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b(resultan)m(ts,)i(w)m(e)f(translate)-24 922 y(these)37 b(requiremen)m(ts)f(in)m(to)h(conditions)f(on)h(the)g(auxiliary)e(p)s (olynomial)f Fm(H)7 b Fo(\()p Fm(x)p Fo(\))37 b(=)f Fm(!)27 b Fl(\000)e Fm(h)p Fo(\()p Fm(x)p Fo(\),)-24 1038 y(prompting)37 b(the)j(study)e(of)h(the)g(p)s(olynomials)e(of)i(minimal)d(degree)k (that)g(ac)m(hiev)m(e)g(maximal)-24 1154 y(p)s(erio)s(ds.)61 b(Based)39 b(on)f(exp)s(erimen)m(tal)f(evidence,)j(w)m(e)e(conjecture)h (that)f(maximal)f(p)s(erio)s(d)f(for)-24 1270 y(all)j(primes)g(can)i(b) s(e)e(ac)m(hiev)m(ed)i(with)f(quadratic)f(p)s(olynomials)f Fm(H)7 b Fo(\()p Fm(x)p Fo(\),)44 b(while)38 b(with)h(a)i(cubic)-24 1386 y(p)s(olynomials)18 b(one)j(can)h(also)f(satisfy)f(certain)h (congruence)h(conditions)d(on)i(m)m(ultipliers)c(\(section)-24 1503 y(4\).)41 b(Finally)-8 b(,)28 b(in)g(section)h(5,)h(w)m(e)f (discuss)e(the)j(relev)-5 b(ance)29 b(of)g(Ceb)s(otarev's)g(densit)m(y) g(theorem)g(to)-24 1619 y(the)h(probabilit)m(y)e(of)j(o)s(ccurrence)f 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e(of)j(the)g(reduced)f(systems)h Fl(L)1781 1052 y Fw(n)1853 1038 y Fo(are)g(uniquely)d(determined)i(b)m(y)h(the)g(m)m(ul-)-24 1154 y(tiplier's)k(parameters)k Fm(T)44 b Fo(and)31 b Fm(s)p Fo(.)45 b(Sp)s(eci\014cally)-8 b(,)30 b(the)j(in)m(v)-5 b(arian)m(t)31 b(circle)g Fm(v)s Fo(\()p Fm(x)p Fo(\))d(=)g Fm(k)35 b Fo(decomp)s(oses)-24 1270 y(in)m(to)j Fm(m)f Fo(=)h(\()p Fm(p)25 b Fl(\000)g Fo(1\))p Fm(=T)52 b Fo(orbits,)39 b(with)e(common)h(p)s(erio)s(d)e Fl(T)22 b Fo(\()p Fm(x)p Fo(\).)64 b(The)38 b(longest)g(cycles)g(lie)f(on)-24 1386 y(the)28 b(unit)f(circle)g(\()p Fm(v)s Fo(\()p Fm(x)p Fo(\))f(=)f(0\),)30 b(with)c(maximal)h(p)s(erio)s(d)f(o)s(ccurring)h (when)g Fm(!)j Fo(is)d(maximal,)h(when)-24 1503 y(the)i Fm(p)p Fo(-adic)f(orbit)g(through)h Fm(x)1017 1517 y Fx(0)1086 1503 y Fo(\014lls)e(densely)g(the)i(circle)g Fl(j)p Fm(x)p Fl(j)25 b Fo(=)g Fl(j)p Fm(x)2267 1517 y Fx(0)2307 1503 y Fl(j)p Fo(.)41 b(If)29 b(the)h(m)m(ultiplier)d(is)i (not)-24 1619 y(maximal,)g(then)h(ev)m(ery)i(circle)d(decomp)s(oses)i (in)m(to)f(\(the)h(closure)f(of)7 b(\))31 b(\014nitely)d(man)m(y)j (orbits.)-24 1784 y Fp(Pro)s(of.)84 b Fo(F)-8 b(orm)m(ula)30 b(\(8\))i(follo)m(ws)d(readily)g(from)h(the)h(estimate)g(of)f (recurrence)g(distance)936 1997 y Fm(!)996 1960 y Fw(T)10 b(p)1083 1936 y Fc(i)1113 1997 y Fm(x)20 b Fl(\000)g Fm(x)25 b Fo(=)g Fm(x)1531 1923 y Fj(\000)1573 1997 y Fm(p)1619 1960 y Fw(s)p Fx(+)p Fw(i)1734 1997 y Fm(\014)h Fo(+)20 b Fm(O)s Fo(\()p Fm(p)2055 1960 y Fx(2)p Fw(s)p Fx(+)p Fw(i)2206 1997 y Fo(\))2241 1923 y Fj(\001)2298 1997 y Fm(:)-24 2202 y Fo(No)m(w,)28 b(if)e(\(9)15 b Fm(i)p Fo(\))28 b(is)e(false,)i(then)e Fm(x)h Fo(and)f Fm(y)j Fo(b)s(elong)d(to)i(di\013eren)m(t)e(circles,)h(so)g Fl(T)c Fo(\()p Fm(x;)15 b(y)s Fo(\))26 b(=)f Fl(1)p Fo(.)39 b(If)26 b(\(9)15 b Fm(i)p Fo(\))-24 2318 y(is)25 b(true,)h(but)f(\(9)15 b Fm(ii)p Fo(\))28 b(is)d(false,)h(then)g Fm(x)f Fo(and)h Fm(y)i 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(the)h(b)s(ene\014t)g(of)g(the)h(follo)m(wing)e(theorem.)1607 5225 y(7)p eop %%Page: 8 8 8 7 bop -24 573 a Fp(Theorem)34 b(1)46 b Fk(cf.)35 b Fo(\([21)q(],)g(theorem)f(3.1\))j Fk(L)-5 b(et)36 b Fm(f)10 b Fo(\()p Fm(x)p Fo(\))31 b(=)f Fm(!)s(x)23 b Fo(+)f Fm(O)s Fo(\()p Fm(x)2287 540 y Fx(2)2326 573 y Fo(\))36 b Fk(b)-5 b(e)36 b(a)g(p)-5 b(ower)37 b(series)f(with)-24 689 y(c)-5 b(o)g(e\016cients)36 b(in)f Fp(Z)620 703 y Fw(p)659 689 y Fk(,)h(with)g Fm(p)f Fk(o)-5 b(dd,)37 b(and)f Fm(T)48 b Fk(and)36 b Fm(s)29 b(<)g Fl(1)35 b Fk(given)g(by)g(\(5\).)50 b(Then)35 b Fm(f)10 b Fo(\()p Fm(x)p Fo(\))35 b Fk(is)g(semi-)-24 805 y(c)-5 b(onjugate)33 b(to)g(its)g(line)-5 b(ar)34 b(p)-5 b(art,)34 b(i.e.,)e(it)g (satis\014es)i(\(3\))f(for)h Fm(\036)e Fk(de\014ne)-5 b(d)34 b(in)e(the)h(domain)1158 1042 y Fm(v)s Fo(\()p Fm(x)p Fo(\))27 b Fm(>)1470 981 y Fo(1)p 1459 1022 66 4 v 1459 1105 a Fm(T)1550 914 y Fj(\022)1617 1042 y Fm(s)20 b Fo(+)1859 981 y(1)p 1781 1022 203 4 v 1781 1105 a Fm(p)g Fl(\000)g 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Fx(1)1607 5225 y Fo(8)p eop %%Page: 9 9 9 8 bop -24 573 a Fo(w)m(e)31 b(\014nd)e(\(cf.)i(equation)f(\(5\)\))440 711 y Fw(k)r Fi(\000)p Fx(1)439 738 y Fj(X)447 933 y Fw(t)p Fx(=1)585 825 y Fm(v)s Fo(\()p Fm(!)727 787 y Fw(t)778 825 y Fl(\000)20 b Fo(1\))83 b(=)1220 711 y Fw(N)1186 738 y Fj(X)1194 933 y Fw(t)p Fx(=1)1333 825 y Fm(v)s Fo(\()p Fm(\021)1463 787 y Fw(t)1513 825 y Fl(\000)20 b Fo(1\))42 b(=)1871 711 y Fw(N)1837 738 y Fj(X)1845 933 y Fw(t)p Fx(=1)1983 696 y Fj(\024)2031 825 y Fm(v)s Fo(\()p Fm(p)p Fo(\))21 b(+)f Fm(v)2369 696 y Fj(\022)2445 763 y Fm(\021)2493 730 y Fw(t)2544 763 y Fl(\000)g Fo(1)p 2445 804 235 4 v 2540 887 a Fm(p)2690 696 y Fj(\023\025)1032 1126 y Fo(=)83 b Fm(N)30 b Fo(+)1411 1013 y Fi(1)1380 1040 y Fj(X)1388 1234 y Fw(t)p Fx(=1)1527 998 y Fj(\026)1590 1065 y Fm(N)p 1590 1105 83 4 v 1594 1189 a(p)1640 1162 y Fw(t)1683 998 y Fj(\027)1776 1126 y Fl(\024)40 b Fm(N)30 b Fo(+)2112 1013 y Fi(1)2081 1040 y Fj(X)2089 1234 y Fw(t)p Fx(=1)2238 1065 y Fm(N)p 2238 1105 V 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Fo(\()p Fm(x)p Fo(\))37 b(w)m(ould)e(not)i(b)s(e)f (a)h(p)s(erm)m(utation)-24 689 y(mo)s(dulo)29 b Fm(p)p Fo(.)40 b(Th)m(us)29 b Fl(j)p Fm(a)21 b Fl(\000)f Fm(b)p Fl(j)892 703 y Fw(p)957 689 y Fo(=)25 b Fm(p)1099 656 y Fi(\000)p Fw(k)1226 689 y Fo(for)31 b(some)f Fm(k)k Fo(with)29 b(1)d Fl(\024)f Fm(k)j(<)d(n)p Fo(.)40 b(But)31 b(then)286 897 y Fm(f)10 b Fo(\()p Fm(a)p Fo(\))20 b Fl(\000)g Fm(f)10 b Fo(\()p Fm(b)p Fo(\))25 b(=)g Fm(f)910 859 y Fi(0)933 897 y Fo(\()p Fm(b)p Fo(\)\()p Fm(a)c Fl(\000)f Fm(b)p Fo(\))g(+)g Fm(O)s Fo(\(\()p Fm(a)h Fl(\000)f Fm(b)p Fo(\))1798 859 y Fx(2)1838 897 y Fo(\))25 b(=)g(\()p Fm(a)c Fl(\000)f Fm(b)p Fo(\)\()p Fm(f)2353 859 y Fi(0)2376 897 y Fo(\()p Fm(b)p Fo(\))h(+)f Fm(O)s Fo(\()p Fm(a)h Fl(\000)e Fm(b)p Fo(\)\))-24 1104 y(giving)1160 1189 y(1)p 1139 1229 89 4 v 1139 1313 a Fm(p)1185 1286 y Fw(k)1262 1250 y Fo(=)25 b Fl(j)p Fm(f)10 b Fo(\()p Fm(a)p Fo(\))21 b Fl(\000)f Fm(f)10 b Fo(\()p Fm(b)p Fo(\))p Fl(j)1857 1264 y Fw(p)1922 1250 y Fl(\024)2051 1189 y Fo(1)p 2028 1229 93 4 v 2028 1313 a Fm(p)2074 1286 y Fw(n)-24 1466 y Fo(whic)m(h)29 b(is)g(imp)s(ossible,)f(since)h Fm(k)g(<)c(n)p Fo(.)p 3210 1396 74 4 v 3210 1462 4 67 v 3280 1462 V 3210 1465 74 4 v 117 1682 a(F)-8 b(rom)32 b(this)f(result,)g(prop)s(osition)e(2.2,)k(and)e(the)h(previous)e (considerations)g(on)i(lifting,)d(w)m(e)-24 1798 y(ha)m(v)m(e)-24 1989 y Fp(Prop)s(osition)36 b(3.2)46 b Fk(L)-5 b(et)31 b Fm(f)10 b Fo(\()p Fm(x)p Fo(\))30 b Fk(b)-5 b(e)31 b(a)g(p)-5 b(ower)32 b(series)f(with)g(c)-5 b(o)g(e\016cients)31 b(in)g Fp(Z)2613 2003 y Fw(p)2652 1989 y Fk(,)g(which)h(r)-5 b(estricts)-24 2105 y(to)37 b(a)f(p)-5 b(ermutation)38 b(mo)-5 b(dulo)39 b Fm(p)p Fk(,)d(with)h(maximal)h(multipliers.)53 b(Then)37 b Fp(Z)2488 2119 y Fw(p)2563 2105 y Fk(de)-5 b(c)g(omp)g(oses)40 b(into)c Fm(p)-24 2221 y Fk(Sie)-5 b(gel)32 b(discs)h(of)g(r)-5 b(adius)34 b Fo(1)p Fm(=p)p Fk(.)-24 2412 y Fo(F)-8 b(or)34 b(these)g(systems,)g(kno)m(wledge)f(of) 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b(w)m(e)f(shall)d(require)i(that)h(the)g(m)m(ultiplier)c(of)j (the)h(\014xed)f(p)s(oin)m(t)-24 4326 y(at)j(zero)g(b)s(e)f (una\013ected)h(b)m(y)f(the)h(nonlinear)d(terms,)i(that)h(is)956 4533 y Fm(F)1027 4496 y Fi(0)1051 4533 y Fo(\(0\))26 b(=)f Fm(!)194 b Fl(\()-15 b(\))191 b Fm(H)7 b Fo(\(0\))27 b(=)d Fm(!)s(;)819 b Fo(\(21\))-24 4741 y(whic)m(h)35 b(also)i(ensures)e(that)j Fm(F)49 b Fo(lea)m(v)m(es)38 b(eac)m(h)g(circle)e(around)f(the)i(origin)e(in)m(v)-5 b(arian)m(t,)38 b(as)f(easily)-24 4857 y(v)m(eri\014ed)29 b(\(cf.)i(equation)g(\(12\)\).)1584 5225 y(11)p eop %%Page: 12 12 12 11 bop 117 573 a Fo(By)31 b(lemma)f(3.1)h(and)f(equation)g(\(20\),)i (w)m(e)f(ha)m(v)m(e)h(that,)f(if)839 769 y Fm(H)7 b Fo(\()p Fm(x)p Fo(\))26 b Fl(6\021)f Fo(0)15 b(\(mo)s(d)31 b Fm(p)p Fo(\))249 b Fm(x)25 b Fo(=)g(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(p)20 b Fl(\000)g Fo(1)702 b(\(22\))-24 965 y(then)23 b Fm(F)247 932 y Fi(0)270 965 y Fo(\()p Fm(x)p Fo(\))h(has)g(no)f(ro)s(ots)h(mo)s (dulo)e Fm(p)p 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b(computation)g(of)g(the)h (resultan)m(t,)-24 689 y(as)38 b(a)h(determinan)m(t,)g(is)f(no)g (faster)g(than)g(the)g(ev)-5 b(aluation)38 b(of)g(the)g(pro)s(duct)f (\(24\).)66 b(Ho)m(w)m(ev)m(er,)-24 805 y(equation)30 b(\(25\))i(leads)e(to)h(useful)e(form)m(ulae.)117 922 y(Let)i Fm(H)7 b Fo(\()p Fm(x)p Fo(\))31 b(b)s(e)f(sp)s(eci\014ed)f(b)m (y)h(\(19\))i(and)e(\(21\),)i(and)e(let)g Fm(\013)2085 936 y Fx(1)2125 922 y Fm(;)15 b(:)g(:)g(:)i(;)e(\013)2385 936 y Fw(n)2463 922 y Fo(b)s(e)29 b(the)i(ro)s(ots)g(of)f Fm(H)7 b Fo(\()p Fm(x)p Fo(\))-24 1038 y(in)26 b(some)i(extension)e(of) i Fp(F)869 1052 y Fw(p)908 1038 y Fo(,)g(the)g(\014eld)e(with)g Fm(p)h Fo(elemen)m(ts.)40 b(Since)26 b Fm(p)14 b Fl(\000)g Fo(1)27 b(is)f(ev)m(en,)j(w)m(e)f(ha)m(v)m(e)g(from)-24 1154 y(the)i(prop)s(erties)f(of)i(the)f(resultan)m(t)662 1423 y Fm(\025)83 b Fl(\021)g Fo(Res\()p Fm(H)7 b Fo(\()p Fm(x)p Fo(\))p Fm(;)15 b(x)1427 1385 y Fw(p)p Fi(\000)p Fx(1)1579 1423 y Fl(\000)20 b Fo(1\))26 b Fl(\021)f Fm(a)1920 1385 y Fw(p)p Fi(\000)p Fx(1)1920 1445 y Fw(n)2101 1309 y(n)2065 1336 y Fj(Y)2066 1532 y Fw(i)p Fx(=1)2181 1423 y Fo(\()p Fm(\013)2274 1378 y Fw(p)p Fi(\000)p Fx(1)2274 1450 y Fw(i)2425 1423 y Fl(\000)20 b Fo(1\))798 1725 y Fl(\021)989 1612 y Fw(n)952 1639 y Fj(Y)953 1834 y Fw(i)p Fx(=1)1068 1725 y Fo(\()p Fm(\013)1161 1681 y Fw(p)p Fi(\000)p Fx(1)1161 1752 y Fw(i)1312 1725 y Fl(\000)g Fo(1\))31 b(\(mo)s(d)f Fm(p)p Fo(\))p Fm(:)1261 b Fo(\(26\))117 2011 y(In)33 b(the)g(case)i(in)d(whic)m(h)g Fm(h)p Fo(\()p Fm(x)p Fo(\))f(=)f Fm(ax)1367 1978 y Fw(n)1447 2011 y Fo(is)j(a)h(monomial,)f(w)m(e)h(ha)m(v)m(e)g Fm(H)7 b Fo(\()p Fm(x)p Fo(\))31 b(=)f Fm(!)25 b Fl(\000)d Fm(ax)3030 1978 y Fw(n)3077 2011 y Fo(,)34 b(and)-24 2128 y(the)e(ro)s(ots)g(of)h Fm(H)7 b Fo(\()p Fm(x)p Fo(\))32 b(are)h Fm(\013)921 2142 y Fw(i)977 2128 y Fo(=)28 b Fm(\020)1123 2095 y Fw(i)1150 2128 y Fm(\013)p Fo(,)33 b(where)f Fm(\013)c Fo(=)1741 2093 y Fc(n)1729 2050 y Fj(p)p 1820 2050 154 4 v 78 x Fm(!)s(=a)33 b 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3797 y Fw(i)p Fx(=1)1318 3626 y Fm(\013)1376 3582 y Fw(p)1376 3653 y(i)1436 3626 y Fl(\000)20 b Fm(\013)1585 3640 y Fw(i)p 1318 3668 296 4 v 1422 3751 a Fm(\013)1480 3765 y Fw(i)1649 3688 y Fl(\021)1754 3627 y Fm(a)1802 3641 y Fw(n)p 1754 3668 96 4 v 1772 3751 a Fm(!)1926 3575 y Fw(n)1890 3602 y Fj(Y)1891 3797 y Fw(i)p Fx(=1)2006 3688 y Fo(\()p Fm(\013)2099 3644 y Fw(p)2099 3716 y(i)2160 3688 y Fl(\000)g Fm(\013)2309 3702 y Fw(i)2337 3688 y Fo(\))1023 3991 y Fl(\021)1186 3930 y Fm(a)1234 3944 y Fw(n)p 1186 3970 V 1204 4053 a Fm(!)1358 3877 y Fw(n)1322 3905 y Fj(Y)1323 4100 y Fw(i)p Fx(=1)1438 3991 y Fo(\()p Fm(\013)1531 4005 y Fw(i)p Fx(+1)1670 3991 y Fl(\000)g Fm(\013)1819 4005 y Fw(i)1847 3991 y Fo(\))15 b(\(mo)s(d)31 b Fm(p)p Fo(\))p Fm(:)877 b Fo(\(27\))-24 4277 y(If)38 b Fm(H)7 b Fo(\()p Fm(x)p Fo(\))39 b(=)429 4209 y Fj(Q)515 4304 y Fw(k)573 4277 y Fm(H)649 4292 y Fw(k)691 4277 y Fo(\()p Fm(x)p Fo(\))g(is)f(not)h(irreducible)c(\(with)i(factors)j Fm(H)2211 4292 y Fw(k)2253 4277 y Fo(\()p Fm(x)p Fo(\))f(of)g(degree)g Fm(n)2872 4292 y Fw(k)2953 4277 y Fl(\025)g Fo(2,)i(for)-24 4410 y(in)m(v)m(ertibilit)m(y\),)e(the)h(ab)s(o)m(v)m(e)g(form)m(ula)e (is)h(generalized)f(b)m(y)i(ordering)d(the)j(ro)s(ots)f Fm(\013)2859 4362 y Fx(\()p Fw(k)r Fx(\))2859 4437 y Fw(i)2996 4410 y Fo(of)h(eac)m(h)-24 4526 y(factor)31 b Fm(H)312 4541 y Fw(k)354 4526 y Fo(\()p Fm(x)p Fo(\),)h(giving)900 4804 y Fm(\025)26 b Fl(\021)1085 4742 y Fm(a)1133 4756 y Fw(n)p 1085 4783 V 1102 4866 a Fm(!)1220 4717 y Fj(Y)1259 4915 y Fw(k)1384 4686 y(n)1427 4698 y Fc(k)1366 4717 y Fj(Y)1367 4913 y Fw(i)p Fx(=1)1483 4804 y Fo(\()p Fm(\013)1576 4756 y Fx(\()p Fw(k)r Fx(\))1576 4831 y Fw(i)p Fx(+1)1715 4804 y Fl(\000)20 b Fm(\013)1864 4756 y Fx(\()p Fw(k)r Fx(\))1864 4831 y Fw(i)1962 4804 y Fo(\))15 b(\(mo)s(d)30 b Fm(p)p Fo(\))p Fm(:)1584 5225 y Fo(14)p eop %%Page: 15 15 15 14 bop 117 573 a Fo(If)34 b Fm(H)7 b Fo(\()p Fm(x)p Fo(\))35 b(is)e(quadratic)h(or)g(cubic)g(\(or)g(if)g(all)f(its)h (factors)h Fm(H)2190 588 y Fw(k)2232 573 y Fo(\()p Fm(x)p Fo(\))g(are\),)h(the)f(quan)m(tit)m(y)g Fm(\025)f Fo(is)-24 689 y(connected)d(to)g(its)f(discriminan)m(t)d(\001,)k(giv)m(en)f(b)m (y)g(\([23)r(],)h(page)g(35\))973 907 y(\001\()p Fm(H)7 b Fo(\))26 b(=)f Fm(a)1372 869 y Fx(2)p Fw(n)p Fi(\000)p Fx(2)1372 929 y Fw(n)1651 821 y Fj(Y)1559 1016 y Fx(1)p Fi(\024)p Fw(i