (term)i(among)f(those)g(with)h Fk(k)j(>)c(p)3088 535 y Fj(0)3160 523 y Fr(in)h(the)515 623 y(normal)30 b(form)h(obtained)f (after)h(the)h(ab)r(o)n(v)n(e)e(transformations.)45 b(Using)31 b(transformations)515 734 y(generated)36 b(b)n(y)h Fk(h)1076 691 y Fj(\(1\))1076 759 y Fl(k)1204 734 y Fm(2)i(W)1387 691 y Fj(\(1\))1380 759 y Fl(k)1515 734 y Fr(:=)f(Ker)o(\()p Fm(L)1864 746 y Fj(0)1902 734 y Fr(\))25 b Fm(\\)h Fr(Ker)n(\()p Fm(L)2262 746 y Fl(p)2296 754 y Fe(0)2334 734 y Fr(\))f Fm(\\)g(V)2522 746 y Fl(k)2600 734 y Fr(\(successiv)n(ely)36 b(for)h Fk(k)k Fr(=)515 854 y(1)p Fk(;)14 b Fr(2)p Fk(;)g(:::)p Fr(\),)25 b(w)n(e)h(can)f(eliminate)h(all)f(terms)h(in)g([)p Fk(W)1990 866 y Fl(p)2024 874 y Fe(1)2061 854 y Fk(;)14 b Fm(W)2187 811 y Fj(\(1\))2180 879 y Fl(k)2276 854 y Fr(],)26 b(i.e.)37 b(in)26 b(the)g(range)e(of)i Fm(M)3144 866 y Fl(p)3178 874 y Fe(1)3214 854 y Fr(,)h(the)515 954 y(restriction)f(of)i Fm(L)1064 966 y Fl(p)1098 974 y Fe(1)1163 954 y Fr(to)f(Ker)o(\()p Fm(L)1487 966 y Fj(0)1525 954 y Fr(\))19 b Fm(\\)f Fr(Ker)o(\()p Fm(L)1872 966 y Fl(p)1906 974 y Fe(0)1944 954 y Fr(\).)681 1053 y(The)25 b(pro)r(cess)g(can)g(ob)n(viously)f(b)r(e)i(con)n(tin)n(ued)f (inde\014nitely)-7 b(,)27 b(un)n(til)f(either)f(all)g(nonlin-)515 1153 y(ear)18 b(terms)g(of)h(degree)e(higher)i(than)f Fk(p)1674 1165 y Fl(q)1730 1153 y Fr(are)f(killed,)k(or)d(Ker)o(\()p Fm(L)2415 1165 y Fj(0)2453 1153 y Fr(\))q Fm(\\)q Fr(Ker)o(\()p Fm(L)2765 1165 y Fl(p)2799 1173 y Fe(0)2836 1153 y Fr(\))q Fm(\\)q Fk(:::)q Fm(\\)q Fr(Ker\()p Fm(L)3275 1165 y Fl(p)3309 1173 y Fd(q)3347 1153 y Fr(\))515 1252 y(is)27 b(empt)n(y)-7 b(.)681 1352 y(Notice)38 b(that)g(in)g(this)g(w)n(a)n(y) -7 b(,)39 b(due)f(to)g(the)g(restriction)f(to)h(k)n(ernels)e(of)i(lo)n (w)n(er)e(order)515 1452 y(homological)25 b(op)r(erators)h(\(i.e.)38 b(due)28 b(to)f(the)h(use)g(of)f Fm(M)2254 1464 y Fl(p)2320 1452 y Fr(rather)g(than)h Fm(L)2821 1464 y Fl(p)2859 1452 y Fr(\),)g(at)g(eac)n(h)f(step)515 1551 y(w)n(e)32 b(are)f(not)i(a\013ecting)f(the)h(terms)f(whic)n(h)g(ha)n(v)n(e)f (already)g(b)r(een)i(simpli\014ed;)i(in)e(facts)f(at)515 1651 y(eac)n(h)27 b(step)g(w)n(e)g(stabilize)h(new)f(terms.)681 1762 y(The)21 b(generators)d Fk(h)1288 1719 y Fj(\()p Fl(j)s Fj(\))1288 1787 y Fl(k)1395 1762 y Fr(are)i(c)n(hosen)g(as)g (solutions)g(to)g(higher)g(homological)f(equations:)515 1882 y(if)i Fk(f)634 1839 y Fj(\()p Fl(j)s Fj(\))625 1907 y Fl(k)740 1882 y Fr(is)f(the)h(term)f(of)h(order)d(\()p Fk(k)6 b Fr(+)t(1\))21 b(after)f(the)g(\014rst)g Fk(j)26 b Fr(rounds)19 b(of)h(further)h(normalization,)515 2002 y(and)27 b Fk(\031)726 1959 y Fj(\()p Fl(j)s Fj(\))723 2027 y Fl(k)841 2002 y Fr(the)h(pro)5 b(jection)27 b(from)g Fm(V)1626 2014 y Fl(k)1694 2002 y Fr(to)h Fm(W)1885 1959 y Fj(\()p Fl(j)s Fj(\))1878 2027 y Fl(k)1971 2002 y Fr(,)g(this)g(is)g (giv)n(en)e(b)n(y)1356 2277 y Fk(\031)1406 2234 y Fj(\()p Fl(j)s Fj(\))1403 2302 y Fl(k)1507 2110 y Fg(2)1507 2260 y(4)1562 2166 y Fj([)p Fl(m=k)q Fj(])1586 2198 y Fg(X)1588 2374 y Fl(s)p Fj(=0)1791 2221 y Fr(1)p 1781 2258 62 4 v 1781 2334 a Fk(s)p Fr(!)1880 2277 y Fm(H)1951 2243 y Fl(s)1987 2277 y Fr(\()p Fk(f)2060 2289 y Fl(m)p Ff(\000)p Fl(sk)2243 2277 y Fr(\))2275 2110 y Fg(3)2275 2260 y(5)2381 2277 y Fr(=)50 b(0)735 b(\(7\))515 2581 y(where)27 b Fm(H)d Fr(=)e Fm(f)p Fk(h)1026 2538 y Fj(\()p Fl(j)s Fj(\))1026 2606 y Fl(k)1113 2581 y Fk(;)14 b(:)p Fm(g)p Fr(;)27 b(that)h(is,)f(see)g(\(5\),)h(b)n(y)g(requiring)e(that)i Fk(\031)2541 2538 y Fj(\()p Fl(j)s Fj(\))2538 2606 y Fl(k)2642 2489 y Fg(\020)2695 2560 y(c)2691 2581 y Fk(W)2781 2538 y Fj(\()p Fl(j)s Fj(\))2769 2606 y Fl(k)2868 2489 y Fg(\021)2941 2581 y Fr(=)22 b(0.)681 2730 y(Notice)29 b(that)h(the)h Fk(h)1319 2687 y Fj(\()p Fl(j)s Fj(\))1319 2755 y Fl(k)1435 2730 y Fr(considered)e(ab)r(o)n(v)n(e)f(are)h(in)h (general)e(not)i(uniquely)g(de\014ned:)515 2829 y(they)c(are)f(unique)i (up)f(to)g(an)g(elemen)n(t)g(in)h(Ker)o(\()p Fm(M)2105 2841 y Fl(p)2139 2849 y Fd(j)2174 2829 y Fr(\),)g(see)f(ab)r(o)n(v)n (e.)35 b(Th)n(us,)26 b(the)h(PRF)f(is)g(in)515 2929 y(general)g(not)i (unique.)681 3029 y(The)41 b(reduced)g(normal)f(form)g(obtained)h (according)e(to)i(this)g(pro)r(cedure)f(will)i(b)r(e)515 3128 y(called)27 b(a)h Fh(Poinc)l(ar)n(\023)-40 b(e)32 b(r)l(enormalize)l(d)f(form)e Fr(\(PRF\))g(for)e(the)h(normal)g(form)f Fk(W)36 b Fr(=)23 b Fk(W)3166 3098 y Fj(\(0\))3255 3128 y Fr(.)39 b(A)515 3228 y(precise)27 b(description)g(of)h(the)g(spaces)f (to)h(whic)n(h)g(the)g(terms)g(non-eliminable)f(in)h(this)g(w)n(a)n(y) 515 3327 y(b)r(elong)35 b({)g(i.e.)60 b(an)35 b(abstract)f(general)g (description)h(of)g(PRFs)g({)g(and)h(a)e(more)h(detailed)515 3427 y(discussion)27 b(are)f(giv)n(en)h(in)h([16)o(,)g(20)o(,)f(21)o (].)515 3657 y Fi(2.2)112 b(The)38 b(LRF)f(approac)m(h)515 3811 y Fr(Let)27 b(us)h(no)n(w)f(consider)g(a)g(di\013eren)n(t)g (further)h(normalization)e(sc)n(heme.)681 3910 y(Consider)18 b(the)i(set)f(of)h(v)n(ector)e(\014elds)h(in)h Fs(R)1968 3880 y Fl(n)2032 3910 y Fr(whic)n(h)f(are)f(in)i(normal)e(form)h(with)h (resp)r(ect)515 4010 y(to)32 b(the)g(giv)n(en)g(linear)f(part)h Fk(A)p Fr(,)h(i.e.)51 b(the)33 b(set)f(of)g Fk(Y)49 b Fm(2)31 b(V)39 b Fr(suc)n(h)32 b(that)g([)p Fk(X)2807 4022 y Fl(A)2861 4010 y Fk(;)14 b(Y)19 b Fr(])30 b(=)h(0.)50 b(It)32 b(is)515 4110 y(ob)n(vious)j(that)h(these)g(form)g(a)g(Lie)g (algebra)f(\(the)i(Lie)f(op)r(eration)f(b)r(eing)h(the)h(standard)515 4209 y(comm)n(utator)26 b(of)i(v)n(ector)e(\014elds\);)i(w)n(e)f (denote)h(this)g(algebra)d(b)n(y)j Fm(G)5 b Fr(.)681 4309 y(Let)25 b(us)g(recall)f(a)h(general)f(c)n(haracterization)e(of)k (v)n(ector)d(\014elds)j(in)f(normal)f(form)h(rele-)515 4408 y(v)-5 b(an)n(t)25 b(in)h(this)g(con)n(text)g([16)o(,)g(19)o(,)g (26)o(,)g(36)o(].)36 b(Consider)25 b(the)h(linear)f(v)n(ector)f (\014eld)i Fk(X)3039 4420 y Fl(A)3093 4408 y Fr(;)h(w)n(e)e(sa)n(y)515 4508 y(that)c(the)h(di\013eren)n(tiable)f(function)h Fk(')h Fr(:)h Fs(R)1832 4478 y Fl(n)1899 4508 y Fm(!)g Fs(R)d Fr(is)g(an)g(in)n(v)-5 b(arian)n(t)20 b(for)h Fk(X)2817 4520 y Fl(A)2892 4508 y Fr(if)h Fk(X)3031 4520 y Fl(A)3085 4508 y Fr(\()p Fk(')p Fr(\))i(=)e(0.)681 4608 y(Denote)j(b)n(y)f Fm(I)1126 4578 y Ff(\003)1164 4608 y Fr(\()p Fk(A)p Fr(\))i(the)f(set)g(of)f(in)n(v)-5 b(arian)n(ts)23 b(for)i Fk(X)2246 4620 y Fl(A)2324 4608 y Fr(whic)n(h)g(are)e(meromorphic)h(\(that)515 4707 y(is,)31 b(can)e(b)r(e)i(expressed)e(as)h(a)f(quotien)n(t)h(of)h(algebraic)d (functions\))j(in)f(the)h Fk(x)f Fr(co)r(ordinates;)515 4807 y(denote)25 b(b)n(y)g Fm(I)6 b Fr(\()p Fk(A)p Fr(\))24 b Fm(\032)f(I)1233 4777 y Ff(\003)1271 4807 y Fr(\()p Fk(A)p Fr(\))j(the)g(set)f(of)g(algebraic)e(in)n(v)-5 b(arian)n(ts)24 b(for)h Fk(X)2704 4819 y Fl(A)2758 4807 y Fr(,)h(and)f(b)n(y)f Fm(I)3123 4819 y Fl(k)3165 4807 y Fr(\()p Fk(A)p Fr(\))f Fm(\032)515 4907 y(I)6 b Fr(\()p Fk(A)p Fr(\))26 b(the)g(set)f(of)h(algebraic)d(in)n(v)-5 b(arian)n(ts)24 b(for)h Fk(X)2000 4919 y Fl(A)2079 4907 y Fr(whic)n(h)h(are)e(functions)i(homogeneous)d(of)515 5006 y(degree)j Fk(k)c Fr(+)c(1)27 b(in)h(the)g Fk(x)g Fr(v)-5 b(ariables.)1926 5255 y(7)p eop %%Page: 8 8 8 7 bop 681 523 a Fr(Let)39 b Fk(G)k Fr(=)f Fk(C)6 b Fr(\()p Fk(A)p Fr(\))40 b(b)r(e)f(the)h(cen)n(tralizer)e(of)h Fk(A)g Fr(in)h(the)f(algebra)e(of)j Fk(n)f Fr(dimensional)515 623 y(matrices;)23 b(let)g(its)g(Lie)g(algebra)e(b)r(e)i(spanned)f(b)n (y)h(matrices)e Fm(f)p Fk(K)2491 635 y Fj(1)2528 623 y Fk(;)14 b(:::;)g(K)2742 635 y Fl(d)2780 623 y Fm(g)22 b Fr(\(w)n(e)h(can)f(alw)n(a)n(ys)515 722 y(assume)34 b Fk(K)880 734 y Fj(1)952 722 y Fr(=)h Fk(I)7 b Fr(,)37 b(and)e(that)g Fk(K)1582 734 y Fl(\013)1664 722 y Fr(=)g Fk(A)h Fr(for)e(some)h Fk(\013)p Fr(,)i(pro)n(vided)d Fk(A)i Fm(6)p Fr(=)f(0;)j(notice)d(that)515 822 y Fk(d)26 b Fm(\024)f Fk(n)p Fr(\).)42 b(W)-7 b(e)30 b(denote)f(b)n(y)g Fk(X)1428 792 y Fj(\()p Fl(\013)p Fj(\))1556 822 y Fr(the)g(v)n(ector)f (\014elds)i(corresp)r(onding)d(to)i(these,)h(i.e.)42 b(giv)n(en)515 922 y(in)28 b(the)g Fk(x)g Fr(co)r(ordinates)e(b)n(y)h Fk(X)1462 891 y Fj(\()p Fl(\013)p Fj(\))1584 922 y Fr(=)22 b(\()p Fk(K)1774 934 y Fl(\013)1821 922 y Fk(x)p Fr(\))1900 891 y Fl(i)1929 922 y Fk(@)1973 934 y Fl(i)2001 922 y Fr(.)681 1021 y(Then)28 b(the)g(most)f(general)f(v)n(ector)g(\014eld)i Fk(W)40 b Fr(in)28 b Fm(G)33 b Fr(can)27 b(b)r(e)h(written)g(as)1529 1281 y Fk(W)62 b Fr(=)1831 1177 y Fl(d)1788 1202 y Fg(X)1785 1378 y Fl(\013)p Fj(=1)1953 1281 y Fk(\026)2003 1293 y Fl(\013)2051 1281 y Fr(\()p Fk(x)p Fr(\))28 b Fk(X)2266 1247 y Fj(\()p Fl(\013)p Fj(\))3273 1281 y Fr(\(8\))515 1540 y(where)34 b Fk(\026)812 1552 y Fl(\013)859 1540 y Fr(\()p Fk(x)p Fr(\))i Fm(2)f(I)1147 1510 y Ff(\003)1186 1540 y Fr(\()p Fk(A)p Fr(\).)59 b(In)35 b(other)f(w)n(ords,)h Fm(G)40 b Fr(is)35 b(con)n(tained)f(in)h(a)f(\014nitely)h(generated)515 1640 y(mo)r(dule)28 b(o)n(v)n(er)d Fm(I)1036 1609 y Ff(\003)1075 1640 y Fr(\()p Fk(A)p Fr(\).)681 1739 y(Notice)33 b(that)h(the)g(v)n (ector)f(\014eld)h Fk(W)45 b Fr(m)n(ust)34 b(b)r(e)g(algebraic)e(in)i (the)g Fk(x)p Fr(,)h(and)f Fk(X)3136 1709 y Fj(\()p Fl(\013)p Fj(\))3268 1739 y Fr(are)515 1839 y(linear)f(in)h Fk(x)p Fr(,)i(so)e(that)g(functions)g Fk(\026)1670 1851 y Fl(\013)1718 1839 y Fr(\()p Fk(x)p Fr(\))g Fm(2)g(I)2003 1809 y Ff(\003)2042 1839 y Fr(\()p Fk(A)p Fr(\))h(ha)n(ving)e(p)r(oles)g(of)h(degree)f Fk(d)h Fm(\025)f Fr(2)h(in)515 1938 y Fk(x)e Fr(=)g(0)g(cannot)h(app)r (ear)f(in)h(\(10\).)53 b(That)33 b(is,)h(only)f(algebraic)e(functions)i (and)g(functions)515 2038 y(with)26 b(simple)g(p)r(oles)f(in)h(the)g (origin)f(can)g(app)r(ear)g(in)h(the)g(actual)f(normal)g(form)g (unfolding:)515 2138 y Fm(G)33 b Fr(is)27 b(not)h(the)g(full)g Fk(G)p Fr(-generated)e(mo)r(dule)i(o)n(v)n(er)e Fm(I)2109 2108 y Ff(\003)2148 2138 y Fr(\()p Fk(A)p Fr(\).)515 2337 y Fs(Example.)34 b Fr(Let)25 b(us)g(brie\015y)g(men)n(tion)g(an)g (example)g(where)g(indeed)g(meromorphic)f(func-)515 2437 y(tions)i(of)h(the)g(in)n(v)-5 b(arian)n(ts)26 b(en)n(ter)g(in)h(the)h (normal)d(form)i(unfolding.)37 b(Consider)25 b(systems)i(in)515 2536 y Fs(R)587 2506 y Fj(3)646 2536 y Fr(with)d(co)r(ordinates)d(\()p Fk(x;)14 b(y)s(;)g(z)t Fr(\);)24 b(let)f(the)g(linear)f(part)h(b)r(e)g (giv)n(en)f(b)n(y)g(the)h(diagonal)f(matrix)515 2636 y Fk(A)h Fr(=)g(diag\()p Fm(\000)p Fr(1)p Fk(;)14 b Fr(1)p Fk(;)g Fr(2\),)24 b(so)g(that)h Fk(X)1562 2648 y Fl(A)1639 2636 y Fr(=)e Fm(\000)p Fk(x@)1883 2648 y Fl(x)1937 2636 y Fr(+)13 b Fk(y)s(@)2103 2648 y Fl(y)2155 2636 y Fr(+)g(2)p Fk(z)t(@)2362 2648 y Fl(z)2398 2636 y Fr(.)36 b(This)25 b(has)f(t)n(w)n(o)g(basic)g(in)n(v)-5 b(ari-)515 2735 y(an)n(ts,)24 b(giv)n(en)g(b)n(y)g(\011)1104 2747 y Fj(1)1163 2735 y Fr(=)f Fk(xy)k Fr(and)e(\011)1590 2747 y Fj(2)1649 2735 y Fr(=)e Fk(x)1784 2705 y Fj(2)1822 2735 y Fk(z)t Fr(.)35 b(W)-7 b(e)24 b(tak)n(e)g(as)g Fk(X)2414 2705 y Fj(\()p Fl(a)p Fj(\))2529 2735 y Fr(the)h(v)n(ectors)d Fk(X)3023 2705 y Fj(\(1\))3135 2735 y Fr(=)h Fk(x@)3314 2747 y Fl(x)3356 2735 y Fr(,)515 2835 y Fk(X)591 2805 y Fj(\(2\))702 2835 y Fr(=)g Fk(y)s(@)878 2847 y Fl(y)918 2835 y Fr(,)g(and)f Fk(X)1196 2805 y Fj(\(3\))1308 2835 y Fr(=)g Fk(z)t(@)1482 2847 y Fl(z)1520 2835 y Fr(.)35 b(It)23 b(is)f(immediate)h(to)f(c)n(hec)n(k)f(that)i(\(\011)2728 2805 y Fj(2)2728 2856 y(1)2765 2835 y Fk(=)p Fr(\011)2872 2847 y Fj(2)2908 2835 y Fr(\))p Fk(X)3016 2805 y Fj(\(3\))3128 2835 y Fr(=)g Fk(y)3260 2805 y Fj(2)3297 2835 y Fk(@)3341 2847 y Fl(z)515 2935 y Fr(and)k(\(\011)773 2947 y Fj(2)810 2935 y Fk(=)p Fr(\011)917 2947 y Fj(1)954 2935 y Fr(\))p Fk(X)1062 2905 y Fj(\(2\))1173 2935 y Fr(=)c Fk(xz)t(@)1395 2947 y Fl(y)1462 2935 y Fr(are)k(p)r(olynomial)g(and)g(resonan)n(t)f (with)j Fk(X)2785 2947 y Fl(A)2838 2935 y Fr(.)37 b Fm(\014)681 3134 y Fr(In)25 b(sev)n(eral)e(cases)h(it)i(happ)r(ens)f(that)g Fm(G)31 b Fr(has)24 b(a)h(more)f(con)n(v)n(enien)n(t)g(structure,)i (i.e.)36 b(the)515 3234 y Fk(\026)565 3246 y Fl(\013)642 3234 y Fr(in)29 b(\(10\))g(can)g(actually)g(b)r(e)h(tak)n(en)f(to)g(b)r (e)h(in)g Fm(I)6 b Fr(\()p Fk(A)p Fr(\),)31 b(and)e(not)h(just)g(in)f Fm(I)2908 3203 y Ff(\003)2947 3234 y Fr(\()p Fk(A)p Fr(\).)43 b(In)30 b(this)515 3333 y(case)j(w)n(e)g(sa)n(y)f(that)i(all)g(the)g(v) n(ector)e(\014elds)i(in)g Fm(G)39 b Fr(are)33 b Fh(quasi-line)l(ar)p Fr(,)j(or)d(that)h(w)n(e)f(ha)n(v)n(e)f(a)515 3433 y Fh(quasi-line)l(ar)k(normal)h(form)p Fr(.)57 b(In)34 b(particular,)h(this)f(is)g(the)h(case)e(when)h Fk(A)g Fr(admits)g(only)515 3532 y(one)27 b(basic)g(in)n(v)-5 b(arian)n(t)26 b(\(see)i(the)g(examples)f(b)r(elo)n(w\).)681 3632 y(If)38 b(the)g(normal)f(form)h(is)f(quasilinear,)i(w)n(e)f(ha)n (v)n(e)e Fm(G)31 b(\\)26 b(V)2519 3644 y Fl(k)q Fj(+1)2684 3632 y Fr(=)39 b Fm(I)2833 3644 y Fl(k)2874 3632 y Fr(\()p Fk(A)p Fr(\))27 b Fm(\012)d Fk(G)p Fr(,)41 b(and)515 3732 y(the)32 b(analysis)e(of)i(the)g(structure)f(of)h Fm(G)37 b Fr(results)31 b(to)h(b)r(e)g(particularly)e(simple,)j(as)e(w) n(e)h(no)n(w)515 3831 y(discuss.)681 3931 y(Call)23 b Fm(X)923 3901 y Ff(\003)911 3951 y Fl(\013)984 3931 y Fr(the)h(algebra)d(spanned)i(b)n(y)g(v)n(ectors)f(whic)n(h)h(are)f (written)h(as)g Fk(X)29 b Fr(=)23 b Fk(s)p Fr(\()p Fk(x)p Fr(\))p Fk(X)3280 3901 y Fj(\()p Fl(\013)p Fj(\))515 4031 y Fr(with)j Fk(s)d Fm(2)h(I)894 4000 y Ff(\003)932 4031 y Fr(\()p Fk(A)p Fr(\);)k(call)d Fm(X)1318 4043 y Fl(\013)1392 4031 y Fr(the)h(algebra)f(spanned)h(b)n(y)f(v)n(ectors)g (as)g(ab)r(o)n(v)n(e)g(with)h Fk(s)d Fm(2)h(I)6 b Fr(\()p Fk(A)p Fr(\))515 4130 y(\(this)28 b(is)f(the)h(mo)r(dule)g(o)n(v)n(er)e Fm(I)6 b Fr(\()p Fk(A)p Fr(\))29 b(generated)d(b)n(y)i Fk(X)2182 4100 y Fj(\()p Fl(\013)p Fj(\))2280 4130 y Fr(\).)681 4230 y(As)f(seen)f(b)r(efore,)h(in)g(general)e(w)n(e)i(ha)n (v)n(e)e Fm(X)2006 4242 y Fj(1)2061 4230 y Fm(\010)16 b Fk(:::)h Fm(\010)g(X)2369 4242 y Fl(d)2431 4230 y Fm(\022)22 b(G)29 b(\032)49 b(X)2781 4200 y Ff(\003)2769 4250 y Fj(1)2837 4230 y Fm(\010)16 b Fk(:::)h Fm(\010)g(X)3157 4200 y Ff(\003)3145 4253 y Fl(d)3195 4230 y Fr(,)27 b(and)515 4329 y(in)h(the)g(\(fa)n(v)n(ourable\))e(quasi-linear)f(case)i(w)n(e)g (actually)g(ha)n(v)n(e)g Fm(G)h Fr(=)23 b Fm(X)2696 4341 y Fj(1)2752 4329 y Fm(\010)18 b Fk(:::)g Fm(\010)g(X)3064 4341 y Fl(d)3103 4329 y Fr(.)681 4429 y(Consider)26 b(no)n(w)h(the)h (comm)n(utation)f(relations)g(b)r(et)n(w)n(een)g(elemen)n(ts)h(of)f (the)h(subalge-)515 4529 y(bras)e Fm(X)766 4499 y Ff(\003)754 4549 y Fl(\013)833 4529 y Fr(and)h Fm(X)1065 4499 y Ff(\003)1053 4552 y Fl(\014)1104 4529 y Fr(;)g(it)h(is)g(immediate)g(to)f(c)n(hec)n (k)g(that)642 4661 y Fg(\002)676 4729 y Fk(\026)726 4741 y Fl(\013)774 4729 y Fr(\(\011\))p Fk(X)979 4698 y Fj(\()p Fl(\013)p Fj(\))1077 4729 y Fk(;)14 b(\033)1161 4741 y Fl(\014)1206 4729 y Fr(\(\011\))p Fk(X)1411 4698 y Fj(\()p Fl(\014)s Fj(\))1508 4661 y Fg(\003)1593 4729 y Fr(=)1708 4661 y Fg(\000)1746 4729 y Fk(\026)1796 4741 y Fl(\013)1844 4729 y Fr(\(\011\))g(\()p Fk(@)5 b(\033)2115 4741 y Fl(\014)2160 4729 y Fk(=@)g( )2305 4741 y Fl(i)2332 4729 y Fr(\))14 b Fk(X)2454 4698 y Fj(\()p Fl(\013)p Fj(\))2552 4729 y Fr(\()p Fk( )2638 4741 y Fl(i)2666 4729 y Fr(\))2698 4661 y Fg(\001)2750 4729 y Fk(X)2826 4698 y Fj(\()p Fl(\014)s Fj(\))725 4832 y Fm(\000)831 4765 y Fg(\000)869 4832 y Fk(\033)916 4844 y Fl(\014)961 4832 y Fr(\(\011\))g(\()p Fk(@)5 b(\026)1235 4844 y Fl(\013)1282 4832 y Fk(=@)g( )1427 4844 y Fl(i)1454 4832 y Fr(\))14 b Fk(X)1576 4802 y Fj(\()p Fl(\014)s Fj(\))1672 4832 y Fr(\()p Fk( )1758 4844 y Fl(i)1786 4832 y Fr(\))1818 4765 y Fg(\001)1871 4832 y Fk(X)1947 4802 y Fj(\()p Fl(\013)p Fj(\))2091 4832 y Fr(+)46 b(\()p Fk(\026)2284 4844 y Fl(a)2324 4832 y Fr(\(\011\))p Fk(\033)2500 4844 y Fl(\014)2546 4832 y Fr(\(\011\)\))2735 4765 y Fg(\002)2769 4832 y Fk(X)2845 4802 y Fj(\()p Fl(\013)p Fj(\))2944 4832 y Fk(;)14 b(X)3057 4802 y Fj(\()p Fl(\014)s Fj(\))3153 4765 y Fg(\003)3229 4832 y Fk(:)3273 4933 y Fr(\(9\))1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fr(Notice)25 b(that)g(when)g Fk(X)1240 493 y Fj(\()p Fl(\014)s Fj(\))1359 523 y Fr(=)d Fk(X)1515 535 y Fl(A)1569 523 y Fr(,)k(b)n(y)e(de\014nition)i Fk(X)2173 493 y Fj(\()p Fl(\014)s Fj(\))2269 523 y Fr(\()p Fk( )2355 535 y Fl(i)2383 523 y Fr(\))d(=)g(0,)i(and)f([)p Fk(X)2873 493 y Fj(\()p Fl(\013)p Fj(\))2972 523 y Fk(;)14 b(X)3085 493 y Fj(\()p Fl(\014)s Fj(\))3181 523 y Fr(])23 b(=)f(0;)515 623 y(th)n(us)27 b(the)h(corresp)r(onding)e(subalgebra)g Fm(X)1849 635 y Fl(\014)1921 623 y Fr(is)i(alw)n(a)n(ys)e(an)h(ab)r (elian)g(ideal)h(in)f Fm(G)5 b Fr(.)515 772 y Fs(Remark.)39 b Fr(Note)29 b(also)f(that,)i(as)e(ob)n(vious)g(from)g(the)i(form)n (ula)e(\(9\))h(ab)r(o)n(v)n(e,)f(the)h(union)g(of)515 872 y(subalgebras)24 b Fm(X)1021 884 y Fl(\013)1064 892 y Fe(1)1117 872 y Fm([)16 b Fk(:::)g Fm([)g(X)1403 884 y Fl(\013)1446 892 y Fd(s)1509 872 y Fr(is)26 b(a)g(subalgebra)e(in)j Fm(G)k Fr(if)c(and)f(only)g(if)h Fm(f)p Fk(X)2858 842 y Fj(\()p Fl(\013)2927 850 y Fe(1)2958 842 y Fj(\))2988 872 y Fk(;)14 b(:::;)g(X)3207 842 y Fj(\()p Fl(\013)3276 850 y Fd(s)3307 842 y Fj(\))3337 872 y Fm(g)515 971 y Fr(span)27 b(a)g(subalgebra)f(in)i Fk(G)p Fr(.)37 b Fm(\014)681 1121 y Fr(It)i(can)e(happ)r(en)i(that)g(w)n(e)f(are)f(able)h(to)h (determine)f(a)g(sequence)g(of)g(subalgebras)515 1220 y Fm(F)575 1232 y Fl(p)636 1220 y Fm(\022)22 b(G)5 b Fr(,)27 b(eac)n(h)d(of)h(them)h(b)r(eing)f(the)g(union)g(of)h Fm(X)2054 1232 y Fl(\013)2126 1220 y Fr(subalgebras,)e(suc)n(h)h(that)g Fm(F)3018 1232 y Fj(0)3078 1220 y Fr(=)e Fm(G)30 b Fr(and)1602 1403 y([)14 b Fm(G)19 b Fk(;)28 b Fm(F)1818 1415 y Fl(p)1870 1403 y Fr(])51 b(=)f Fm(F)2119 1415 y Fl(p)p Fj(+1)2269 1403 y Fr(;)939 b(\(10\))515 1586 y(if)29 b(this)g(terminates)g(in)g (zero)f(w)n(e)h(sa)n(y)f(that)h Fm(G)34 b Fr(has)29 b(a)f Fh(quasi-nilp)l(otent)i Fr(structure.)40 b(Notice)515 1685 y(that)28 b(the)g(factor)e(algebras)g(\000)1451 1697 y Fl(p)1512 1685 y Fr(:=)d Fm(F)1683 1697 y Fl(p)1721 1685 y Fk(=)p Fm(F)1823 1697 y Fl(p)p Fj(+1)1972 1685 y Fr(are)k(in)h(general)e Fh(not)57 b Fr(ab)r(elian.)681 1785 y(By)31 b(the)h(ab)r(o)n(v)n(e)f(remark,)g Fm(G)37 b Fr(can)31 b(ha)n(v)n(e)g(a)g(quasi-nilp)r(oten)n(t)g(structure)h (only)f(if)h Fk(G)g Fr(is)515 1885 y(nilp)r(oten)n(t.)k(The)24 b(c)n(hain)g(of)g(subalgebras)e Fm(F)1880 1897 y Fl(p)1941 1885 y Fm(\032)h(G)29 b Fr(can)24 b(then)h(b)r(e)f(read)g(o\013)g(the)g (descending)515 1984 y(cen)n(tral)g(series)g Fk(G)1072 1996 y Fl(p)1135 1984 y Fr(of)h Fk(G)p Fr(;)i(recall)d(that)h(the)g (factor)f(algebras)f Fk(\015)2477 1996 y Fl(p)2539 1984 y Fr(=)f Fk(G)2691 1996 y Fl(p)2730 1984 y Fk(=G)2837 1996 y Fl(p)p Fj(+1)2984 1984 y Fr(for)j(this)g(are)515 2084 y(ab)r(elian.)45 b(The)30 b(subalgebras)f(\000)1521 2096 y Fl(p)1589 2084 y Fr(in)n(tro)r(duced)h(ab)r(o)n(v)n(e)f(are)h (therefore)f(mo)r(duli)i(o)n(v)n(er)e Fm(I)6 b Fr(\()p Fk(A)p Fr(\))515 2183 y(generated)26 b(b)n(y)h(ab)r(elian)h (subalgebras)d Fk(\015)1788 2195 y Fl(p)1854 2183 y Fr(of)j Fk(G)p Fr(.)681 2383 y(Assume)23 b(no)n(w)g Fm(G)29 b Fr(is)23 b(quasi-nilp)r(oten)n(t.)35 b(In)24 b(this)g(case)e(w)n(e)h (can)g(\014rst)g(w)n(ork)f(with)i(gener-)515 2482 y(ators)19 b(in)i(\000)858 2494 y Fj(1)917 2482 y Fr(and)f(simplify)i(terms)e(in)h (\000)1745 2494 y Fj(1)1804 2482 y Fr(\(e.g.)34 b(b)n(y)21 b(follo)n(wing)f(the)h(PRF)g(algorithm)e(within)515 2582 y(the)32 b(set)f(\000)847 2594 y Fj(1)884 2582 y Fr(;)j(this)e(allo)n (ws)e(to)h(w)n(ork)f(with)j(more)d(familiar)h(pro)5 b(jection)31 b(and)g(homological)515 2682 y(equations)24 b(than)h(if)h(setting)f (the)h(problem)f(in)g(a)g(completely)g(Lie)g(algebraic)f(framew)n (ork\),)515 2781 y(then)e(consider)f(generators)f(in)i(\000)1556 2793 y Fj(2)1616 2781 y Fr(and)g(simplify)g(the)h(corresp)r(onding)d (terms)h(b)r(eing)i(guar-)515 2881 y(an)n(teed)k(that)h(\000)1012 2893 y Fj(1)1077 2881 y Fr(terms)f(are)g(not)g(c)n(hanged,)g(and)g(so)g (on.)681 2980 y(Notice)22 b(that)h(in)g(this)f(case)g(w)n(e)g(are)g({)g (roughly)f(sp)r(eaking)g({)i(just)g(using)f(the)h(nilp)r(oten)n(t)515 3080 y(structure)e(of)g(\(the)i(\014nite)f(dimensional)f(group\))g Fk(G)p Fr(,)i(rather)e(than)g(the)h(one)f(of)h(\(the)g(in\014nite)515 3180 y(dimensional)27 b(algebra\))f Fm(G)5 b Fr(.)681 3279 y(Needless)31 b(to)h(sa)n(y)-7 b(,)32 b(this)h(approac)n(h)d(is)i (particularly)e(con)n(v)n(enien)n(t)h(when)h(the)h(\000)3198 3291 y Fl(p)3268 3279 y Fr(are)515 3379 y(generated)26 b(b)n(y)h(a)h(single)f(elemen)n(t)g(of)h Fk(G)p Fr(.)681 3479 y(The)g(situation)g(depicted)h(ab)r(o)n(v)n(e)d(is)i(met)h(in)f (applications:)38 b(e.g.,)27 b(it)i(applies)f(to)g(an)n(y)515 3578 y(non)n(trivial)h(t)n(w)n(o-dimensional)g(case)h(and)h(sev)n(eral) e(three-dimensional)g(ones)h([22].)46 b(More)515 3678 y(generally)-7 b(,)26 b(it)i(alw)n(a)n(ys)e(applies)h(when)h(there)f (is)h(only)f(one)g(basic)g(in)n(v)-5 b(arian)n(t.)681 3778 y(In)28 b(the)h(follo)n(wing)e(sections)h(w)n(e)g(will)h(consider) e(some)h(simple)g(examples)g(where)g(the)515 3877 y(LRF)23 b(is)h(easily)e(computed,)j(and)e(it)h(turns)f(out)h(to)f(b)r(e)h (de\014nitely)g(simpler)f(than)g(the)h(PRF.)515 4152 y Fq(3)134 b(Example)46 b(I)515 4334 y Fr(Let)21 b(us)g(consider)g(a)g (system)g(in)g Fs(R)1569 4303 y Fj(3)1628 4334 y Fr(\(w)n(e)g(use)g(co) r(ordinates)f Fk(x;)14 b(y)s(;)g(z)t Fr(\))21 b(with)g(linear)g(part)g (giv)n(en)515 4433 y(b)n(y)1536 4628 y Fk(A)51 b Fr(=)1765 4461 y Fg(0)1765 4610 y(@)1851 4528 y Fr(0)83 b Fm(\000)p Fr(1)114 b(0)1851 4628 y(1)h(0)147 b(0)1851 4727 y(0)115 b(0)g Fm(\000)p Fr(1)2285 4461 y Fg(1)2285 4610 y(A)515 4882 y Fr(It)30 b(is)g(easy)f(to)h(see)g(that)g(this)g(has)g(only)f (one)h(basic)f(in)n(v)-5 b(arian)n(t)29 b(\011)e(:=)g(\()p Fk(x)2802 4852 y Fj(2)2860 4882 y Fr(+)19 b Fk(y)2988 4852 y Fj(2)3025 4882 y Fr(\).)45 b(As)30 b(an)n(y)515 4982 y(meromorphic)e(function)i(of)g(\011)g(is)f(either)h(algebraic)e (or)h(has)g(a)g(p)r(ole)h(of)g(degree)e Fk(d)f Fm(\025)f Fr(2)k(in)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fr(the)34 b(origin,)h(w)n(e)f(deduce)g(that)g(the)h (most)f(general)e(v)n(ector)h(\014eld)h(in)h(normal)e(form)h(with)515 623 y(resp)r(ect)27 b(to)h(this)f(linear)g(part)g(is)1243 867 y Fk(W)63 b Fr(=)50 b Fk(X)1568 879 y Fl(A)1668 867 y Fr(+)1806 763 y Ff(1)1779 788 y Fg(X)1779 967 y Fl(k)q Fj(=1)1913 867 y Fk(a)1957 879 y Fl(k)1998 867 y Fk(X)2067 879 y Fl(k)2126 867 y Fr(+)18 b Fk(b)2245 879 y Fl(k)2286 867 y Fk(Y)2334 879 y Fl(k)2393 867 y Fr(+)g Fk(c)2512 879 y Fl(k)2553 867 y Fk(Z)2610 879 y Fl(k)3231 867 y Fr(\(11\))515 1129 y(where)27 b(\(with)h Fk(k)e Fm(\025)d Fr(0\))1396 1158 y Fg(8)1396 1233 y(<)1396 1382 y(:)1484 1233 y Fk(X)1553 1245 y Fl(k)1616 1233 y Fr(:=)51 b(\011)1820 1203 y Fl(k)1874 1233 y Fr(\()p Fk(x@)1997 1245 y Fl(x)2058 1233 y Fr(+)18 b Fk(y)s(@)2229 1245 y Fl(y)2269 1233 y Fr(\))1484 1332 y Fk(Y)1532 1344 y Fl(k)1596 1332 y Fr(:=)50 b(\011)1799 1302 y Fl(k)1853 1332 y Fr(\()p Fm(\000)p Fk(y)s(@)2038 1344 y Fl(x)2098 1332 y Fr(+)18 b Fk(x@)2272 1344 y Fl(y)2313 1332 y Fr(\))1484 1432 y Fk(Z)1541 1444 y Fl(k)1604 1432 y Fr(:=)50 b(\011)1807 1402 y Fl(k)1862 1432 y Fr(\()p Fk(z)t(@)1981 1444 y Fl(z)2019 1432 y Fr(\))377 b(.)515 1573 y(This)41 b(form)g(can)f(also)g (b)r(e)i(easily)e(deduced)h(b)n(y)g(explicit)g(computation)g(applying)g (the)515 1672 y(de\014nition)35 b(of)f(resonan)n(t)f(v)n(ector)h (\014eld.)58 b(Ob)n(viously)-7 b(,)35 b Fk(X)2308 1684 y Fl(A)2396 1672 y Fr(=)g Fk(Y)2544 1684 y Fj(0)2604 1672 y Fm(\000)23 b Fk(Z)2749 1684 y Fj(0)2786 1672 y Fr(.)58 b(W)-7 b(e)35 b(denote)f(b)n(y)515 1772 y Fk(\026;)14 b(\027)q(;)g(\033)30 b Fr(the)e(\014rst)g Fk(k)e Fm(\025)c Fr(1)27 b(suc)n(h)h(that)g Fk(a)1710 1784 y Fl(k)1750 1772 y Fk(;)14 b(b)1823 1784 y Fl(k)1864 1772 y Fk(;)g(c)1937 1784 y Fl(k)2005 1772 y Fr(are)27 b(nonzero.)681 1872 y(The)g Fk(X)920 1884 y Fl(k)961 1872 y Fk(;)14 b(Y)1046 1884 y Fl(k)1087 1872 y Fk(;)g(Z)1181 1884 y Fl(k)1249 1872 y Fr(satisfy)27 b(the)h(comm)n(utation)f(relations)896 2049 y([)p Fk(X)988 2061 y Fl(k)1029 2049 y Fk(;)14 b(X)1135 2061 y Fl(m)1198 2049 y Fr(])23 b(=)f(2\()p Fk(m)d Fm(\000)f Fk(k)s Fr(\))p Fk(X)1727 2061 y Fl(k)q Fj(+)p Fl(m)1905 2049 y Fk(;)41 b Fr([)p Fk(Y)2040 2061 y Fl(k)2082 2049 y Fk(;)14 b(Y)2167 2061 y Fl(m)2230 2049 y Fr(])23 b(=)f(0)28 b Fk(;)41 b Fr([)p Fk(Z)2577 2061 y Fl(k)2618 2049 y Fk(;)14 b(Z)2712 2061 y Fl(m)2774 2049 y Fr(])24 b(=)e(0)896 2148 y([)p Fk(X)988 2160 y Fl(k)1029 2148 y Fk(;)14 b(Y)1114 2160 y Fl(m)1177 2148 y Fr(])23 b(=)g(2)p Fk(mY)1474 2160 y Fl(k)q Fj(+)p Fl(m)1652 2148 y Fk(;)41 b Fr([)p Fk(X)1808 2160 y Fl(k)1849 2148 y Fk(;)14 b(Z)1943 2160 y Fl(m)2006 2148 y Fr(])23 b(=)f(2)p Fk(mZ)2311 2160 y Fl(k)q Fj(+)p Fl(m)2489 2148 y Fk(;)42 b Fr([)p Fk(Y)2625 2160 y Fl(k)2666 2148 y Fk(;)14 b(Z)2760 2160 y Fl(m)2822 2148 y Fr(])23 b(=)g(0)3231 2099 y(\(12\))681 2327 y(Denoting)30 b(b)n(y)f Fm(X)12 b Fk(;)i Fm(Y)7 b Fk(;)14 b Fm(Z)37 b Fr(the)30 b(algebras)e(spanned)i(b)n(y)f(the)h Fk(X)2591 2339 y Fl(k)2632 2327 y Fr(,)g(the)h Fk(Y)2879 2339 y Fl(k)2950 2327 y Fr(and)e(the)h Fk(Z)3315 2339 y Fl(k)3356 2327 y Fr(,)515 2427 y(w)n(e)c(ha)n(v)n(e)g(that)i Fm(G)g Fr(=)23 b Fm(X)30 b(\010)17 b(Y)25 b(\010)17 b(Z)7 b Fr(,)27 b(and)g(that)g Fm(Y)e(\010)17 b(Z)34 b Fr(is)27 b(an)g(ab)r(elian)g(ideal)f(in)i Fm(G)5 b Fr(.)37 b(W)-7 b(e)27 b(can)515 2527 y(th)n(us)g(apply)h(the)g(LRF)g(pro)r(cedure)e (discussed)h(ab)r(o)n(v)n(e.)681 2626 y(W)-7 b(e)34 b(\014rst)g(op)r (erate)f(on)h Fm(X)12 b Fr(,)36 b(with)f(generators)c(also)i(in)i Fm(X)46 b Fr(\(th)n(us)34 b Fk(H)2840 2638 y Fl(k)2915 2626 y Fr(=)g Fk(\013)3067 2638 y Fl(k)3107 2626 y Fk(X)3176 2638 y Fl(k)3217 2626 y Fr(\);)k(in)515 2726 y(this)24 b(w)n(a)n(y)g(w)n(e)g(can)g(eliminate)g(all)g(terms)g(except)h(the)g Fk(X)2270 2738 y Fl(\026)2338 2726 y Fr(and)g(the)f Fk(X)2705 2738 y Fj(2)p Fl(\026)2807 2726 y Fr(ones,)h(as)e(implied)515 2825 y(b)n(y)k(\(12\).)37 b(In)27 b(doing)g(this)h(w)n(e)g(mo)r(dify)g (terms)f(in)h Fm(Y)d(\010)19 b(Z)7 b Fr(.)681 2925 y(Ha)n(ving)30 b(p)r(erformed)g(this)h(\014rst)f(step,)i(w)n(e)f(pass)f(to)g(consider) g(the)h Fm(Y)38 b Fr(and)31 b Fm(Z)37 b Fr(terms,)515 3025 y(op)r(erating)27 b(with)i(generators)e(in)i Fm(Y)d(\010)19 b(Z)35 b Fr(\(th)n(us)29 b Fk(H)2124 3037 y Fl(k)2190 3025 y Fr(=)24 b Fk(\014)2326 3037 y Fl(k)2367 3025 y Fk(Y)2415 3037 y Fl(k)2475 3025 y Fr(+)19 b Fk(\015)2602 3037 y Fl(k)2643 3025 y Fk(Z)2700 3037 y Fl(k)2740 3025 y Fr(\).)40 b(It)29 b(is)g(clear)e(from)515 3124 y(\(12\))c(that)h(w)n (e)f(can)g(eliminate)h(all)g(terms)f(with)h Fk(k)i(>)c(\026)p Fr(,)j(but)f(no)g(lo)n(w)n(est)e(order)g(ones.)35 b(Th)n(us)515 3224 y(w)n(e)27 b(end)h(up)g(with)g(a)f(LRF)h(giv)n(en)f(b)n(y)905 3451 y Fg(c)901 3472 y Fk(W)63 b Fr(=)50 b Fk(X)1226 3484 y Fl(A)1326 3472 y Fr(+)c Fk(a)1481 3484 y Fl(\026)1525 3472 y Fk(X)1594 3484 y Fl(\026)1657 3472 y Fr(+)17 b Fg(b)-45 b Fk(a)1784 3484 y Fj(2)p Fl(\026)1861 3472 y Fk(X)1930 3484 y Fj(2)p Fl(\026)2054 3472 y Fr(+)2207 3365 y Fl(\026)2167 3394 y Fg(X)2165 3572 y Fl(k)q Fj(=)p Fl(\027)2298 3450 y Fg(b)2303 3472 y Fk(b)2339 3484 y Fl(k)2380 3472 y Fk(Y)2428 3484 y Fl(k)2515 3472 y Fr(+)2670 3365 y Fl(\026)2630 3394 y Fg(X)2626 3572 y Fl(k)q Fj(=)p Fl(\033)2767 3472 y Fg(b)g Fk(c)2804 3484 y Fl(k)2844 3472 y Fk(Z)2901 3484 y Fl(k)2970 3472 y Fr(;)515 3730 y(the)34 b(hat)f(on)h(constan)n(ts)e(mean)h(that)h(these)g(are)e(not)i (the)g(same)f(as)g(in)h(the)g(initial)f(form)515 3829 y(\(11\),)26 b(and)g(ob)n(viously)f(a)g(sum)i(with)f(lo)n(w)n(er)f (limit)i(greater)d(than)i(the)h(higher)e(limit)i(should)515 3929 y(just)h(b)r(e)g(mean)n(t)f(as)g(zero.)681 4028 y(As)21 b(sho)n(wn)f(b)n(y)h(this)g(example,)h(the)f(computations)g (required)f(for)g(the)h(determination)515 4128 y(of)27 b(the)h(general)e(LRF)i(are)f(actually)g(v)n(ery)f(simple.)515 4403 y Fq(4)134 b(Example)46 b(I)t(I)515 4584 y Fr(W)-7 b(e)30 b(w)n(an)n(t)e(no)n(w)h(to)g(consider)g(an)g(example)g(where)g (the)h(algebra)d Fm(G)2640 4596 y Fl(A)2724 4584 y Fr(of)i(resonan)n(t) f(v)n(ector)515 4684 y(\014elds)39 b(has)f(not)i(the)f(optimal)g (structure)g(for)f(LRF)i(reduction,)h(i.e.)72 b(the)39 b(\000)3031 4696 y Fl(p)3109 4684 y Fr(are)f(not)515 4784 y(generated)26 b(b)n(y)h(a)h(single)f(elemen)n(t)g(of)h Fk(G)g Fr(\(see)f(section)h(2\).)681 4883 y(W)-7 b(e)19 b(consider)g(a)g(system)g(in)g Fs(R)1619 4853 y Fj(4)1676 4883 y Fr(with)h(linear)e(part)h(corresp)r(onding)e(to)j(t)n(w)n(o)e (oscillators)515 4983 y(with)33 b(nonzero)f(and)h(non-resonan)n(t)e (frequencies)h Fk(\013)i Fr(and)e Fk(\014)t Fr(,)j Fm(j)p Fk(\013=\014)t Fm(j)e(62)f Fs(Q)p Fr(,)i(i.e.)53 b(in)34 b(blo)r(c)n(k)1905 5255 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fr(notation)1210 652 y Fk(A)51 b Fr(=)1439 535 y Fg(\022)1514 602 y Fk(\013J)123 b Fr(0)1547 702 y(0)115 b Fk(\014)t(J)1823 535 y Fg(\023)1981 652 y Fk(;)70 b(J)31 b Fr(=)2238 535 y Fg(\022)2313 602 y Fr(0)83 b Fm(\000)p Fr(1)2313 702 y(1)115 b(0)2558 535 y Fg(\023)2660 652 y Fr(;)515 831 y(this)28 b Fk(A)g Fr(has)f(eigen)n(v)-5 b(alues)26 b Fk(\025)e Fr(=)e Fm(\006)p Fk(i\013;)14 b Fm(\006)p Fk(i\014)t Fr(.)36 b(W)-7 b(e)28 b(use)g(co)r(ordinates)e(\()p Fk(x;)14 b(y)s(;)g(z)t(;)g(w)r Fr(\),)28 b(so)f(that)1129 980 y Fk(X)1198 992 y Fl(A)1303 980 y Fr(=)50 b Fk(\013)14 b Fr(\()p Fm(\000)p Fk(y)s(@)1670 992 y Fl(x)1730 980 y Fr(+)k Fk(x@)1904 992 y Fl(y)1945 980 y Fr(\))46 b(+)g Fk(\014)18 b Fr(\()p Fm(\000)p Fk(w)r(@)2401 992 y Fl(z)2458 980 y Fr(+)g Fk(z)t(@)2628 992 y Fl(w)2681 980 y Fr(\))28 b Fk(:)515 1129 y Fr(It)f(is)h(immediate)f(to)g(see)g (that)h(the)g(system)f(is)g(simply)g(resonan)n(t)f(and)h(admits)h(t)n (w)n(o)e(inde-)515 1229 y(p)r(enden)n(t)k(basic)e(in)n(v)-5 b(arian)n(ts,)28 b Fk( )1504 1241 y Fj(1)1567 1229 y Fr(=)e Fk(x)1705 1199 y Fj(2)1762 1229 y Fr(+)19 b Fk(y)1890 1199 y Fj(2)1956 1229 y Fr(and)29 b Fk( )2173 1241 y Fj(2)2236 1229 y Fr(=)c Fk(z)2369 1199 y Fj(2)2425 1229 y Fr(+)19 b Fk(w)2570 1199 y Fj(2)2608 1229 y Fr(.)42 b(On)29 b(the)g(other)g(hand,)515 1329 y(the)36 b(linear)e(space)h Fm(G)1183 1341 y Fj(1)1256 1329 y Fr(of)g(linear)g(v)n(ector)f (\014elds)i(comm)n(uting)f(with)h Fk(X)2780 1341 y Fl(A)2869 1329 y Fr(is)f(spanned)g(b)n(y)515 1428 y Fk(X)591 1398 y Fj(\()p Fl(\013)p Fj(\))716 1428 y Fr(=)27 b(\()p Fk(M)921 1440 y Fl(\013)968 1428 y Fk(x)p Fr(\))p Fm(r)p Fr(,)k(with)g Fk(\013)c Fr(=)g(1)p Fk(;)14 b(:::;)g Fr(4)29 b(and)g(w)n(e)h(can)f(c)n (ho)r(ose)g(the)h(matrices)g Fk(M)3060 1440 y Fl(\013)3136 1428 y Fr(e.g.)44 b(as)515 1528 y(\(in)26 b(blo)r(c)n(k)g(notation,)g (with)h Fk(I)33 b Fr(the)26 b(t)n(w)n(o-dimensional)f(iden)n(tit)n(y)h (matrix)g(and)f Fk(J)35 b Fr(as)25 b(ab)r(o)n(v)n(e\))685 1727 y Fk(M)766 1739 y Fj(1)826 1727 y Fr(=)913 1610 y Fg(\022)988 1677 y Fk(I)90 b Fr(0)989 1777 y(0)83 b(0)1170 1610 y Fg(\023)1272 1727 y Fk(;)42 b(M)1418 1739 y Fj(2)1478 1727 y Fr(=)1565 1610 y Fg(\022)1640 1677 y Fr(0)84 b(0)1640 1777 y(0)f Fk(I)1822 1610 y Fg(\023)1924 1727 y Fk(;)42 b(M)2070 1739 y Fj(3)2130 1727 y Fr(=)2217 1610 y Fg(\022)2292 1677 y Fk(J)91 b Fr(0)2299 1777 y(0)d(0)2485 1610 y Fg(\023)2587 1727 y Fk(;)42 b(M)2733 1739 y Fj(4)2793 1727 y Fr(=)2880 1610 y Fg(\022)2955 1677 y Fr(0)89 b(0)2955 1777 y(0)83 b Fk(J)3148 1610 y Fg(\023)515 1937 y Fr(It)28 b(is)f(also)g(immediate) g(to)h(c)n(hec)n(k)f(that)g Fk(X)1841 1907 y Fj(\()p Fl(\013)p Fj(\))1940 1937 y Fr(\()p Fk( )2026 1949 y Fl(i)2054 1937 y Fr(\))c(=)g(0)k(except)h(for)f Fk(X)2730 1907 y Fj(\(1\))2818 1937 y Fr(\()p Fk( )2904 1949 y Fj(1)2942 1937 y Fr(\))c(=)g(2)p Fk( )3181 1949 y Fj(1)3245 1937 y Fr(and)515 2037 y Fk(X)591 2007 y Fj(\(2\))679 2037 y Fr(\()p Fk( )765 2049 y Fj(2)803 2037 y Fr(\))g(=)g(2)p Fk( )1042 2049 y Fj(2)1079 2037 y Fr(.)681 2137 y(W)-7 b(e)40 b(will)g(write)f Fk(X)1305 2106 y Fl(\013)1298 2160 y(k)q(m)1441 2137 y Fr(:=)j Fk( )1628 2106 y Fl(k)1625 2157 y Fj(1)1669 2137 y Fk( )1726 2106 y Fl(m)1723 2157 y Fj(2)1789 2137 y Fk(X)1865 2106 y Fj(\()p Fl(\013)p Fj(\))1964 2137 y Fr(;)k(these)39 b(span)g Fm(G)5 b Fr(,)43 b(for)c Fk(k)s(;)14 b(m)40 b Fr(non-negativ)n(e)515 2236 y(in)n(tegers.)35 b(W)-7 b(e)27 b(also)f(write,)h(with)h(the)f (notation)f(in)n(tro)r(duced)h(ab)r(o)n(v)n(e,)f Fm(G)i Fr(=)23 b Fm(X)2956 2248 y Fj(1)3010 2236 y Fm(\010)17 b Fk(:::)g Fm(\010)f(X)3318 2248 y Fj(4)3356 2236 y Fr(;)515 2336 y(note)27 b(that)h Fm(X)938 2348 y Fj(3)994 2336 y Fm(\010)18 b(X)1136 2348 y Fj(4)1201 2336 y Fr(is)28 b(an)f(ab)r(elian)g(subalgebra)f(in)i Fk(G)p Fr(.)681 2435 y(The)f(non)n(trivial)g(comm)n(utation)g(relations)f(are)h(giv)n (en)g(b)n(y)614 2513 y Fg(\002)649 2580 y Fk(X)725 2550 y Fj(1)718 2603 y Fl(k)q(m)817 2580 y Fk(;)14 b(X)930 2550 y Fj(1)923 2600 y Fl(pq)994 2513 y Fg(\003)1079 2580 y Fr(=)50 b(2\()p Fk(p)18 b Fm(\000)h Fk(k)s Fr(\))14 b Fk(X)1580 2550 y Fj(1)1573 2603 y Fl(k)q Fj(+)p Fl(p;m)p Fj(+)p Fl(q)1916 2580 y Fk(;)2008 2513 y Fg(\002)2043 2580 y Fk(X)2119 2550 y Fj(2)2112 2603 y Fl(k)q(m)2211 2580 y Fk(;)g(X)2324 2550 y Fj(2)2317 2600 y Fl(pq)2388 2513 y Fg(\003)2473 2580 y Fr(=)50 b(2\()p Fk(q)22 b Fm(\000)c Fk(m)p Fr(\))c Fk(X)2999 2550 y Fj(2)2992 2603 y Fl(k)q Fj(+)p Fl(p;m)p Fj(+)p Fl(q)614 2618 y Fg(\002)649 2685 y Fk(X)725 2655 y Fj(1)718 2709 y Fl(k)q(m)817 2685 y Fk(;)g(X)930 2655 y Fj(2)923 2706 y Fl(pq)994 2618 y Fg(\003)1079 2685 y Fr(=)50 b(2)p Fk(p)14 b(X)1368 2655 y Fj(2)1361 2709 y Fl(k)q Fj(+)p Fl(p;m)p Fj(+)p Fl(q)1681 2685 y Fm(\000)32 b Fr(2)p Fk(mX)1969 2655 y Fj(1)1962 2709 y Fl(k)q Fj(+)p Fl(p;m)p Fj(+)p Fl(q)614 2723 y Fg(\002)649 2791 y Fk(X)725 2761 y Fj(1)718 2814 y Fl(k)q(m)817 2791 y Fk(;)14 b(X)930 2761 y Fl(\013)923 2811 y(pq)994 2723 y Fg(\003)1051 2791 y Fr(=)23 b(2)p Fk(pX)1299 2761 y Fl(\013)1292 2814 y(k)q Fj(+)p Fl(p;m)p Fj(+)p Fl(q)1635 2791 y Fk(;)1727 2723 y Fg(\002)1762 2791 y Fk(X)1838 2761 y Fj(2)1831 2814 y Fl(k)q(m)1930 2791 y Fk(;)14 b(X)2043 2761 y Fl(\013)2036 2811 y(pq)2107 2723 y Fg(\003)2164 2791 y Fr(=)23 b(2)p Fk(q)s(X)2410 2761 y Fl(\013)2403 2814 y(k)q Fj(+)p Fl(p;m)p Fj(+)p Fl(q)2746 2791 y Fr(\()p Fk(\013)h Fr(=)e(3)p Fk(;)14 b Fr(4\))515 2941 y(It)37 b(is)g(easy)f(to)h(see)g(that)h(there)f(is)g (no)f(w)n(a)n(y)g(to)h(separate)f(the)i(algebra)d(as)h(the)i(sum)f(of) 515 3041 y(one-dimensional)32 b(mo)r(duli)i(o)n(v)n(er)e Fm(I)6 b Fr(\()p Fk(A)p Fr(\).)56 b(Th)n(us,)35 b(the)f(b)r(est)g(w)n (e)f(can)g(do)h(in)g(the)g(spirit)f(of)515 3140 y(the)26 b(LRF)h(approac)n(h)d(is)i(to)g(act)f(\014rst)h(in)h(the)f Fm(X)1990 3152 y Fj(1)2043 3140 y Fm(\010)15 b(X)2182 3152 y Fj(2)2246 3140 y Fr(subalgebra.)34 b(Once)26 b(this)g(has)g(b)r (een)515 3240 y(reduced,)36 b(sa)n(y)d(with)j(lo)n(w)n(est)d(nonlinear) h(term)g(in)h Fm(G)2189 3252 y Fl(\026)2234 3240 y Fr(,)i(w)n(e)d(can)g (eliminate)h(all)f(terms)h(in)515 3340 y(\()p Fm(X)606 3352 y Fj(3)666 3340 y Fm(\010)21 b(X)811 3352 y Fj(4)849 3340 y Fr(\))h Fm(\\)g(G)1029 3352 y Fl(m)1126 3340 y Fr(for)32 b(all)h Fk(m)f(>)g(\026)p Fr(,)i(as)e(follo)n(ws)g (immediately)i(from)e(the)i(comm)n(utation)515 3439 y(relations.)681 3539 y(Note)i(that)h(the)f(c)n(hoice)g(of)g(terms)g(to)g(b)r(e)h (eliminated)f(in)h Fm(X)2640 3551 y Fj(1)2702 3539 y Fm(\010)23 b(X)2849 3551 y Fj(2)2923 3539 y Fr(is)36 b(to)h(a)e(large)515 3638 y(exten)n(t)27 b(arbitrary)-7 b(,)26 b(and)i(corresp)r(ondingly)d(the)j(LRF)g(is)f(not)h(unique.)515 3907 y Fq(5)134 b(Example)46 b(I)t(I)t(I)515 4089 y Fr(W)-7 b(e)29 b(will)f(no)n(w)g(consider)f(the)i(LRF)g(pro)r(cedure)f(for)f(a) h(simple)h(system)f(in)h(full)g(detail,)g(i.e.)515 4189 y(aim)22 b(at)g(pro)r(ducing)g(completely)g(explicit)h(form)n(ulas)e (for)h(the)h(renormalized)d(form)i(and)h(for)515 4288 y(the)g(renormalizing)e(transformation,)h(thanks)h(to)f(the)i (computational)e(simplicit)n(y)h(of)g(the)515 4388 y(LRF)h(pro)r (cedure.)34 b(W)-7 b(e)24 b(will)g(also)f(compare)g(explicitely)g(the)h (LRF)g(and)g(PRF)g(reductions.)681 4488 y(Let)38 b(us)g(consider)f(a)h (t)n(w)n(o-dimensional)e(system)i(\(in)g Fs(R)2492 4457 y Fj(2)2567 4488 y Fr(with)h(co)r(ordinates)e Fk(x;)14 b(y)s Fr(\))515 4587 y(with)28 b(linear)f(part)g(giv)n(en)g(b)n(y)1702 4732 y Fk(A)d Fr(=)1875 4615 y Fg(\022)1950 4682 y Fr(0)83 b(0)1950 4782 y(0)g(1)2130 4615 y Fg(\023)515 4907 y Fr(i.e.)40 b(corresp)r(onding)27 b(to)i(the)g(v)n(ector)f(\014eld)h Fk(y)s(@)1963 4919 y Fl(y)2002 4907 y Fr(.)41 b(W)-7 b(e)29 b(note)g(immediately)g(that)g(here)f Fk(A)h Fr(is)515 5006 y(diagonal,)d(so)h(that)h Fk(A)23 b Fr(=)g Fk(A)1387 4976 y Fj(+)1442 5006 y Fr(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(only)i (one)f(basic)g(in)n(v)-5 b(arian)n(t)26 b Fk( )s Fr(\()p Fk(x;)14 b(y)s Fr(\))24 b(=)f Fk(x)p Fr(.)1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fi(5.1)112 b(Normal)36 b(forms)515 676 y Fr(It)30 b(is)g(easy)g(to)g(see)g(that)g(the)h(k)n(ernel)e(of)h Fm(L)1849 688 y Fj(0)1917 676 y Fr(is)g(spanned)g(b)n(y)g(the)g(arra)n (ys)e(of)i(v)n(ector)f(\014elds)515 776 y(\(with)f Fk(k)e Fm(\025)d Fr(0\))1089 959 y Fk(X)1158 971 y Fl(k)1221 959 y Fr(:=)g Fk(x)1379 924 y Fl(k)q Fj(+1)1518 959 y Fk(@)1562 971 y Fl(x)1655 959 y Fm(2)g(W)1815 971 y Fl(k)1911 959 y Fr(and)56 b Fk(Y)2149 971 y Fl(k)2213 959 y Fr(:=)22 b Fk(x)2370 924 y Fl(k)2412 959 y Fk(y)16 b(@)2513 971 y Fl(y)2604 959 y Fm(2)23 b(W)2764 971 y Fl(k)515 1141 y Fr(\(with)h(this)g(notation)g(the)g(linear)f(part)g(considered)g (here)g(is)h(giv)n(en)f(b)n(y)g Fk(Y)2792 1153 y Fj(0)2830 1141 y Fr(\).)36 b(These)23 b(v)n(ector)515 1241 y(\014elds)k(satisfy)h (the)g(comm)n(utation)f(relations)727 1423 y([)p Fk(X)819 1435 y Fl(k)860 1423 y Fk(;)14 b(X)966 1435 y Fl(m)1028 1423 y Fr(])24 b(=)e(\()p Fk(m)d Fm(\000)f Fk(k)s Fr(\))c Fk(X)1530 1435 y Fl(k)q Fj(+)p Fl(m)1708 1423 y Fk(;)69 b Fr([)p Fk(X)1892 1435 y Fl(k)1933 1423 y Fk(;)14 b(Y)2018 1435 y Fl(m)2081 1423 y Fr(])23 b(=)g Fk(m)14 b(Y)2350 1435 y Fl(m)2441 1423 y Fk(;)69 b Fr([)p Fk(Y)2604 1435 y Fl(k)2645 1423 y Fk(;)14 b(Y)2730 1435 y Fl(m)2793 1423 y Fr(])23 b(=)g(0)k Fk(:)212 b Fr(\(13\))515 1606 y(W)-7 b(e)26 b(denote)g(b)n(y)f Fm(X)39 b Fr(the)26 b(algebra)e(spanned)i(b)n(y)f(the)i Fk(X)2216 1576 y Ff(0)2209 1630 y Fl(k)2249 1606 y Fk(s)p Fr(,)g(b)n(y)e Fm(Y)33 b Fr(the)27 b(algebra)d(spanned)h(b)n(y)515 1706 y(the)j Fk(Y)706 1718 y Fl(k)747 1706 y Fr('s;)f(ob)n(viously)f Fm(G)j Fr(=)22 b Fm(X)31 b(\010)18 b(Y)7 b Fr(.)38 b(Note)27 b(that)h Fm(Y)35 b Fr(is)28 b(an)f(ab)r(elian)g(ideal)g(in)h Fk(G)p Fr(.)681 1805 y(The)d(\(standard\))f(normal)g(form)h(corresp)r (onding)e(to)i(the)g(linear)f(part)h(considered)f(in)515 1905 y(this)k(section)f(will)h(th)n(us)f(b)r(e)h(giv)n(en)f(b)n(y)g(a)h (v)n(ector)e(\014eld)1430 2150 y Fk(W)35 b Fr(=)22 b Fk(Y)1678 2162 y Fj(0)1734 2150 y Fr(+)1845 2046 y Ff(1)1818 2071 y Fg(X)1817 2249 y Fl(k)q Fj(=1)1938 2150 y Fr(\()p Fk(a)2014 2162 y Fl(k)2055 2150 y Fk(X)2124 2162 y Fl(k)2183 2150 y Fr(+)c Fk(b)2302 2162 y Fl(k)2343 2150 y Fk(Y)2391 2162 y Fl(k)2432 2150 y Fr(\))767 b(\(14\))515 2407 y(dep)r(ending)28 b(on)f(the)h(t)n(w)n(o)f(in\014nite)h(sequences)f(of)h(real)e(constan)n (ts)h Fk(a)2655 2419 y Fl(k)2696 2407 y Fk(;)14 b(b)2769 2419 y Fl(k)2809 2407 y Fr(.)681 2506 y(W)-7 b(e)31 b(will)h(denote)f (b)n(y)g Fk(\026)g Fr(\(resp)r(ectiv)n(ely)-7 b(,)32 b(b)n(y)e Fk(\027)5 b Fr(\))32 b(the)g(\014rst)f Fk(k)h Fm(\025)c Fr(1)j(suc)n(h)g(that)g Fk(a)3174 2518 y Fl(k)3244 2506 y Fm(6)p Fr(=)d(0)515 2606 y(\(resp)r(ectiv)n(ely)-7 b(,)40 b(suc)n(h)e(that)h Fk(b)1454 2618 y Fl(k)1535 2606 y Fm(6)p Fr(=)h(0\).)69 b(It)39 b(results)e(that)i(for)e(the)i (sak)n(e)e(of)h(our)g(presen)n(t)515 2706 y(discussion,)33 b(it)g(is)f(of)g(in)n(terest)g(to)h(consider)e(the)i(case)e Fk(\027)37 b(<)31 b(\026)h Fr(\(for)g(a)g(full)h(discussion)f(of)515 2805 y(this)j(system,)h(whatev)n(er)d Fk(\026)i Fr(and)f Fk(\027)5 b Fr(,)37 b(see)d([22)o(]\).)58 b(W)-7 b(e)35 b(will)g(refer)f(to)g(the)h(case)f Fk(\027)39 b Fr(=)c(1)f(as)515 2905 y(nondegenerate,)26 b(and)h(to)h Fk(\027)g(>)23 b Fr(1)k(as)g(degenerate.)515 3137 y Fi(5.2)112 b(The)38 b(PRF)f(reduction)f(sc)m(heme)515 3290 y Fr(W)-7 b(e)34 b(w)n(an)n(t)g(no)n(w)f(to)h(consider)f(the)i(PRF)f(corresp)r(onding)e (to)i(the)g(linear)f(part)h(giv)n(en)f(b)n(y)515 3390 y Fk(A)p Fr(.)78 b(In)41 b(the)h(spirit)f(of)g(PRF,)g(w)n(e)g(should)g (act)g(on)g(the)g(NF)h(\(14\))f(with)h(Lie-P)n(oincar)n(\023)-39 b(e)515 3490 y(transformations)17 b(generated)i(b)n(y)g(homogeneous)f (functions)h Fk(h)2480 3502 y Fl(m)2566 3490 y Fm(2)24 b Fr(Ker)o(\()p Fm(L)2868 3502 y Fj(0)2906 3490 y Fr(\))r Fm(\\)r Fk(V)3045 3502 y Fl(m)3109 3490 y Fr(.)34 b(These)515 3589 y(will)28 b(corresp)r(ond)e(to)h(the)h(action)f(of)h(v)n(ector)e (\014elds)i(of)f(the)h(form)f Fk(H)2647 3601 y Fl(m)2733 3589 y Fr(=)c Fk(\013X)2943 3601 y Fl(m)3025 3589 y Fr(+)18 b Fk(\014)t(Y)3207 3601 y Fl(m)3270 3589 y Fr(.)681 3689 y(W)-7 b(e)31 b(ha)n(v)n(e)f(then)i(to)e(consider)g Fm(L)1703 3701 y Fj(1)1741 3689 y Fr(;)j(this)e(dep)r(ends)h(on)e(the)i(co)r (e\016cien)n(ts)e(of)h(the)h(qua-)515 3789 y(dratic)27 b(part)g Fk(W)1013 3801 y Fj(1)1078 3789 y Fr(of)h(the)g(v)n(ector)e (\014eld)i Fk(W)12 b Fr(,)28 b(whic)n(h)f(w)n(e)g(write)h(as)f Fk(W)2638 3801 y Fj(1)2698 3789 y Fr(=)c Fk(a)2830 3801 y Fj(1)2867 3789 y Fk(X)2936 3801 y Fj(1)2992 3789 y Fr(+)18 b Fk(b)3111 3801 y Fj(1)3148 3789 y Fk(Y)3196 3801 y Fj(1)3233 3789 y Fr(.)681 3888 y(Under)j(our)f(assumption)h (that)g Fk(\027)28 b(<)23 b(\026)p Fr(,)f(necessarily)e Fk(a)2377 3900 y Fj(1)2437 3888 y Fr(=)j(0.)34 b(In)21 b(the)h(nondegenerate)515 3988 y(case,)27 b Fk(b)750 4000 y Fj(1)809 3988 y Fm(6)p Fr(=)c(0,)k(while)h(in)g(the)g (degenerate)e(one,)i Fk(b)2073 4000 y Fj(1)2133 3988 y Fr(=)22 b(0.)681 4088 y(In)30 b(the)g(nondegenerate)e(case)h(w)n(e)g (ha)n(v)n(e)f Fk(W)2056 4100 y Fj(1)2121 4088 y Fr(=)e Fk(b)2248 4100 y Fj(1)2285 4088 y Fk(Y)2333 4100 y Fj(1)2370 4088 y Fr(.)43 b(W)-7 b(e)30 b(notice)g(that)g([)p Fk(Y)3081 4100 y Fj(1)3118 4088 y Fk(;)14 b(X)3224 4100 y Fl(k)3265 4088 y Fr(])26 b(=)515 4187 y Fm(\000)p Fk(Y)628 4199 y Fl(k)q Fj(+1)778 4187 y Fr(and)f([)p Fk(Y)1008 4199 y Fj(1)1045 4187 y Fk(;)14 b(Y)1130 4199 y Fl(k)1171 4187 y Fr(])23 b(=)g(0;)i(therefore)g(Ker)o(\()p Fm(M)2008 4199 y Fj(1)2045 4187 y Fr(\))e(=)g Fm(Y)7 b Fr(.)36 b(On)25 b(the)h(other)e(hand,)i(Ran)o(\()p Fm(M)3309 4199 y Fj(1)3347 4187 y Fr(\))515 4287 y(also)31 b(is)h(giv)n(en)g(b)n (y)g Fm(Y)7 b Fr(,)34 b(and)e(Ker)o(\()p Fm(M)1667 4251 y Fj(+)1667 4309 y(1)1722 4287 y Fr(\))f(=)g Fm(X)12 b Fr(.)52 b(In)32 b(this)h(case)e(w)n(e)h(also)g(ha)n(v)n(e)f(to)h (consider)515 4386 y(higher)d(order)f(parts)i(of)f Fk(W)12 b Fr(;)31 b(the)g(\014rst)e(step)h(of)g(the)g(PRF)g(pro)r(cedure)f(can) g(eliminate)h(all)515 4486 y(terms)d(in)h(Ran\()p Fm(M)1124 4498 y Fj(1)1161 4486 y Fr(\))g(and)f(th)n(us)h(w)n(e)f(will)h(only)f (consider)g(terms)g(in)h(Ker)o(\()p Fm(M)2944 4450 y Fj(+)2944 4508 y(1)2999 4486 y Fr(\).)681 4586 y(Let)e Fk(\026)h Fr(b)r(e)g(as)e(ab)r(o)n(v)n(e,)h(and)g(let)g Fk(W)1731 4598 y Fl(\026)1799 4586 y Fr(=)d Fk(a)1931 4598 y Fl(\026)1975 4586 y Fk(X)2044 4598 y Fl(\026)2115 4586 y Fr(\(all)k(the)f Fk(Y)2451 4598 y Fl(k)2519 4586 y Fr(parts)f(with)i Fk(k)f Fm(\025)d Fr(2)j(can)g(b)r(e)515 4685 y(eliminated,)33 b(as)e(just)h(recalled\).)49 b(No)n(w)31 b Fm(M)1898 4697 y Fl(\026)1974 4685 y Fr(is)h(the)g(restriction)f(of)h Fm(L)2767 4697 y Fl(\026)2843 4685 y Fr(to)g(Ker)o(\()p Fm(M)3215 4697 y Fj(1)3252 4685 y Fr(\))e(=)515 4785 y(Ker)o(\()p Fm(L)738 4797 y Fj(0)776 4785 y Fr(\))d Fm(\\)h Fr(Ker)o(\()p Fm(L)1141 4797 y Fj(1)1179 4785 y Fr(\):)64 b(indeed)41 b(the)g Fm(L)1787 4797 y Fl(m)1892 4785 y Fr(with)g(1)k Fk(<)g(m)g(<)g(\026)c Fr(are)f(zero)g(and)h(put)g (no)515 4885 y(restriction.)53 b(W)-7 b(e)33 b(ha)n(v)n(e)g([)p Fk(X)1400 4897 y Fl(\026)1444 4885 y Fk(;)14 b(Y)1529 4897 y Fl(k)1570 4885 y Fr(])32 b(=)h Fk(k)s(Y)1817 4897 y Fl(k)q Fj(+)p Fl(\026)1982 4885 y Fr(and)g(th)n(us)h(Ker)o(\()p Fm(M)2604 4897 y Fl(\026)2648 4885 y Fr(\))f(=)f Fm(f)p Fr(0)p Fm(g)p Fr(:)47 b(no)33 b(further)515 4984 y(normalization)26 b(is)h(p)r(ossible)h(within)g(the)g(PRF)f(sc)n(heme.)1905 5255 y(12)p eop %%Page: 13 13 13 12 bop 681 523 a Fr(Th)n(us)27 b(the)h(PRF)g(is)f(giv)n(en)g(in)h (this)g(case)e(b)n(y)1368 747 y Fg(c)1364 768 y Fk(W)63 b Fr(=)50 b Fk(Y)1668 780 y Fj(0)1738 768 y Fr(+)32 b Fk(b)1871 780 y Fj(1)1921 768 y Fk(Y)1969 780 y Fj(1)2039 768 y Fr(+)2163 664 y Ff(1)2136 689 y Fg(X)2136 867 y Fl(k)q Fj(=2)2283 768 y Fg(b)-45 b Fk(a)2328 780 y Fl(k)2369 768 y Fk(X)2438 780 y Fl(k)2507 768 y Fk(;)701 b Fr(\(15\))515 1027 y(where)27 b(the)i(hats)f(on)g(constan)n(ts)e Fg(b)-45 b Fk(a)1608 1039 y Fl(k)1677 1027 y Fr(indicate)28 b(that)g(co)r (e\016cien)n(ts)g(are)f(in)i(general)e(not)h(the)515 1127 y(same)f(as)g(those)g(of)g(the)h(initial)g(NF)g(\(14\).)681 1226 y(W)-7 b(e)25 b(an)n(ticipate)f(that)g(the)h(LRF)g(reduction)f(sc) n(heme)g(can)g(giv)n(e)g(a)g(\014nite)h(dimensional)515 1326 y(normal)h(form)i(for)f(this)h(case,)e(see)i(b)r(elo)n(w.)681 1525 y(The)35 b(previous)f(discussion)h(can)g(easily)f(b)r(e)i (generalized)e(to)h(the)h(degenerate)e(case)515 1625 y(where)28 b(1)23 b Fk(<)h(\027)30 b(<)24 b(\026)p Fr(,)k(with)h Fk(\026)g Fr(and)f Fk(\027)34 b Fr(de\014ned)28 b(ab)r(o)n(v)n(e;)g (notice)g(that)h(at)f(least)g(one)g(of)g(these)515 1724 y(has)22 b(to)g(exist)g(and)g(b)r(e)h(\014nite,)h(or)e(the)h(system)f (w)n(ould)g(already)f(b)r(e)h(linear)g(and)g(th)n(us)h(trivial.)515 1824 y(Here)k(the)h(NF)g(is)1114 2069 y Fk(W)63 b Fr(=)50 b Fk(Y)1418 2081 y Fj(0)1502 2069 y Fr(+)1613 1961 y Fl(\026)p Ff(\000)p Fj(1)1615 1990 y Fg(X)1613 2169 y Fl(k)q Fj(=)p Fl(\027)1752 2069 y Fk(b)1788 2081 y Fl(k)1828 2069 y Fk(Y)1876 2081 y Fl(\026)1967 2069 y Fr(+)2107 1965 y Ff(1)2080 1990 y Fg(X)2078 2169 y Fl(k)q Fj(=)p Fl(\027)2203 2069 y Fr(\()p Fk(a)2279 2081 y Fl(k)2320 2069 y Fk(X)2389 2081 y Fl(k)2448 2069 y Fr(+)18 b Fk(b)2567 2081 y Fl(k)2607 2069 y Fk(Y)2655 2081 y Fl(k)2696 2069 y Fr(\))28 b Fk(:)452 b Fr(\(16\))515 2331 y(No)n(w)41 b Fm(L)775 2343 y Fl(\027)816 2331 y Fr(\()p Fk(H)917 2343 y Fl(k)958 2331 y Fr(\))47 b(=)e Fk(b)1183 2343 y Fl(\027)1224 2331 y Fr([)p Fk(Y)1295 2343 y Fl(\027)1337 2331 y Fk(;)14 b(\013)1427 2343 y Fl(k)1468 2331 y Fk(X)1537 2343 y Fl(k)1605 2331 y Fr(+)27 b Fk(\014)1744 2343 y Fl(k)1785 2331 y Fk(Y)1833 2343 y Fl(k)1874 2331 y Fr(])46 b(=)g Fm(\000)p Fk(\027)5 b(b)2201 2343 y Fl(\027)2242 2331 y Fk(\013)2295 2343 y Fl(k)2336 2331 y Fk(Y)2384 2343 y Fl(\027)t Fj(+)p Fl(k)2513 2331 y Fr(,)45 b(and)c(therefore)f(w) n(e)h(can)515 2430 y(eliminate)c(all)h(the)g Fk(Y)1211 2442 y Fl(\027)t Fj(+)p Fl(k)1378 2430 y Fr(terms)f(simply)h(b)n(y)f(c) n(ho)r(osing,)i(with)f(the)g(same)f(notation)g(as)515 2540 y(b)r(efore,)25 b Fk(\013)838 2552 y Fl(k)902 2540 y Fr(=)e Fm(\000)1050 2518 y Fg(e)1055 2540 y Fk(b)1091 2552 y Fl(\027)t Fj(+)p Fl(k)1219 2540 y Fk(=)p Fr(\()p Fk(\027)5 b(b)1375 2552 y Fl(\027)1416 2540 y Fr(\);)27 b(w)n(e)e(cannot)g(eliminate)h(an)n(y)f(of)g(the)h Fk(X)2703 2552 y Fl(k)2769 2540 y Fr(terms.)36 b(Th)n(us,)26 b(the)515 2639 y(PRF)h(in)h(the)g(degenerate)e(case)h(is)1346 2863 y Fg(c)1343 2884 y Fk(W)62 b Fr(=)51 b Fk(Y)1647 2896 y Fj(0)1730 2884 y Fr(+)46 b Fk(b)1877 2896 y Fl(\027)1918 2884 y Fk(Y)1966 2896 y Fl(\027)2053 2884 y Fr(+)2195 2780 y Ff(1)2168 2805 y Fg(X)2164 2984 y Fl(k)q Fj(=)p Fl(\026)2305 2884 y Fg(e)-45 b Fk(a)2350 2896 y Fl(k)2391 2884 y Fk(X)2460 2896 y Fl(k)2528 2884 y Fk(:)680 b Fr(\(17\))515 3152 y(Similarly)23 b(to)h(what)g(happ)r(ens)f(for)h(the)g (nondegenerate)e(case,)i(the)g(LRF)g(pro)r(cedure)f(giv)n(es)515 3252 y(b)r(etter)28 b(results)f(in)h(this)g(case.)515 3484 y Fi(5.3)112 b(The)38 b(LRF)f(reduction)g(sc)m(heme)515 3638 y Fr(In)f(the)h(previous)f(computations,)i(w)n(e)e(ha)n(v)n(e)g (follo)n(w)n(ed)f(the)i(general)e(PRF)i(sc)n(heme)f(for)515 3737 y(further)23 b(normalizing)f(the)i(standard)e(NF)i(\(14\);)h(this) f(ga)n(v)n(e)d(an)j(in\014nite)g(PRF)f(in)h(b)r(oth)g(the)515 3837 y(degenerate)i(and)i(nondegenerate)e(cases.)681 3937 y(Ho)n(w)n(ev)n(er)k(one)h(can)h(tak)n(e)f(adv)-5 b(an)n(tage)30 b(of)i(the)h(sp)r(eci\014c)f(Lie)g(algebraic)e (structure)h(of)515 4036 y Fm(G)d Fr(=)23 b Fm(X)h(\010)12 b(Y)7 b Fr(,)25 b(em)n(b)r(o)r(died)g(in)g(\(13\),)g(to)f(obtain)g(a)g (more)g(drastical)f(reduction:)35 b(indeed,)25 b(one)515 4136 y(can)j(obtain)h(a)g(reduction)g(to)g(a)f(\014nite)i(normal)e (form)h(\(the)h(Lie)f(renormalized)e(form\),)j(as)515 4235 y(w)n(e)d(no)n(w)f(discuss.)36 b(W)-7 b(e)28 b(use)f(the)h(same)e (notation)h(as)g(in)g(discussing)f(the)i(degenerate)e(case)515 4335 y(ab)r(o)n(v)n(e.)681 4435 y(W)-7 b(e)22 b(\014rst)g(op)r(erate)g (a)f(sequence)h(of)g(normalizations)e(with)j(generators)d Fk(h)2950 4392 y Fj(\()p Fl(a)p Fj(\))2950 4460 y Fl(k)3065 4435 y Fr(=)i Fk(\013)3205 4447 y Fl(k)3246 4435 y Fk(X)3315 4447 y Fl(k)3356 4435 y Fr(,)515 4534 y(whic)n(h)32 b(w)n(e)h(c)n(ho)r (ose)e(so)h(as)g(to)h(eliminate)f(higher)g(order)g Fk(X)2389 4546 y Fl(k)2462 4534 y Fr(terms,)i(i.e.)52 b Fk(X)2950 4546 y Fl(k)3023 4534 y Fr(for)33 b Fk(k)h(>)d(\026)515 4634 y Fr(\(as)e(w)n(e)g(kno)n(w,)h(this)f(is)h(not)g(p)r(ossible)f (for)g Fk(k)g Fr(=)d(2)p Fk(\026)p Fr(\).)43 b(Notice)30 b(this)g(will)f(c)n(hange)g(not)g(only)515 4734 y(the)d(\(co)r (e\016cien)n(ts)g(of)g(the\))g Fk(X)1442 4746 y Fl(k)1509 4734 y Fr(terms,)g(but)g(the)g(\(co)r(e\016cien)n(ts)g(of)g(the\))g Fk(Y)2818 4746 y Fl(k)2885 4734 y Fr(terms)g(as)f(w)n(ell;)515 4833 y(ho)n(w)n(ev)n(er,)g(no)j(terms)f(of)g(degree)g Fk(k)f(<)d(\027)32 b Fr(will)c(b)r(e)g(pro)r(duced.)1905 5255 y(13)p eop %%Page: 14 14 14 13 bop 681 523 a Fr(In)29 b(this)g(w)n(a)n(y)-7 b(,)28 b(w)n(e)g(arriv)n(e)f(at)i(a)f(partially)g(reduced)h(form)f(\(the)i (tilde)f(indicates)g(that)515 623 y(the)f(co)r(e\016cien)n(ts)f(are)g (not)g(the)h(same)f(as)g(the)h(initial)g(ones,)f(but)h(not)g(y)n(et)f (\014nal\))1160 837 y Fg(f)1156 858 y Fk(W)63 b Fr(=)50 b Fk(Y)1460 870 y Fj(0)1544 858 y Fr(+)c Fk(a)1699 870 y Fl(\026)1743 858 y Fk(X)1812 870 y Fl(\026)1875 858 y Fr(+)17 b Fg(e)-45 b Fk(a)2002 870 y Fj(2)p Fl(\026)2079 858 y Fk(X)2148 870 y Fj(2)p Fl(\026)2272 858 y Fr(+)2412 754 y Ff(1)2385 779 y Fg(X)2382 958 y Fl(k)q Fj(=)p Fl(\027)2516 836 y Fg(e)2521 858 y Fk(b)2557 870 y Fl(k)2598 858 y Fk(Y)2646 870 y Fl(k)2714 858 y Fk(:)681 1101 y Fr(Once)25 b(this)h(has)f(b)r(een)h(done,)f(w)n(e)g(pass)g(to)h(consider)e(a)h (second)g(sequence)g(of)g(normal-)515 1212 y(izations)k(with)i (generators)d Fk(h)1467 1169 y Fj(\()p Fl(b)p Fj(\))1467 1237 y Fl(k)1580 1212 y Fr(=)f Fk(\014)1719 1224 y Fl(k)1760 1212 y Fk(Y)1808 1224 y Fl(k)1849 1212 y Fr(.)45 b(As)30 b Fm(Y)38 b Fr(is)30 b(an)g(ideal)g(in)h Fm(G)5 b Fr(,)31 b(the)g Fk(X)2963 1224 y Fl(k)3034 1212 y Fr(terms)f(are)515 1312 y(una\013ected.)51 b(On)33 b(the)f(other)g(side,)i Fm(Y)40 b Fr(is)32 b(ab)r(elian,)h(and)g(so)e(only)h(the)h Fk(X)2846 1324 y Fl(\026)2923 1312 y Fr(and)f Fk(X)3158 1324 y Fj(2)p Fl(\026)3268 1312 y Fr(are)515 1411 y(actually)g(activ)n (e)g(in)g(these)h(transformations:)45 b(that)33 b(is,)h(w)n(e)e(can)g (only)h(eliminate)f(terms)515 1511 y Fk(Y)563 1523 y Fl(\026)p Fj(+1)716 1511 y Fr(and)25 b(higher)f(\(it)i(is)e(clear)g(b)n (y)h(the)g(comm)n(utation)f(relations)g(that)h(these)g(can)f(alw)n(a)n (ys)515 1611 y(b)r(e)k(eliminated\).)681 1710 y(In)j(this)h(w)n(a)n(y)f (w)n(e)g(arriv)n(e)e(at)j(the)g(LRF:)g(this)f(is)h(a)f(NF)h(dep)r (ending)g(on)f(\()p Fk(\026)22 b Fm(\000)e Fk(\027)27 b Fr(+)20 b(3\))515 1810 y(constan)n(ts)26 b(\(recall)h(w)n(e)g (assumed)g Fk(\026)d(>)e(\027)5 b Fr(\),)29 b(of)e(the)h(form)1160 2028 y Fg(c)1156 2049 y Fk(W)63 b Fr(=)50 b Fk(Y)1460 2061 y Fj(0)1544 2049 y Fr(+)c Fk(a)1699 2061 y Fl(\026)1743 2049 y Fk(X)1812 2061 y Fl(\026)1875 2049 y Fr(+)17 b Fg(b)-45 b Fk(a)2002 2061 y Fj(2)p Fl(\026)2079 2049 y Fk(X)2148 2061 y Fj(2)p Fl(\026)2272 2049 y Fr(+)2425 1941 y Fl(\026)2385 1970 y Fg(X)2382 2149 y Fl(k)q Fj(=)p Fl(\027)2516 2027 y Fg(b)2521 2049 y Fk(b)2557 2061 y Fl(k)2598 2049 y Fk(Y)2646 2061 y Fl(k)2714 2049 y Fk(:)494 b Fr(\(18\))515 2316 y(It)32 b(is)f(also)g(clear)f(b)n(y)h(this)h (discussion)f(that)h(actually)2246 2295 y Fg(b)2251 2316 y Fk(b)2287 2328 y Fl(k)2357 2316 y Fr(=)2446 2295 y Fg(e)2451 2316 y Fk(b)2487 2328 y Fl(k)2528 2316 y Fr(,)f Fg(b)-45 b Fk(a)2627 2328 y Fj(2)p Fl(\026)2735 2316 y Fr(=)28 b Fg(e)-45 b Fk(a)2873 2328 y Fj(2)p Fl(\026)2950 2316 y Fr(.)49 b(Note)32 b(that)515 2416 y(the)c(n)n(um)n(b)r(er)f(of)h (constan)n(ts)e(\()p Fk(\026)19 b Fm(\000)f Fk(\027)24 b Fr(+)18 b(3\))27 b(agrees)f(with)i(that)g(computed)g(b)n(y)f(Bruno)g ([11)o(].)681 2516 y(It)33 b(should)f(b)r(e)g(stressed)g(that)h(this)f (LRF)h(is)f Fh(not)66 b Fr(a)32 b(PRF,)h(as)e(can)h(b)r(e)h(c)n(hec)n (k)n(ed)e(b)n(y)515 2615 y(comparing)26 b(this)i(with)g(\(15\))f(and)h (\(17\))f(ab)r(o)n(v)n(e,)f(or)h(comparing)f(the)i(de\014nition)g(of)f (PRF.)681 2731 y(Indeed,)22 b(with)f(the)g(notation)f(emplo)n(y)n(ed)f (in)i([20)o(,)f(21)o(],)i(the)f(spaces)f Fk(F)2809 2688 y Fj(\()p Fl(k)q Fj(\))2797 2756 y Fl(k)2924 2731 y Fr(:=)j Fk(F)3100 2701 y Fj(\()p Fl(k)q Fj(\))3197 2731 y Fm(\\)t(W)3338 2743 y Fl(k)515 2851 y Fr(with)30 b Fk(\027)j(<)27 b(k)j Fm(\024)d Fk(\026)k Fr(reduce)f(to)g(m)n(ultiples)g(of)g Fk(X)2014 2863 y Fl(k)2055 2851 y Fr(.)45 b(Here)30 b(w)n(e)g(ha)n(v)n (e)f(therefore)g Fk(W)3070 2863 y Fl(k)3139 2851 y Fm(62)e Fk(F)3286 2808 y Fj(\()p Fl(k)q Fj(\))3274 2876 y Fl(k)515 2951 y Fr(for)g Fk(\027)h(<)23 b(k)j Fm(\024)c Fk(\026)p Fr(,)28 b(and)g(th)n(us)f(the)h(LRF)g(cannot)f(b)r(e)h(a)f(PRF.)515 3181 y Fi(5.4)112 b(Explicit)34 b(reduction)j(\(nondegenerate)h(case\)) 515 3334 y Fr(The)28 b(reductions)g(describ)r(ed)g(in)h(previous)e (subsections)h(can)g(b)r(e)h(explicitely)f(p)r(erformed;)515 3434 y(detailed)h(computations)f(are)g(rep)r(orted)g(in)h([22)o(],)g (while)g(here)g(w)n(e)f(just)i(giv)n(e)e(results.)40 b(W)-7 b(e)515 3533 y(write)30 b(the)i(normal)e(form)g Fk(W)43 b Fr(in)31 b(the)g(form)g(\(14\))f(and)h(consider)f(further)h (normalization)515 3633 y(up)d(to)f(order)f(six.)681 3733 y(Let)38 b(us)g(\014rst)g(consider)f(the)i(PRF)f(reduction.)68 b(In)39 b(the)f(nondegenerate)f(case)g(w)n(e)515 3832 y(will)h(tak)n(e,)h(for)e(the)h(sak)n(e)f(of)g(simplicit)n(y)-7 b(,)41 b Fk(\014)1937 3844 y Fl(k)2017 3832 y Fr(=)f(0;)i(w)n(e)37 b(c)n(ho)r(ose)g Fk(\013)2688 3844 y Fj(1)2765 3832 y Fr(=)i Fm(\000)p Fk(b)2970 3844 y Fj(2)3007 3832 y Fk(=b)3085 3844 y Fj(1)3121 3832 y Fr(,)h Fk(\013)3237 3844 y Fj(2)3314 3832 y Fr(=)515 3932 y(\()p Fk(b)583 3902 y Fj(2)583 3952 y(2)636 3932 y Fm(\000)16 b Fk(b)753 3944 y Fj(1)790 3932 y Fk(b)826 3944 y Fj(3)863 3932 y Fr(\))p Fk(=b)973 3902 y Fj(2)973 3952 y(1)1009 3932 y Fr(,)27 b Fk(\013)1112 3944 y Fj(3)1173 3932 y Fr(=)22 b Fm(\000)p Fr(\(2)p Fk(b)1435 3902 y Fj(3)1435 3952 y(2)1488 3932 y Fm(\000)15 b Fr(3)p Fk(b)1646 3944 y Fj(1)1683 3932 y Fk(b)1719 3944 y Fj(2)1756 3932 y Fk(b)1792 3944 y Fj(3)1845 3932 y Fr(+)g Fk(b)1961 3902 y Fj(2)1961 3952 y(1)1998 3932 y Fk(b)2034 3944 y Fj(4)2071 3932 y Fr(\))p Fk(=b)2181 3902 y Fj(3)2181 3952 y(1)2218 3932 y Fr(,)27 b Fk(\013)2321 3944 y Fj(4)2381 3932 y Fr(=)c(\(9)p Fk(b)2579 3902 y Fj(4)2579 3952 y(2)2632 3932 y Fm(\000)15 b Fr(18)p Fk(b)2832 3944 y Fj(1)2868 3932 y Fk(b)2904 3902 y Fj(2)2904 3952 y(2)2941 3932 y Fk(b)2977 3944 y Fj(3)3030 3932 y Fr(+)h(3)p Fk(b)3189 3902 y Fj(2)3189 3952 y(1)3225 3932 y Fk(b)3261 3902 y Fj(2)3261 3952 y(3)3314 3932 y Fr(+)515 4031 y(8)p Fk(b)593 4001 y Fj(2)593 4052 y(1)629 4031 y Fk(b)665 4043 y Fj(2)702 4031 y Fk(b)738 4043 y Fj(4)793 4031 y Fm(\000)i Fr(2)p Fk(b)954 4001 y Fj(3)954 4052 y(1)991 4031 y Fk(b)1027 4043 y Fj(5)1064 4031 y Fr(\))p Fk(=)p Fr(\(2)p Fk(b)1248 4001 y Fj(4)1248 4052 y(1)1284 4031 y Fr(\).)37 b(In)28 b(this)g(w)n(a)n(y)e(w)n(e)i(obtain)852 4192 y Fg(f)848 4213 y Fk(W)938 4183 y Fj(\(5\))1078 4213 y Fr(=)110 b Fk(Y)1301 4225 y Fj(0)1385 4213 y Fr(+)46 b Fk(b)1532 4225 y Fj(1)1582 4213 y Fk(Y)1630 4225 y Fj(1)1714 4213 y Fr(+)f Fk(a)1868 4225 y Fj(2)1919 4213 y Fk(X)1988 4225 y Fj(2)2072 4213 y Fr(+)g([)p Fk(a)2249 4225 y Fj(3)2305 4213 y Fm(\000)18 b Fk(a)2432 4225 y Fj(2)2469 4213 y Fk(b)2505 4225 y Fj(2)2542 4213 y Fk(=b)2620 4225 y Fj(1)2656 4213 y Fr(])c Fk(X)2762 4225 y Fj(3)2827 4213 y Fr(+)1226 4312 y(+)27 b([)p Fk(a)1385 4324 y Fj(4)1440 4312 y Fm(\000)18 b Fr(2)p Fk(a)1609 4324 y Fj(3)1646 4312 y Fk(b)1682 4324 y Fj(2)1719 4312 y Fk(=b)1797 4324 y Fj(1)1852 4312 y Fr(+)g Fk(a)1979 4324 y Fj(2)2016 4312 y Fk(b)2052 4282 y Fj(2)2052 4333 y(2)2089 4312 y Fk(=b)2167 4282 y Fj(2)2167 4333 y(1)2203 4312 y Fr(])c Fk(X)2309 4324 y Fj(4)2374 4312 y Fr(+)1226 4412 y(+)27 b([)p Fk(a)1385 4424 y Fj(5)1440 4412 y Fm(\000)18 b Fr(3)p Fk(a)1609 4424 y Fj(4)1646 4412 y Fk(b)1682 4424 y Fj(2)1719 4412 y Fk(=b)1797 4424 y Fj(1)1852 4412 y Fr(+)g(4)p Fk(a)2021 4424 y Fj(3)2058 4412 y Fk(b)2094 4382 y Fj(2)2094 4433 y(2)2130 4412 y Fk(=b)2208 4382 y Fj(2)2208 4433 y(1)2245 4412 y Fr(+)1336 4512 y Fm(\000)p Fk(a)1445 4524 y Fj(3)1482 4512 y Fk(b)1518 4524 y Fj(3)1555 4512 y Fk(=b)1633 4524 y Fj(1)1688 4512 y Fm(\000)g Fr(2)p Fk(a)1857 4524 y Fj(2)1893 4512 y Fk(b)1929 4524 y Fj(2)1966 4512 y Fk(b)2002 4524 y Fj(3)2039 4512 y Fk(=b)2117 4482 y Fj(2)2117 4532 y(1)2172 4512 y Fr(+)g Fk(a)2299 4524 y Fj(2)2336 4512 y Fk(b)2372 4524 y Fj(4)2409 4512 y Fk(=b)2487 4524 y Fj(1)2523 4512 y Fr(])c Fk(X)2629 4524 y Fj(5)2712 4512 y Fr(+)46 b Fk(O)r Fr(\(6\))28 b Fk(:)681 4807 y Fr(Let)d(us)g(no)n(w)f(pass)h(to)f(consider)g(the)i(LRF)f (reduction,)g(and)g(p)r(erform)g(detailed)g(com-)515 4907 y(putations)i(according)f(to)h(the)h(LRF)g(sc)n(heme)f(for)g(the)h (nondegenerate)e(case;)g(that)i(is,)f(w)n(e)515 5006 y(\014rst)g(tak)n(e)g(care)f(of)i(the)g Fk(X)1348 5018 y Fl(k)1416 5006 y Fr(terms)g(\(up)g(to)f Fk(k)f Fr(=)d(5\),)k(and)h (then)g(of)f(the)h Fk(Y)2818 5018 y Fl(k)2887 5006 y Fr(ones.)1905 5255 y(14)p eop %%Page: 15 15 15 14 bop 681 523 a Fr(With)28 b(a)f(transformation)f Fk(h)1578 535 y Fj(1)1639 523 y Fr(=)c Fk(\013)1779 535 y Fj(1)1817 523 y Fk(X)1886 535 y Fj(1)1923 523 y Fr(,)27 b(the)h Fk(W)2194 535 y Fj(3)2260 523 y Fr(term)f(reads)1039 685 y Fg(f)1036 706 y Fk(W)1114 718 y Fj(3)1202 706 y Fr(=)51 b([)p Fk(a)1385 718 y Fj(3)1440 706 y Fr(+)18 b Fk(a)1567 718 y Fj(2)1604 706 y Fk(\013)1657 718 y Fj(1)1695 706 y Fr(])c Fk(X)1801 718 y Fj(3)1884 706 y Fr(+)1995 638 y Fg(\002)2029 706 y Fk(b)2065 718 y Fj(3)2120 706 y Fr(+)k(2)p Fk(b)2281 718 y Fj(2)2318 706 y Fk(\013)2371 718 y Fj(1)2427 706 y Fr(+)g Fk(b)2546 718 y Fj(1)2583 706 y Fk(\013)2636 671 y Fj(2)2636 726 y(1)2673 638 y Fg(\003)2722 706 y Fk(Y)2770 718 y Fj(3)2835 706 y Fk(:)515 888 y Fr(W)-7 b(e)41 b(disregard)f(the)i Fk(Y)1251 900 y Fj(3)1329 888 y Fr(term)g(and)f(c)n(ho)r(ose)f Fk(\013)2047 900 y Fj(1)2126 888 y Fr(so)g(to)i(eliminate)f(the)h Fk(X)2956 900 y Fj(3)3034 888 y Fr(term,)j(i.e.)515 988 y Fk(\013)568 1000 y Fj(1)644 988 y Fr(=)38 b Fm(\000)p Fk(a)856 1000 y Fj(3)892 988 y Fk(=a)978 1000 y Fj(2)1015 988 y Fr(.)65 b(After)37 b(computing)g(the)g(e\013ect)g(of)g(this)g(on) g(higher)f(order)g(terms,)i(w)n(e)515 1088 y(could)28 b(p)r(erform)g(a)g(transformation)f(with)i(generator)e Fk(h)2298 1100 y Fj(2)2359 1088 y Fr(=)e Fk(\013)2502 1100 y Fj(2)2539 1088 y Fk(X)2608 1100 y Fj(2)2645 1088 y Fr(.)40 b(Ho)n(w)n(ev)n(er,)27 b(w)n(e)h(kno)n(w)515 1187 y(that)37 b(there)f(will)h(b)r(e)g(no)g(w)n(a)n(y)e(to)i (eliminate)g(the)g Fk(X)2216 1199 y Fj(4)2289 1187 y Fr(term,)i(so)d(w)n(e)h(set)g Fk(\013)2957 1199 y Fj(2)3032 1187 y Fr(=)h(0.)64 b(W)-7 b(e)515 1287 y(p)r(erform)30 b(a)g(transformation)f(with)i(generator)e Fk(h)2086 1299 y Fj(3)2151 1287 y Fr(=)f Fk(\013)2297 1299 y Fj(3)2334 1287 y Fk(X)2403 1299 y Fj(3)2440 1287 y Fr(.)46 b(With)32 b(this,)g(the)f Fk(W)3140 1299 y Fj(5)3208 1287 y Fr(term)515 1386 y(reads)936 1457 y Fg(f)932 1478 y Fk(W)1010 1490 y Fj(5)1099 1478 y Fr(=)110 b([2)p Fk(a)1383 1448 y Fj(3)1383 1499 y(3)1420 1478 y Fk(=a)1506 1448 y Fj(2)1506 1499 y(2)1561 1478 y Fm(\000)18 b Fr(3)p Fk(a)1730 1490 y Fj(3)1766 1478 y Fk(a)1810 1490 y Fj(4)1847 1478 y Fk(=a)1933 1490 y Fj(2)1989 1478 y Fr(+)g Fk(a)2116 1490 y Fj(5)2171 1478 y Fm(\000)g Fk(a)2298 1490 y Fj(2)2335 1478 y Fk(\013)2388 1490 y Fj(3)2426 1478 y Fr(])p Fk(X)2518 1490 y Fj(5)2582 1478 y Fr(+)1246 1578 y(+)27 b([)p Fk(a)1405 1548 y Fj(4)1405 1598 y(3)1443 1578 y Fk(b)1479 1590 y Fj(1)1515 1578 y Fk(=a)1601 1548 y Fj(4)1601 1598 y(2)1657 1578 y Fm(\000)18 b Fr(4)p Fk(a)1826 1548 y Fj(3)1826 1598 y(3)1862 1578 y Fk(b)1898 1590 y Fj(2)1935 1578 y Fk(=a)2021 1548 y Fj(3)2021 1598 y(2)2076 1578 y Fr(+)g(6)p Fk(a)2245 1548 y Fj(2)2245 1598 y(3)2282 1578 y Fk(b)2318 1590 y Fj(3)2355 1578 y Fk(=a)2441 1548 y Fj(2)2441 1598 y(2)2496 1578 y Fm(\000)g Fr(4)p Fk(a)2665 1590 y Fj(3)2701 1578 y Fk(b)2737 1590 y Fj(4)2774 1578 y Fk(=a)2860 1590 y Fj(2)2897 1578 y Fr(+)1246 1677 y(+)83 b Fk(b)1430 1689 y Fj(5)1485 1677 y Fm(\000)18 b Fr(2)p Fk(a)1654 1689 y Fj(3)1691 1677 y Fk(b)1727 1689 y Fj(1)1764 1677 y Fk(\013)1817 1689 y Fj(3)1854 1677 y Fk(=a)1940 1689 y Fj(2)1995 1677 y Fr(+)g(2)p Fk(b)2156 1689 y Fj(2)2192 1677 y Fk(\013)2245 1689 y Fj(3)2283 1677 y Fr(])p Fk(Y)2354 1689 y Fj(5)2419 1677 y Fk(:)515 1833 y Fr(Again)k(w)n(e)h(only)f(aim)h(at)g (eliminating)g(the)g Fk(X)1937 1845 y Fj(5)1997 1833 y Fr(term,)h(and)f(th)n(us)g(w)n(e)g(c)n(ho)r(ose)e Fk(\013)2979 1845 y Fj(3)3040 1833 y Fr(=)h([\(2)p Fk(a)3268 1803 y Fj(3)3268 1853 y(3)3314 1833 y Fm(\000)515 1932 y Fr(3)p Fk(a)601 1944 y Fj(2)637 1932 y Fk(a)681 1944 y Fj(3)719 1932 y Fk(a)763 1944 y Fj(4)813 1932 y Fr(+)14 b Fk(a)936 1902 y Fj(2)936 1953 y(2)998 1932 y Fk(a)1042 1944 y Fj(5)1079 1932 y Fr(\))p Fk(=)p Fr(\()p Fk(a)1229 1902 y Fj(3)1229 1953 y(2)1266 1932 y Fr(\)].)37 b(W)-7 b(e)25 b(will)h(b)r(e)f(satis\014ed)g(with)h(this)f(order)f(of)h (normalization)f(for)515 2032 y(the)k Fk(X)727 2044 y Fl(k)795 2032 y Fr(terms,)f(and)h(tak)n(e)f(no)n(w)g(care)f(of)i(the)g Fk(Y)2025 2044 y Fl(k)2093 2032 y Fr(ones.)681 2132 y(W)-7 b(e)28 b(\014rst)f(op)r(erate)g(a)g(transformation)f(with)i(generator)e Fk(h)2535 2144 y Fj(1)2595 2132 y Fr(=)c Fk(\014)2729 2144 y Fj(1)2766 2132 y Fk(Y)2814 2144 y Fj(1)2852 2132 y Fr(;)28 b(w)n(e)f(get)1117 2293 y Fg(f)1114 2314 y Fk(W)1192 2326 y Fj(3)1280 2314 y Fr(=)1396 2247 y Fg(\002)1430 2314 y Fk(a)1474 2280 y Fj(2)1474 2335 y(3)1511 2314 y Fk(b)1547 2326 y Fj(1)1584 2314 y Fk(=a)1670 2280 y Fj(2)1670 2335 y(2)1725 2314 y Fm(\000)18 b Fr(2)p Fk(a)1894 2326 y Fj(3)1931 2314 y Fk(b)1967 2326 y Fj(2)2004 2314 y Fk(=a)2090 2326 y Fj(2)2145 2314 y Fr(+)g Fk(b)2264 2326 y Fj(3)2319 2314 y Fm(\000)g Fk(a)2446 2326 y Fj(2)2483 2314 y Fk(\014)2530 2326 y Fj(1)2568 2247 y Fg(\003)2644 2314 y Fk(Y)2692 2326 y Fj(3)2757 2314 y Fk(:)515 2497 y Fr(By)33 b(c)n(ho)r(osing)g Fk(\014)1042 2509 y Fj(1)1113 2497 y Fr(=)g([\()p Fk(a)1310 2467 y Fj(2)1310 2518 y(3)1347 2497 y Fk(b)1383 2509 y Fj(1)1443 2497 y Fm(\000)22 b Fr(2)p Fk(a)1616 2509 y Fj(2)1652 2497 y Fk(a)1696 2509 y Fj(3)1734 2497 y Fk(b)1770 2509 y Fj(2)1829 2497 y Fr(+)g Fk(a)1960 2467 y Fj(2)1960 2518 y(2)1997 2497 y Fk(b)2033 2509 y Fj(3)2070 2497 y Fr(\))p Fk(=)p Fr(\()p Fk(a)2220 2467 y Fj(3)2220 2518 y(2)2258 2497 y Fr(\)])34 b(w)n(e)f(eliminate)h(this.)56 b(W)-7 b(e)35 b(com-)515 2597 y(pute)c(the)f(e\013ect)h(on)f(higher)g(order)f(term,)i(and)g (then)f(consider)g(a)g(transformation)f(with)515 2696 y(generator)c Fk(h)932 2708 y Fj(2)992 2696 y Fr(=)e Fk(\014)1127 2708 y Fj(2)1164 2696 y Fk(Y)1212 2708 y Fj(2)1250 2696 y Fr(;)k(with)h(these,)g(w)n(e)f(ha)n(v)n(e)906 2866 y Fg(f)902 2887 y Fk(W)980 2899 y Fj(4)1069 2887 y Fr(=)1244 2820 y Fg(\002)1278 2887 y Fk(a)1322 2899 y Fj(4)1378 2887 y Fm(\000)18 b Fk(a)1505 2857 y Fj(2)1505 2908 y(3)1542 2887 y Fk(=a)1628 2899 y Fj(2)1665 2820 y Fg(\003)1741 2887 y Fk(X)1810 2899 y Fj(4)1875 2887 y Fr(+)1244 2987 y(+)27 b(\(1)p Fk(=a)1496 2957 y Fj(3)1496 3007 y(2)1533 2987 y Fr(\))g([)p Fk(a)1659 2957 y Fj(3)1659 3007 y(3)1697 2987 y Fk(b)1733 2999 y Fj(1)1788 2987 y Fr(+)18 b(3)p Fk(a)1957 2999 y Fj(2)1994 2987 y Fk(a)2038 2957 y Fj(2)2038 3007 y(3)2075 2987 y Fk(b)2111 2999 y Fj(2)2166 2987 y Fm(\000)g Fr(3)p Fk(a)2335 2999 y Fj(2)2372 2987 y Fk(a)2416 2999 y Fj(3)2453 2987 y Fr(\()p Fk(a)2529 2999 y Fj(4)2566 2987 y Fk(b)2602 2999 y Fj(1)2658 2987 y Fr(+)g Fk(a)2785 2999 y Fj(2)2822 2987 y Fk(b)2858 2999 y Fj(3)2895 2987 y Fr(\)+)1272 3086 y(+)p Fk(a)1381 3056 y Fj(2)1381 3107 y(2)1417 3086 y Fr(\()p Fk(a)1493 3098 y Fj(5)1531 3086 y Fk(b)1567 3098 y Fj(1)1622 3086 y Fr(+)g Fk(a)1749 3098 y Fj(2)1786 3086 y Fr(\()p Fk(b)1854 3098 y Fj(4)1910 3086 y Fm(\000)g Fr(2)p Fk(a)2079 3098 y Fj(2)2115 3086 y Fk(\014)2162 3098 y Fj(2)2200 3086 y Fr(\)\))c(])28 b Fk(Y)2377 3098 y Fj(4)2442 3086 y Fk(:)515 3275 y Fr(W)-7 b(e)34 b(w)n(an)n(t)g(to)g(eliminate)g(the)h Fk(Y)1546 3287 y Fj(4)1618 3275 y Fr(term,)h(and)e(th)n(us)g(w)n(e)g(c) n(ho)r(ose)f Fk(\014)2651 3287 y Fj(2)2722 3275 y Fr(=)h(\(1)p Fk(=)p Fr(2)p Fk(a)3023 3245 y Fj(4)3023 3296 y(2)3059 3275 y Fr(\))14 b(\()p Fk(a)3181 3245 y Fj(3)3181 3296 y(3)3219 3275 y Fk(b)3255 3287 y Fj(1)3314 3275 y Fm(\000)515 3375 y Fr(3)p Fk(a)601 3387 y Fj(2)637 3375 y Fk(a)681 3387 y Fj(3)719 3375 y Fk(a)763 3387 y Fj(4)800 3375 y Fk(b)836 3387 y Fj(1)885 3375 y Fr(+)e Fk(a)1006 3345 y Fj(2)1006 3395 y(2)1042 3375 y Fk(a)1086 3387 y Fj(5)1123 3375 y Fk(b)1159 3387 y Fj(1)1208 3375 y Fr(+)g(3)p Fk(a)1371 3387 y Fj(2)1407 3375 y Fk(a)1451 3345 y Fj(2)1451 3395 y(3)1489 3375 y Fk(b)1525 3387 y Fj(2)1573 3375 y Fm(\000)g Fr(3)p Fk(a)1736 3345 y Fj(2)1736 3395 y(2)1773 3375 y Fk(a)1817 3387 y Fj(3)1854 3375 y Fk(b)1890 3387 y Fj(3)1939 3375 y Fr(+)g Fk(a)2060 3345 y Fj(3)2060 3395 y(2)2096 3375 y Fk(b)2132 3387 y Fj(4)2169 3375 y Fr(\).)36 b(Again)24 b(w)n(e)g(tak)n(e)g(in)n(to)g(accoun)n(t)g(the)515 3474 y(e\013ect)j(of)f(this)h(on)f(higher)f(order)g(terms,)i(and)f (pass)f(to)i(consider)e(a)h(transformation)f(with)515 3574 y(generator)g Fk(h)932 3586 y Fj(4)992 3574 y Fr(=)e Fk(\014)1127 3586 y Fj(4)1164 3574 y Fk(Y)1212 3586 y Fj(4)1250 3574 y Fr(;)k(w)n(e)h(get)583 3744 y Fg(f)580 3765 y Fk(W)658 3777 y Fj(5)746 3765 y Fr(=)111 b([)p Fm(\000)p Fr(2)p Fk(a)1096 3735 y Fj(4)1096 3785 y(3)1132 3765 y Fk(b)1168 3777 y Fj(1)1205 3765 y Fk(=a)1291 3735 y Fj(4)1291 3785 y(2)1346 3765 y Fr(+)18 b(5)p Fk(a)1515 3735 y Fj(2)1515 3785 y(3)1551 3765 y Fk(a)1595 3777 y Fj(4)1633 3765 y Fk(b)1669 3777 y Fj(1)1705 3765 y Fk(=a)1791 3735 y Fj(3)1791 3785 y(2)1847 3765 y Fm(\000)g Fr(2)p Fk(a)2016 3777 y Fj(3)2052 3765 y Fk(a)2096 3777 y Fj(5)2133 3765 y Fk(b)2169 3777 y Fj(1)2206 3765 y Fk(=a)2292 3735 y Fj(2)2292 3785 y(2)2347 3765 y Fm(\000)g Fr(2)p Fk(a)2516 3735 y Fj(3)2516 3785 y(3)2553 3765 y Fk(b)2589 3777 y Fj(2)2626 3765 y Fk(=a)2712 3735 y Fj(3)2712 3785 y(2)2767 3765 y Fm(\000)g Fr(4)p Fk(a)2936 3777 y Fj(3)2973 3765 y Fk(a)3017 3777 y Fj(4)3054 3765 y Fk(b)3090 3777 y Fj(2)3127 3765 y Fk(=a)3213 3735 y Fj(2)3213 3785 y(2)3249 3765 y Fr(+)922 3864 y(+2)p Fk(a)1073 3876 y Fj(5)1109 3864 y Fk(b)1145 3876 y Fj(2)1182 3864 y Fk(=a)1268 3876 y Fj(2)1323 3864 y Fr(+)g(7)p Fk(a)1492 3834 y Fj(2)1492 3885 y(3)1528 3864 y Fk(b)1564 3876 y Fj(3)1601 3864 y Fk(=a)1687 3834 y Fj(2)1687 3885 y(2)1742 3864 y Fm(\000)g Fk(a)1869 3876 y Fj(4)1907 3864 y Fk(b)1943 3876 y Fj(3)1979 3864 y Fk(=a)2065 3876 y Fj(2)2120 3864 y Fm(\000)h Fr(4)p Fk(a)2290 3876 y Fj(3)2326 3864 y Fk(b)2362 3876 y Fj(4)2399 3864 y Fk(=a)2485 3876 y Fj(2)2540 3864 y Fr(+)f Fk(b)2659 3876 y Fj(5)2714 3864 y Fm(\000)g Fr(3)p Fk(a)2883 3876 y Fj(2)2920 3864 y Fk(\014)2967 3876 y Fj(3)3004 3864 y Fr(])28 b Fk(Y)3103 3876 y Fj(5)515 4053 y Fr(whic)n(h)38 b(can)h(b)r(e)g(eliminated)f(b)n(y)h(c)n(ho)r (osing)e Fk(\014)1989 4065 y Fj(3)2068 4053 y Fr(=)k Fm(\000)p Fr(\(1)p Fk(=)p Fr(\(3)p Fk(a)2473 4023 y Fj(5)2473 4074 y(2)2508 4053 y Fr(\)\))14 b(\(2)p Fk(a)2704 4023 y Fj(4)2704 4074 y(3)2742 4053 y Fk(b)2778 4065 y Fj(1)2840 4053 y Fm(\000)26 b Fr(5)p Fk(a)3017 4065 y Fj(2)3053 4053 y Fk(a)3097 4023 y Fj(2)3097 4074 y(3)3135 4053 y Fk(a)3179 4065 y Fj(4)3216 4053 y Fk(b)3252 4065 y Fj(1)3314 4053 y Fr(+)515 4153 y(2)p Fk(a)601 4123 y Fj(2)601 4173 y(2)637 4153 y Fk(a)681 4165 y Fj(3)719 4153 y Fk(a)763 4165 y Fj(5)800 4153 y Fk(b)836 4165 y Fj(1)883 4153 y Fr(+)10 b(2)p Fk(a)1044 4165 y Fj(2)1081 4153 y Fk(a)1125 4123 y Fj(3)1125 4173 y(3)1162 4153 y Fk(b)1198 4165 y Fj(2)1245 4153 y Fr(+)g(4)p Fk(a)1406 4123 y Fj(2)1406 4173 y(2)1443 4153 y Fk(a)1487 4165 y Fj(3)1524 4153 y Fk(a)1568 4165 y Fj(4)1605 4153 y Fk(b)1641 4165 y Fj(2)1688 4153 y Fm(\000)g Fr(2)p Fk(a)1849 4123 y Fj(3)1849 4173 y(2)1886 4153 y Fk(a)1930 4165 y Fj(5)1967 4153 y Fk(b)2003 4165 y Fj(2)2050 4153 y Fm(\000)g Fr(7)p Fk(a)2211 4123 y Fj(2)2211 4173 y(2)2248 4153 y Fk(a)2292 4123 y Fj(2)2292 4173 y(3)2329 4153 y Fk(b)2365 4165 y Fj(3)2412 4153 y Fr(+)g Fk(a)2531 4123 y Fj(3)2531 4173 y(2)2568 4153 y Fk(a)2612 4165 y Fj(4)2649 4153 y Fk(b)2685 4165 y Fj(3)2733 4153 y Fr(+)g(4)p Fk(a)2894 4123 y Fj(3)2894 4173 y(2)2930 4153 y Fk(a)2974 4165 y Fj(3)3011 4153 y Fk(b)3047 4165 y Fj(4)3095 4153 y Fm(\000)g Fk(a)3214 4123 y Fj(4)3214 4173 y(2)3251 4153 y Fk(b)3287 4165 y Fj(5)3324 4153 y Fr(\).)681 4252 y(Summarizing,)25 b(and)h(ha)n(ving)e(tak)n(en)h(in)n (to)g(accoun)n(t)g(all)g(higher)g(order)f(e\013ects)i(\(up)g(to)515 4352 y(order)g(six\),)i(w)n(e)f(ha)n(v)n(e)f(reac)n(hed)h(the)h(LRF)628 4526 y Fg(c)625 4547 y Fk(W)63 b Fr(=)50 b Fk(Y)929 4559 y Fj(0)985 4547 y Fr(+)18 b Fk(b)1104 4559 y Fj(1)1141 4547 y Fk(Y)1189 4559 y Fj(1)1245 4547 y Fr(+)g Fk(a)1372 4559 y Fj(2)1409 4547 y Fk(X)1478 4559 y Fj(2)1533 4547 y Fr(+)g([)p Fk(b)1675 4559 y Fj(2)1731 4547 y Fm(\000)g Fr(\()p Fk(a)1890 4559 y Fj(3)1927 4547 y Fk(b)1963 4559 y Fj(1)2000 4547 y Fk(=a)2086 4559 y Fj(2)2123 4547 y Fr(\)])p Fk(Y)2226 4559 y Fj(2)2282 4547 y Fr(+)g([)p Fk(a)2432 4559 y Fj(4)2487 4547 y Fm(\000)h Fr(\()p Fk(a)2647 4513 y Fj(2)2647 4568 y(3)2684 4547 y Fk(=a)2770 4559 y Fj(2)2807 4547 y Fr(\)])p Fk(X)2931 4559 y Fj(4)2986 4547 y Fr(+)46 b Fk(O)r Fr(\(6\))1905 5255 y(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fq(App)t(endix)43 b(A.)515 672 y(Bruno)h (alternativ)l(e)j(de\014nition)f(and)e(example.)515 854 y Fr(In)38 b(his)g(recen)n(t)g(b)r(o)r(ok)g([13)o(])h(\(and)f(b)r (efore)g(this)g(is)h([12)o(]\),)i(A.D.)e(Bruno)f(has)f(dealt)i(with)515 954 y(PRFs.)44 b(Unfortunately)30 b(he)h(rep)r(orts)e(a)h(de\014nition) g(for)g(PRFs)g(whic)n(h)g(is)g(not)g(equiv)-5 b(alen)n(t)515 1054 y(to)27 b(the)h(one)e(con)n(tained)h(in)h([20)o(,)f(21)o(])h (\(and)f(rep)r(orted)g(ab)r(o)n(v)n(e\),)f(so)h(that)g(his)g(commen)n (ts)g(on)515 1153 y(PRFs)34 b(can)h(impro)n(v)n(e)e(confusion)h(rather) g(than)h(clarifying)f(the)h(issue,)h(esp)r(ecially)e(since)515 1253 y(the)28 b(di\013erence)f(in)h(de\014nitions)g(is)f(not)h(made)f (clear)g(\(nor)g(men)n(tioned\).)681 1352 y(The)i(de\014nition)h(of)f (PRFs)g(giv)n(en)g(in)g(Bruno's)f(w)n(orks)g(\(whic)n(h)i(I)f(tak)n(e)g (from)f(section)515 1452 y(V.22)j(of)g([13)o(]\))h(is)f(as)g(follo)n (ws,)g(once)g(translated)g(to)g(the)h(notation)f(used)g(in)h(the)g (presen)n(t)515 1552 y(pap)r(er)f(\(the)g(de\014nition)h(giv)n(en)f(in) g(the)h(\014rst)f(review)f([12)o(])i(is)f(sligh)n(tly)f(di\013eren)n (t,)j(but)f(the)515 1651 y(di\013erence)27 b(is)h(unessen)n(tial)f (here\).)681 1751 y(W)-7 b(e)20 b(consider)f(the)h(v)n(ector)f(p)r(o)n (w)n(er)g(series)f Fk(f)1999 1763 y Fl(k)2040 1751 y Fr(;)k(let)f Fk(F)2251 1763 y Fl(m)2337 1751 y Fr(=)2425 1689 y Fg(P)2513 1709 y Fl(m)2513 1776 y(k)q Fj(=0)2651 1751 y Fk(f)2692 1763 y Fl(k)2733 1751 y Fr(;)h(the)f(op)r(erators)d (\011)3339 1763 y Fl(k)515 1851 y Fr(are)28 b(then)i(de\014ned)f(as)g (\011)1302 1863 y Fl(m)1365 1851 y Fr(\()p Fk(H)7 b Fr(\))26 b(=)f(\()p Fk(H)1729 1820 y Fl(j)1764 1851 y Fk(@)1808 1863 y Fl(j)1843 1851 y Fr(\))p Fk(F)1928 1863 y Fl(m)2012 1851 y Fm(\000)19 b Fr(\()p Fk(F)2193 1820 y Fl(j)2181 1871 y(m)2245 1851 y Fk(@)2289 1863 y Fl(j)2324 1851 y Fr(\))p Fk(H)7 b Fr(;)30 b(in)f(the)h(presen)n(t)f(notations,)515 1950 y(w)n(e)e(ha)n(v)n(e)1605 2096 y(\011)1670 2108 y Fl(m)1784 2096 y Fr(=)50 b Fm(\000)2022 1992 y Fl(m)1992 2017 y Fg(X)1991 2195 y Fl(k)q Fj(=0)2140 2096 y Fm(L)2197 2108 y Fl(k)2265 2096 y Fk(:)900 b Fr(\()p Fk(A:)p Fr(1\))515 2320 y(Bruno)33 b(considers)f(then)i(the)g(adjoin)n(t)g(\011)1833 2289 y Ff(\003)1833 2340 y Fl(m)1929 2320 y Fr(of)g(the)g(op)r(erators) e(\011)2618 2332 y Fl(m)2680 2320 y Fr(,)k(and)d(declares)g(that)515 2419 y(PRFs)27 b(ha)n(v)n(e)f(the)i(prop)r(ert)n(y)f(that)1643 2602 y(\011)1708 2568 y Ff(\003)1708 2622 y Fl(k)q Ff(\000)p Fj(1)1834 2602 y Fr(\()p Fk(F)1919 2614 y Fl(k)1960 2602 y Fr(\))51 b(=)f(0)28 b(;)937 b(\()p Fk(A:)p Fr(2\))515 2784 y(this)30 b(is)g(not)f(equiv)-5 b(alen)n(t)30 b(to)g(m)n(y)f (original)g(de\014nition.)44 b(In)30 b(facts,)g(this)g(condition)g (implies)515 2884 y(that)d(eac)n(h)e(term)i Fk(G)1142 2896 y Fl(k)1209 2884 y Fr(is)g(in)g(Ker)o(\()p Fm(L)1611 2854 y Fj(+)1611 2905 y Fl(p)1666 2884 y Fr(\))g(for)f(all)g(the)h Fk(p)c(<)g(k)s Fr(;)j(in)h(m)n(y)f(de\014nition)h([20)o(,)g(21)o(])g (this)515 2984 y(applies)e(to)h Fm(M)990 2954 y Fj(+)990 3004 y Fl(p)1071 2984 y Fr(rather)f(than)h Fm(L)1568 2954 y Fj(+)1568 3004 y Fl(p)1624 2984 y Fr(,)g(whic)n(h)g(ob)n (viously)f(mak)n(es)g(a)h(substan)n(tial)f(di\013erence.)681 3083 y(Bruno)i(considers)f(then)i(an)f(example,)g(i.e.)37 b(the)28 b(system)1366 3266 y(_)-38 b Fk(x)52 b Fr(=)e Fk(x)1612 3232 y Fj(3)1705 3266 y Fk(;)84 b Fr(_)-38 b Fk(y)54 b Fr(=)c Fk(y)s Fr(\(1)18 b(+)g Fk(x)h Fr(+)f Fk(x)2422 3232 y Fj(2)2459 3266 y Fr(\))28 b(;)646 b(\()p Fk(A:)p Fr(3\))515 3449 y(according)26 b(to)h([13)o(])h(the)g(PRF)f (for)g(this)h(is)g(giv)n(en)f(b)n(y)g(\()p Fk(a)2270 3461 y Fj(2)2330 3449 y Fm(6)p Fr(=)c(0\))1298 3631 y(_)-37 b Fk(x)51 b Fr(=)f Fk(a)1541 3643 y Fj(2)1578 3631 y Fk(x)1625 3597 y Fj(3)1682 3631 y Fr(+)18 b Fk(\013x)1865 3597 y Fj(5)1958 3631 y Fk(;)84 b Fr(_)-38 b Fk(y)54 b Fr(=)c Fk(y)s Fr(\(1)18 b(+)g Fk(\014)t(x)p Fr(\))579 b(\()p Fk(A:)p Fr(4\))515 3814 y(with)28 b(no)f(higher)g(order)f (terms.)37 b(In)28 b([12)o(])f(the)h(PRF)g(is)f(instead)h(claimed)f(to) h(b)r(e)1459 3997 y(_)-38 b Fk(x)52 b Fr(=)e Fk(x)1705 3962 y Fj(3)1798 3997 y Fk(;)84 b Fr(_)-38 b Fk(y)54 b Fr(=)c Fk(y)s Fr(\(1)18 b(+)g Fk(x)p Fr(\))28 b Fk(;)739 b Fr(\()p Fk(A:)p Fr(5\))515 4179 y(again)29 b(with)h(no)g(higher)f (order)g(term)h(\(the)h(di\013erence)f(b)r(et)n(w)n(een)f(these)i(t)n (w)n(o)e(expressions)515 4279 y(is)k(nev)n(er)f(men)n(tioned,)j(nor)e (explained,)i(in)e(b)r(oth)h(of)f([12)o(,)h(13)o(]\).)55 b(In)33 b(b)r(oth)h(cases,)g(Bruno)515 4378 y(do)r(es)27 b(not)h(explain)g(ho)n(w)f(these)h(expressions)e(are)h(obtained.)37 b(Ho)n(w)n(ev)n(er,)27 b(b)r(oth)h(\(A.4\))g(and)515 4478 y(\(A.5\))k(are)g(ob)n(viously)e(di\013eren)n(t)j(from)f(the)g 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b(One)21 b(can)h(consider)f(the)h(descending)g(cen)n(tral)f(series)g (\(DCS\))i(of)515 1950 y Fm(G)28 b Fr([27)o(,)23 b(31)o(].)36 b(W)-7 b(e)23 b(recall)f(this)h(is)g(de\014ned)g(b)n(y)f Fm(G)1940 1962 y Fj(0)2001 1950 y Fr(=)h Fm(G)28 b Fr(and)23 b Fm(G)2372 1962 y Fl(k)q Fj(+1)2520 1950 y Fr(=)g([)p Fm(G)5 b Fk(;)14 b Fm(G)2771 1962 y Fl(k)2812 1950 y Fr(];)25 b(as)d(w)n(ell)h(kno)n(wn)515 2050 y(the)31 b(factor)f(algebras)f(\000)1281 2062 y Fl(k)1351 2050 y Fr(=)f Fm(G)1493 2062 y Fl(k)1534 2050 y Fk(=)p Fm(G)1625 2062 y Fl(k)q Fj(+1)1781 2050 y Fr(are)i(ab)r(elian.)47 b(It)31 b(is)g(then)g(p)r(ossible)g(to)g(eliminate)515 2149 y(terms)23 b(b)n(y)h(inner)g(automorphisms)e(of)i Fm(G)30 b Fr(\(that)24 b(is,)h(b)n(y)e(acting)h(on)f Fm(G)30 b Fr(with)24 b(v)n(ector)f(\014elds)h(in)515 2249 y Fm(G)5 b Fr(\))29 b(pro)r(ceeding)f(along)g Fm(G)1321 2261 y Fl(k)1362 2249 y Fr(,)h(i.e.)41 b(\014ltering)28 b(the)h(Lie)g(algebra)e Fm(G)5 b Fr(.)41 b(This)28 b(is)h(relev)-5 b(an)n(t)28 b(to)h(our)515 2349 y(problem)e(since)g(Lie-P)n(oincar)n (\023)-39 b(e)24 b(transformations)i(reduce)h(to)g(inner)h(op)r (erations)e(on)h Fm(G)5 b Fr(.)681 2448 y(This)35 b(approac)n(h)e(is,)k (of)e(course,)h(particularly)e(con)n(v)n(enien)n(t)g(when)h Fm(G)40 b Fr(is)35 b(nilp)r(oten)n(t;)515 2548 y(actually)21 b(if)h(w)n(e)g(consider)f(nonlinear)g(v)n(ector)f(\014elds)i(resonan)n (t)e(with)j(a)e(giv)n(en)g Fk(A)p Fr(,)j(i.e.)35 b Fm(G)3230 2518 y Ff(\003)3291 2548 y Fr(:=)515 2648 y Fm(G)5 b(n)p Fr(\()p Fm(G)26 b(\\)21 b(V)845 2660 y Fj(0)882 2648 y Fr(\),)33 b(and)e Fm(G)1189 2617 y Ff(\003)1184 2671 y Fl(k)1258 2648 y Fr(its)h(DCS,)g(w)n(e)e(ha)n(v)n(e)g Fm(G)1976 2617 y Ff(\003)1971 2671 y Fl(k)2036 2648 y Fm(\\)21 b(V)2163 2660 y Fl(m)2255 2648 y Fr(=)28 b Fm(;)j Fr(for)g Fk(m)e(<)f(k)s Fr(,)k(and)f(th)n(us)g Fm(G)3253 2617 y Ff(\003)3323 2648 y Fr(is)515 2747 y(nilp)r(oten)n(t)d(of)f (in\014nite)i(order.)35 b(W)-7 b(e)28 b(can)g(reduce)f(to)g(the)h(more) 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b(will)h(pro)r (duce)e(a)h(sequence)g(of)g(v)n(ector)e(\014elds)j Fk(W)2377 3413 y Fj(\()p Fl(j)s Fj(\))2365 3468 y Ff(\003)2463 3456 y Fr(,)g(eac)n(h)f(b)r(eing)g(the)g(result)g(of)515 3556 y(the)23 b(\014rst)f Fk(j)28 b Fr(steps)22 b(of)h(further)f (normalization.)34 b(Let)23 b Fk(\021)2190 3568 y Fl(j)2247 3556 y Fr(b)r(e)g(the)g(op)r(erator)e(on)h Fm(G)28 b Fr(de\014ned)23 b(b)n(y)515 3675 y Fk(\021)556 3687 y Fl(j)591 3675 y Fr(\()p Fk(H)7 b Fr(\))26 b(=)f Fk(e)886 3644 y Fl(H)948 3675 y Fk(W)1038 3631 y Fj(\()p Fl(j)s Fj(\))1026 3686 y Ff(\003)1125 3675 y Fk(e)1164 3644 y Ff(\000)p Fl(H)1279 3675 y Fr(;)30 b(let)f Fk(\031)1500 3687 y Fl(j)1565 3675 y Fr(b)r(e)g(the)g(op)r(erator)f(of)h(pro)5 b(jection)28 b(from)h Fm(G)34 b Fr(to)29 b(the)g(range)515 3793 y(of)e Fk(\021)650 3805 y Fl(j)685 3793 y Fr(.)37 b(W)-7 b(e)28 b(can)g(reduce)f(the)h(normal)e(form)i Fk(W)35 b Fr(=)22 b Fk(W)2215 3750 y Fj(\(0\))2203 3805 y Ff(\003)2332 3793 y Fr(as)27 b(follo)n(ws.)681 3912 y(As)22 b(the)g(\014rst)g(step,)i(consider)d(a)g Fk(H)1753 3882 y Fj(\(0\))1865 3912 y Fm(2)j(G)j Fr(and)22 b(require)f(that)2628 3891 y Fg(f)2625 3912 y Fk(W)35 b Fr(=)22 b Fk(W)2915 3869 y Fj(\(1\))2903 3924 y Ff(\003)3050 3912 y Fr(:=)g Fk(\021)3201 3924 y Fj(0)3239 3912 y Fr(\()p Fk(H)7 b Fr(\))515 4012 y(is)38 b(suc)n(h)f(that)i Fk(\031)1044 4024 y Fj(0)1081 4012 y Fr([)p Fk(\037)1156 4024 y Fj(0)1193 4012 y Fk(\021)1234 4024 y Fj(0)1272 4012 y Fr(\()p Fk(H)1380 3982 y Fj(\(0\))1469 4012 y Fr(\)])i(=)f(0.)67 b(This)38 b(determines)g(\(non)g(uniquely\))h Fk(H)3093 3982 y Fj(\(0\))3182 4012 y Fr(,)h(and)515 4127 y(pro)r(duces)27 b(a)g Fk(W)1023 4084 y Fj(\(1\))1011 4139 y Ff(\003)1112 4127 y Fr(.)681 4227 y(A)n(t)d(further)h(steps,)f(w)n(e)g(ha)n(v)n(e)f (the)i(same)f(setting;)h(the)g(\\homological)c(equations")i(on)515 4327 y(Lie)k(algebras)f(to)h(b)r(e)h(solv)n(ed)f(at)g(eac)n(h)g(step)h (will)g(b)r(e)1465 4518 y Fk(\031)1512 4530 y Fl(j)1575 4426 y Fg(h)1628 4518 y Fk(\037)1680 4530 y Fl(j)1729 4426 y Fg(\020)1778 4518 y Fk(\021)1819 4530 y Fl(j)1855 4518 y Fr(\()p Fk(H)1963 4484 y Fj(\()p Fl(j)s Fj(\))2049 4518 y Fr(\))2081 4426 y Fg(\021i)2221 4518 y Fr(=)51 b(0)27 b(;)754 b(\()p Fk(B)t(:)p Fr(1\))515 4739 y(this)28 b(determines)f Fk(H)1173 4709 y Fj(\()p Fl(j)s Fj(\))1260 4739 y Fr(.)37 b(Eac)n(h)26 b Fk(W)1616 4696 y Fj(\()p Fl(j)s Fj(+1\))1604 4751 y Ff(\003)1815 4739 y Fr(is)h(then)h (determined)g(as)793 4911 y Fk(W)883 4868 y Fj(\()p Fl(j)s Fj(+1\))871 4922 y Ff(\003)1105 4911 y Fr(=)50 b Fk(\021)1261 4923 y Fl(j)1296 4911 y Fr(\()p Fk(H)1404 4876 y Fj(\()p Fl(j)s Fj(\))1491 4911 y Fr(\))h(:=)g(exp)o([)p Fk(H)1938 4876 y Fj(\()p Fl(j)s Fj(\))2025 4911 y Fr(])28 b(exp[)p Fk(W)2316 4868 y Fj(\()p Fl(j)s Fj(\))2304 4922 y Ff(\003)2403 4911 y Fr(])f(exp[)p Fm(\000)p Fk(H)2744 4876 y Fj(\()p Fl(j)s Fj(\))2830 4911 y Fr(])h Fk(:)279 b Fr(\()p Fk(B)t(:)p Fr(2\))1905 5255 y(17)p eop %%Page: 18 18 18 17 bop 515 523 a Fq(References)556 705 y Fr([1])41 b(A.)27 b(Algaba,)f(E.)g(F)-7 b(reire)25 b(and)i(E.)f(Gamero,)f(\\Hyp)r (ernormal)g(forms)h(for)f(equilibria)h(of)685 805 y(v)n(ector)20 b(\014elds.)i(Co)r(dimension)f(one)g(linear)f(degeneracies",)h Fh(R)l(o)l(cky)j(Mount.)g(J.)g(Math.)685 904 y Fs(29)k Fr(\(1999\),)e(13-45)556 1037 y([2])41 b(V.I.)i(Arnold,)j Fh(Ge)l(ometric)l(al)f(metho)l(ds)f(in)g(the)f(the)l(ory)h(of)h(or)l (dinary)g(di\013er)l(ential)685 1137 y(e)l(quations)p Fr(,)28 b(Springer,)f(Berlin)g(1983)556 1269 y([3])41 b(V.I.)27 b(Arnold)f(and)g(Y)-7 b(u.S.)27 b(Il'y)n(ashenk)n(o,)e Fh(Or)l(dinary)k(di\013er)l(ential)h(e)l(quations)p Fr(;)d(in:)37 b(En-)685 1369 y(cyclopaedia)25 b(of)h(Mathematical)g(Sciences)f(v)n (ol.)h(1)f({)h(Dynamical)g(Systems)g(I,)g(\(D.V.)685 1469 y(Anoso)n(v)h(and)g(V.I.)h(Arnold)f(eds.\),)h(pp.)g(1-148,)e (Springer,)g(Berlin)i(1988)556 1602 y([4])41 b(A.)36 b(Baider,)f(\\Unique)g(normal)e(forms)i(for)f(v)n(ector)f(\014elds)i (and)f(hamiltonians",)i Fh(J.)685 1701 y(Di\013.)31 b(Eqs.)d Fs(78)f Fr(\(1989\),)g(33)556 1834 y([5])41 b(A.)29 b(Baider)f(and)g (R.C.)h(Ch)n(urc)n(hill,)f(\\Uniqueness)g(and)h(non-uniqueness)f(of)g (normal)685 1934 y(forms)k(for)g(v)n(ector)f(\014elds",)i Fh(Pr)l(o)l(c.)i(R)l(oyal)f(So)l(c.)h(Edinbur)l(gh)e Fs(108A)f Fr(\(1988\),)g(27-33;)685 2033 y(Unique)f(normal)f(forms)g (for)g(planar)g(v)n(ector)f(\014elds",)i Fh(Math.)j(Z.)d Fs(199)f Fr(\(1988\),)g(303-)685 2133 y(310)556 2266 y([6])41 b(A.)25 b(Baider)d(and)i(J.)g(Sanders,)g(\\F)-7 b(urther)23 b(reduction)g(of)h(the)g(T)-7 b(ak)n(ens-Bogdano)n(v)20 b(nor-)685 2365 y(mal)34 b(form",)h Fh(J.)g(Di\013.)h(Eqs.)f Fs(99)e Fr(\(1992\),)h(205-244;)g(\\Unique)f(normal)g(forms:)49 b(the)685 2465 y(nilp)r(oten)n(t)29 b(hamiltonian)e(case",)f Fh(J.)k(Di\013.)g(Eqs.)f Fs(92)e Fr(\(1991\),)f(282-304)556 2598 y([7])41 b(G.R.)35 b(Belitskii,)h(\\Equiv)-5 b(alence)33 b(and)h(normal)f(forms)g(of)h(germs)f(of)i(smo)r(oth)f(map-)685 2697 y(pings",)27 b Fh(R)n(uss.)i(Math.)i(Surv.)d Fs(33)f Fr(\(1978\),)f(107)556 2830 y([8])41 b(G.)h(Benettin,)j(L.)c(Galgani)g (and)g(A.)g(Giorgilli,)j(\\A)d(pro)r(of)g(of)g(the)h(Kolmogoro)n(v)685 2930 y(theorem)33 b(on)h(in)n(v)-5 b(arian)n(t)32 b(tori)h(using)g (canonical)f(transformations)g(de\014ned)i(b)n(y)f(the)685 3029 y(Lie)28 b(metho)r(d",)g Fh(Nuovo)i(Cimento)g(B)e Fs(79)f Fr(\(1984\),)g(201)556 3162 y([9])41 b(H.W.)g(Bro)r(er,)h(\\F) -7 b(ormal)39 b(normal)g(form)h(theorems)f(for)h(v)n(ector)f(\014elds)h (and)g(some)685 3262 y(consequences)23 b(for)g(bifurcations)g(in)g(the) h(v)n(olume)f(preserving)f(case",)h(in)h Fh(Dynamic)l(al)685 3362 y(systems)i(and)g(turbulenc)l(e)p Fr(,)e(D.A.)g(Rand)f(and)g(L.S.) h(Y)-7 b(oung)23 b(eds.,)h Fh(L)l(e)l(ct.)i(Notes)g(Math.)685 3461 y Fs(898)p Fr(,)h(Springer,)g(Berlin)g(1981)515 3594 y([10])40 b(H.W.)27 b(Bro)r(er)d(and)i(F.)g(T)-7 b(ak)n(ens,)25 b(\\F)-7 b(ormally)24 b(symmetric)i(normal)f(forms)g (and)g(gener-)685 3694 y(icit)n(y",)i Fh(Dynamics)k(R)l(ep)l(orte)l(d)d Fs(2)f Fr(\(1989\),)f(39-59)515 3827 y([11])40 b(A.D.)35 b(Bruno,)f(\\Lo)r(cal)e(in)n(v)-5 b(arian)n(ts)32 b(of)i(di\013eren)n (tial)f(equations",)h Fh(Math.)i(Notes)e Fs(14)685 3926 y Fr(\(1973\),)27 b(844-848)515 4059 y([12])40 b(A.D.)29 b(Bruno,)e(reviews)f(1999a:34111)d(and)28 b(2000h:37071,)23 b Fh(Mathematic)l(al)32 b(R)l(eviews)515 4192 y Fr([13])40 b(A.D.)23 b(Bruno,)f Fh(Power)k(ge)l(ometry)f(in)f(algebr)l(aic)j(and)e (di\013er)l(ential)g(e)l(quations)p Fr(,)f(North-)685 4291 y(Holland,)k(Amsterdam)f(2000)515 4424 y([14])40 b(G.)23 b(Chen)g(and)g(J.)f(Della)h(Dora,)g(\\F)-7 b(urther)22 b(reduction)g(of)h(normal)e(forms)h(for)h(dynam-)685 4524 y(ical)28 b(systems",)e Fh(J.)k(Di\013.)g(Eqs.)f Fs(166)e Fr(\(2000\),)f(79-106)515 4657 y([15])40 b(L.O.)24 b(Ch)n(ua)g(and)h(H.)f(Kokubu,)h(\\Normal)e(forms)h(for)g(nonlinear)f (v)n(ector)g(\014elds.)h(P)n(art)685 4756 y(I:)29 b(theory",)e Fh(IEEE)k(T)-6 b(r)l(ans.)31 b(Cir)l(c.)h(Syst.)c Fs(35)g Fr(\(1988\),)f(863-888)e(\\Normal)i(forms)h(for)685 4856 y(nonlinear)h(v)n(ector)f(\014elds.)h(P)n(art)f(I)r(I:)i (applications",)f Fh(IEEE)j(T)-6 b(r)l(ans.)31 b(Cir)l(c.)i(Syst.)d Fs(36)685 4956 y Fr(\(1989\),)d(51-70)1905 5255 y(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fr([16])40 b(G.)28 b(Cicogna)e(and)i(G.)f(Gaeta,)g Fh(Symmetry)j(and)g(p)l(erturb)l(ation)f(the)l(ory)i(in)e(nonline)l(ar) 685 623 y(dynamics)p Fr(,)g(Springer,)e(Berlin)g(1999)515 756 y([17])40 b(A.)32 b(Deprit,)h(\\Canonical)d(transformations)g(dep)r (ending)i(on)f(a)h(small)f(parameter",)685 855 y Fh(Cel.)g(Me)l(ch.)f Fs(1)d Fr(\(1969\),)f(12-30)515 988 y([18])40 b(H.)32 b(Dulac,)g(\\Solution)f(d'un)g(syst)n(\023)-39 b(eme)30 b(d')n(\023)-39 b(equations)30 b(di\013)n(\023)-39 b(eren)n(tielles)30 b(dans)g(le)h(v)n(oisi-)685 1088 y(nage)c(des)h(v)-5 b(aleurs)26 b(singuli)n(\023)-39 b(eres",)25 b Fh(Bul)t(l.)31 b(So)l(c.)f(Math.)h(F)-6 b(r)l(anc)l(e)28 b Fs(40)f Fr(\(1912\),)f (324-383)515 1220 y([19])40 b(C.)30 b(Elphic)n(k,)f(E.)h(Tirap)r(egui,) f(M.)h(Brac)n(het,)f(P)-7 b(.)29 b(Coullet)g(and)h(G.)g(Io)r(oss,)f (\\A)g(simple)685 1320 y(global)d(c)n(haracterization)f(for)h(normal)g (forms)g(of)h(singular)f(v)n(ector)f(\014elds",)i Fh(Physic)l(a)685 1420 y(D)h Fs(29)f Fr(\(1987\),)f(95-127;)f(addendum,)j Fh(Physic)l(a)k(D)27 b Fs(32)g Fr(\(1988\),)g(488)515 1553 y([20])40 b(G.)e(Gaeta,)h(\\Reduction)e(of)g(P)n(oincar)n(\023)-39 b(e)35 b(normal)h(forms",)i Fh(L)l(ett.)h(Math.)h(Phys.)f Fs(42)685 1652 y Fr(\(1997\),)27 b(103-114)515 1785 y([21])40 b(G.)34 b(Gaeta,)g(\\P)n(oincar)n(\023)-39 b(e)30 b(renormalized)i (forms",)h Fh(A)n(nn.)i(I.H.P.)h(\(Phys.)g(The)l(o.\))g Fs(70)685 1885 y Fr(\(1999\),)27 b(461-514)515 2017 y([22])40 b(G.)24 b(Gaeta,)g(\\P)n(oincar)n(\023)-39 b(e)20 b(renormalized)i (forms)h(and)g(regular)f(singular)g(p)r(oin)n(ts)i(of)f(v)n(ec-)685 2117 y(tor)k(\014elds)h(in)g(the)g(plane",)f(preprin)n(t)g Fh(mp-ar)l(c)j(01-17)f Fr(\(2001\))515 2250 y([23])40 b(G.)21 b(Gaeta,)h(\\P)n(oincar)n(\023)-39 b(e)17 b(normal)i(forms)h (and)h(compact)f(simple)g(Lie)h(groups")e(\(revised)685 2350 y(v)n(ersion\),)27 b(preprin)n(t)g Fh(mp-ar)l(c)j(01-18)f Fr(\(2001\))515 2482 y([24])40 b(P)-7 b(.)25 b(Glendinning,)i Fh(Stability,)i(instability)g(and)f(chaos:)39 b(an)28 b(intr)l(o)l(duction)g(to)f(the)h(the-)685 2582 y(ory)k(of)g(nonline)l (ar)f(di\013er)l(ential)i(e)l(quations)p Fr(,)c(Cam)n(bridge)f(Univ)n (ersit)n(y)g(Press,)g(Cam-)685 2682 y(bridge)f(1994)515 2814 y([25])40 b(J.)20 b(Guc)n(k)n(enheimer)e(and)i(P)-7 b(.)19 b(Holmes,)i Fh(Nonline)l(ar)h(oscil)t(lations,)27 b(dynamic)l(al)d(systems,)685 2914 y(and)31 b(bifur)l(c)l(ation)f(of)h (ve)l(ctor)f(\014elds)p Fr(,)e(Springer,)f(Berlin)g(1983)515 3047 y([26])40 b(G.)23 b(Io)r(oss)d(and)i(M.)g(Adelmey)n(er,)h(\\T)-7 b(opics)20 b(in)j(bifurcation)e(theory)g(and)h(applications,)685 3147 y(W)-7 b(orld)28 b(Scien)n(ti\014c,)g(Singap)r(ore)e(1992)515 3279 y([27])40 b(A.A.)29 b(Kirillo)n(v,)d Fh(Elements)k(of)g(the)g(the) l(ory)g(of)h(r)l(epr)l(esentations)p Fr(,)d(Springer)f(1984)515 3412 y([28])40 b(H.)33 b(Kokubu,)f(H.)g(Ok)-5 b(a)32 b(and)f(D.)i(W)-7 b(ang,)32 b(\\Linear)f(grading)f(functions)i(and)g (further)685 3512 y(reduction)c(of)f(normal)g(forms",)f Fh(J.)k(Di\013.)g(Eqs.)f Fs(132)e Fr(\(1996\),)f(293-318)515 3645 y([29])40 b(J-C.)c(v)-5 b(an)37 b(der)f(Meer,)i Fh(The)h(Hamiltonian)g(Hopf)g(bifur)l(c)l(ation)p Fr(;)j(Lecture)36 b(Notes)g(in)685 3744 y(Mathematics)28 b Fs(1160)p Fr(,)e(Springer,)h (Berlin)g(1985)515 3877 y([30])40 b(Y)-7 b(u.A.)35 b(Mitrop)r(olosky)d (and)h(A.K.)h(Lopatin,)h Fh(Nonline)l(ar)h(me)l(chanics,)i(gr)l(oups)d (and)685 3977 y(symmetry)p Fr(,)28 b(Klu)n(w)n(er,)f(Dordrec)n(h)n(t)f (1995)515 4110 y([31])40 b(M.A.)35 b(Naimark)e(and)g(A.I.)i(Stern,)g Fh(The)l(ory)i(of)f(gr)l(oup)g(r)l(epr)l(esentations)p Fr(,)g(Springer)685 4209 y(1982)515 4342 y([32])k(J.)27 b(Sc)n(heurle)f(and)h(S.)g(W)-7 b(alc)n(her,)26 b(\\On)g(normal)g(form) g(computations",)g(forthcoming)685 4442 y(pap)r(er)i(\(2001\))515 4575 y([33])40 b(F.)27 b(T)-7 b(ak)n(ens,)25 b(\\Singularities)g(of)i (v)n(ector)e(\014elds",)h Fh(Publ.)j(Math.)h(I.H.E.S.)e Fs(43)d Fr(\(1974\),)685 4674 y(47-100)515 4807 y([34])40 b(S.)34 b(Ushiki,)g(\\Normal)e(forms)g(for)g(singularities)g(of)h(v)n (ector)f(\014elds",)i Fh(Jap.)h(J.)g(Appl.)685 4907 y(Math.)29 b Fs(1)f Fr(\(1984\),)e(1-34)1905 5255 y(19)p eop %%Page: 20 20 20 19 bop 515 523 a Fr([35])40 b(F.)46 b(V)-7 b(erh)n(ulst,)141 b Fh(Nonline)l(ar)47 b(di\013er)l(ential)g(e)l(quations)g(and)f (dynamic)l(al)i(systems)p Fr(,)685 623 y(Springer,)27 b(Berlin)g(1989,)f(1996)515 756 y([36])40 b(S.)34 b(W)-7 b(alc)n(her,)33 b(\\On)f(di\013eren)n(tial)h(equations)f(in)h(normal)f (form",)h Fh(Math.)j(A)n(nn.)d Fs(291)685 855 y Fr(\(1991\),)27 b(293-314)515 988 y([37])40 b(S.)32 b(W)-7 b(alc)n(her,)31 b(\\On)g(transformation)e(in)n(to)i(normal)f(form",)i Fh(J.)h(Math.)h(A)n(nal.)g(Appl.)685 1088 y Fs(180)27 b Fr(\(1993\),)g(617-632)1905 5255 y(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0102261208768--