On Cyclotomic Polynomials with1 Coefficients

Size: px
Start display at page:

Download "On Cyclotomic Polynomials with1 Coefficients"

Transcription

1 On Cyclotomic Polynomials with1 Coefficients Peter Borwein and Kwok-Kwong Stephen Choi CONTENTS 1. Introduction 2. Cyclotomic Polynomials with Odd Coefficients 3. Cyclotomic Littlewood Polynomials 4. Cyclotomic Littlewood Polynomials of Odd Degree Acknowledgment References ResearchofP.Borweinissupported,inpart,byNSERCof Keywords:Cyclotomicpolynomials,Mahlermeasure,Littlewood FellowandtheInstitute'ssupportisgratefullyacknowledged. Canada. MathematicsSubjectClassication(1991):Primary11R09; K.-K.ChoiisaPacicInstituteofMathematicsPostdoctoral Secondary:11Y99 We characterize all cyclotomic polynomials of even degree with coefficients restricted to the setf+1, 1g. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd coefficients. The characterization is as follows. A polynomial P(x) with coefficients1 of even degree N 1 is cyclotomic if and only if P(x) =ol p1 (x) ol p2 (x p1 )ol where N = p 1 p 2p r pr (x p1p2p r 1 ), and the p i are primes, not necessarily distinct, and where ol p (x) := (x p 1)/ (x 1) is the p-th cyclotomic polynomial. We conjecture that this characterization also holds for polynomials of odd degree with1 coefficients. This conjecture is based on substantial computation plus a number of special cases. Weareinterestedinstudyingpolynomialswithcoef- Central cientsrestrictedtothesetf+1; 1g.Thisparticularsetofpolynomialshasdrawnmuchattentionand to this paper is a careful analysis of the effect of Graeffe s root squaring algorithm on cyclotomic polynomials. questionsandsowecallthesepolynomialslittle- thereareanumberofdicultoldquestionsconcerningit.littlewood[1968]raisedanumberofthese 1. INTRODUCTION woodpolynomialsanddenotethembyl.alittle- woodpolynomialofdegreenhasl2normonthe raisedconcerncomparingthebehaviorofthesepolynomialsinothernormstothel2norm.oneofthe unitcircleequaltopn+1:manyofthequestions isttwopositiveconstantsc1andc2sothatforeach polynomialscanbe\at".specically,dothereex- olderandmoreintriguingoftheseaskswhethersuch nthereisalittlewoodpolynomialofdegreenwith C1pn+1<jp(z)j<C2pn+1cAKPeters, Ltd /1999 $0.50 per page Experimental Mathematics 8:4, page 399

2 beenopenformorethanfortyyears,isdiscussed foreachzofmodulus1?thisproblem,whichhas calledrudin{shapiropolynomials.itisstillun- knownwhetherthereisasequencesatisfyingjust in[borwein1998],wherethereisanextensivebibliography.theupperboundissatisedbytheso- 400 Experimental Mathematics, Vol. 8 (1999), No. 4 timization"). ofthe\veryhardestproblemsincombinatorialop- thelowerbound(thisproblemhasbeencalledone hasbeenstudiedfromanumberofpointsofview. TheproblemofminimizingtheL4norm(orequivalentlyofmaximizingtheso-called\meritfactor") ThesizeoftheLpnormofLittlewoodpolynomials hasalsoattractedalotofattention. greenhavel4normasymptoticallyclosetopn+1? Thistooisstillopenandisdiscussedin[Borwein 1998]. Inparticular,canLittlewoodpolynomialsofde- themahlermeasureoflittlewoodpolynomials.the MahlermeasureisjusttheL0normonthecircleand onewouldexpectthistobecloselyrelatedtothe Mahler[1963]raisedthequestionofmaximizing coursetheminimumpossiblemahlermeasurefora minimizationproblemforthel4normabove.of Littlewoodpolynomialis1andthisisachievedby anycyclotomicpolynomial.inthispaperwedene anddenotebyn(x)then-thirreduciblecyclotomic polynomial,thatis,theminimumpolynomialofa acyclotomicpolynomialasanymonicpolynomial primitiven-throotofunity. withintegercoecientsandallrootsofmodulus1, withcoecients1ofevendegreen 1iscyclotomicifandonlyif degree.specically,weshowthatapolynomialp(x) ingthecyclotomiclittlewoodpolynomialsofeven Thispaperaddressesthequestionofcharacteriz- wheren=p1p2prandthepiareprimes(not necessarilydistinct).the\if"partisobvioussince pi(x)hascoecients1. P(x)=p1(x)p2(xp1)pr(xp1p2pr 1); ofsuchpolynomials. Graee'srootsquaringalgorithm.Ittranspiresthat Thisanalysisisbasedonacarefultreatmentof Wealsogiveanexplicitformulaforthenumber greehavethesamexedpointoniteratinggra- ee'srootsquaringalgorithm.thisalsoallowsus allcyclotomiclittlewoodpolynomialsofaxedde- tocharacterizeallcyclotomicpolynomialswithodd characterizationofcyclotomiclittlewoodpolynomialsofevendegreealsoholdsforodddegree.oneof Substantialcomputations,aswellasanumberof specialcases,leadustoconjecturethattheabove coecients. thecaseswecanhandleiswhennisapowerof2. isreallytheobservationthatthecyclotomiclittle- thisiscarefullyerasedinthenalexposition).it pectsofthispaper.(asisperhapsusual,muchof Itisworthcommentingontheexperimentalasrithmthatiscritical.ThisallowsforcomputatiosentiallybyinvertingGraee'srootsquaringalgo- overallcyclotomiclittlewoodsuptodegreeseveral woodpolynomialscanbeexplicitlyconstructedes- turesofthepaperandsuggestedtheroutetosome itwasthesecalculationsthatallowedfortheconjec- hundred(withexhaustivesearchfailingfarearlier), oftheresults. aconstructionwhichisofinterestinitself.indeed Section3looksatcyclotomicLittlewoodpolynomialswithacompleteanalysisoftheevendegreecase. Thepaperisorganizedasfollows.Section2examinescyclotomicpolynomialswithoddcoecients. andotherevidencetosupporttheconjecturethat theoddcasebehavesliketheevencase. Thelastsectionpresentssomenumericalevidence ofirreduciblecyclotomicpolynomials.todothis, Inthissection,wediscussthefactorizationofcyclotomicpolynomialswithoddcoecientsasaproduct pisaprimenumber.themostusefulcaseisp=2 werstconsiderthefactorizationoverzp[x],where becauseeverylittlewoodpolynomialreducestothe Dirichletkernel1+x++xN 1 inz2[x].inzp[x],n(x)isnolongerirreducible ingeneralbutn(x)andm(x)arestillrelatively primetoeachother. arerelativelyprimeinzp[x]. tegersrelativelyprimetop.thenn(x)andm(x) 2.1.Supposenandmaredistinctpositivein- 2. CYCLOTOMIC POLYNOMIALS WITH ODD COEFFICIENTS

3 Proof.Supposeeandfarethesmallestpositiveintegerssuchthat pe1(modn)andpf1(modm): LetFpkbetheeldoforderpk.ThenFpecontains Borwein and Choi: On Cyclotomic Polynomials with1 Coefficients 401 isaproductof'(n)=eirreduciblefactorsofdegreee exactly'(n)elementsofordernandoverzp,n(x) bedenedoverthequotientoftwomonicpolynomialsinz[x]bytp[(p=q)(x)]:=tp[p(x)]=tp[q(x)]. isalsoamonicpolynomialinz[x].weextendtpto identities[borweinanderdelyi1995,p.5],tp[p(x)] foreveryp(x)=qni=1(x i)inz[x].bynewton's haveacommonfactorinzp[x]sincetheirirreducible foranelementinfpeofordernoverzp;see[lidl andniederreiter1983].son(x)andm(x)cannot andeachirreduciblefactorisaminimalpolynomial rootsofp.alsoweletmpbethenaturalprojection befactorsofpolynomialswithoddcoecients. factorsareminimalpolynomialsofdierentorders. fromz[x]ontozp[x]: polynomialwhoserootsarethep-thpowersofthe Thisoperatorobviouslytakesapolynomialtothe Thefollowinglemmatellswhichm(x)canpossibly Thisprovesourlemma. Proof.Sincem(x)dividesP(x),som(x)alsodividesP(x)inZ2[x].However,inZ2[x],P(x)equals coecientsofdegreen 1.Ifm(x)dividesP(x), thenmdivides2n. 2.2.SupposeP(x)isapolynomialwithodd tivelyprimetopandi2.then Mp[P(x)]=P(x)(modp): (ii)tp[pn(x)]=n(x)p 1, (i)tp[n(x)]=n(x), 2.3.Supposethatnisapositiveintegerrela- to1+x++xn 1andcanbefactoredas wheren=2tm,t0andmisodd.inviewof P(x)=1(x) 1YdjM2t permutestherootsofn(x).toprove(ii)and(iii), (iii)tp[pin(x)]=pi 1n(x)p. 2.1,d1(x)andd2(x)arerelativelyprime d(x); weconsider Proof.(i)istrivialbecauseif(n;p)=1thenTpjust Tp[P(xp)]=Tp"NYj=1(xp i)# inz2[x]ifd1andd2aredistinctoddintegers.soif misodd,thenm(x)isafactorintherighthand =Tp"NYj=1pYl=1(x e2il=p1=p i)# sideof(2{1)andhencem=dforsomedjm.on (2 1) theotherhand,ifmisevenandm=2lm0where inz2[x].thus,inviewof(2{1),wemusthavem0= l1andm0isodd,then =NYj=1pYl=1(x i) dfordjmandlt+1.henceinbothcaseswe m(x)=2m0(x2l 1)=m0(x2l 1)=m0(x)2l 1 Thus(ii)and(iii)followfrom(i)andtheequalities =P(x)p: writtenas P(x),withoddcoecientsofdegreeN 1canbe havemdivides2n. Inviewof2.2,everycyclotomicpolynomial, vergeinanitenumberofstepstoaxedpointof pn(x)=n(xp)=n(x)andpin(x)=pi 1n(xp) wherethee(d)arenonnegativeintegers. P(x)=Ydj2Ne(d) ForeachprimepletTpbetheoperatordened d(x); TpandwedenethistobethexedpointofP(x) [Hungerford1974,x5.8]. withrespecttotp. WhenP(x)iscyclotomic,theiteratesTnp[P(x)]con- inz[x],thenmp[tp[p(x)]]=mp[p(x)]; 2.4.IfP(x)isamoniccyclotomicpolynomial overallmonicpolynomialsinz[x]by Tp[P(x)]:=NYi=1(x pi) (2 2) mialsn(x).letnbeanintegerrelativelyprime sucestoconsidertheprimitivecyclotomicpolyno- inzp[x]. Proof.SincebothTpandMparemultiplicative,it (2 3)

4 top.then(2{3)istrueforp(x)=n(x)by(i)of 2.3.ForP(x)=pn(x),wehave by(ii)of2.3.however, Mp[Tp[pn(x)]]=Mp[n(x)p 1]=Mp[n(x)]p Experimental Mathematics, Vol. 8 (1999), No. 4 Mp[pn(x)]=Mp[n(xp)] Mp[n(x)]=Mp[n](xp) inzp[x].thisprovesthat(2{3)isalsotruefor =Mp[n(x)]p 1 P(x)=pn(x).Finally,P(x)=pin(x)implies Mp[Tp[pin(x)]]=Mp[pi 1n(x)p] by(iii)of2.3.thiscompletestheproofof =Mp[pin(x)] =Mp[pi 1n(xp)] theconverseisalsotrue. Mp[P(x)]=Mp[Q(x)].Thenextresultshowsthat 2.4showsthatTp[P(x)]=Tp[Q(x)]implies ourlemma. samexedpointwithrespecttoiterationoftp. Zp[x]ifandonlyifbothP(x)andQ(x)havethe polynomialsinz[x]andmp[p(x)]=mp[q(x)]in 2.5.P(x)andQ(x)aremoniccyclotomic Proof.Suppose P(x)=Yd2De(d) d(x)e(pd) pd(x)e(ptd) ptd(x) Theorem andq(x)=yd2de(d)0 d(x)e(pd)0 pd(x)e(ptd)0 integersrelativelyprimetop.thenusingparts(i){ wheret;e(j);e(j)00anddisasetofpositive ptd(x); (iii)of2.3,wehaveforlt Tlp[Q(x)]=Yd2Dd(x)f(d)0; Tlp[P(x)]=Yd2Dd(x)f(d); wheref(d)=e(d)+(p 1)tXj=1pj 1e(pjd); (2 4) f(d)0=e(d)0+(p 1)tXj=1pj 1e(pjd)0: From2.4,wehave foranylt.fromthisand(2{4), Mp[Tlp[P(x)]]=Mp[P(x)]=Mp[Q(x)] Yd2DMp[d(x)]f(d)=Yd2DMp[d(x)]f(d)0: =Mp[Tlp[Q(x)]]; andmp[d0(x)]arerelativelyprimeifd6=d0.sowe However,with2.1,weknowthatMp[d(x)] musthavef(d)=f(d)0foralld2dandhencefrom (2{4),P(x)andQ(x)havethesamexedpointwith respecttotp. FromTheorem2.5,wecancharacterizethemonic dertheprojectionmp.theyallhavethesamexed cyclotomicpolynomialsbytheirimagesinzp[x]un- pointundertp.inparticular,whenp=2wehave: xedpointunderiterationoft2.specically,ifn= oddcoecientsofthedegreen 1havethesame 2.6.Allmoniccyclotomicpolynomialswith occursatthe(t+1)-thstepoftheiterationandequals 2tMwheret0andMisoddthenthexedpoint 2.5andthefactthat Proof.TherstpartfollowsdirectlyfromTheorem (xm 1)2t(x 1) 1: coecientsofdegreen 1.IfN=2tM,thenfrom inz2[x]ifp(x)isamonicpolynomialwithodd M2[P(x)]=1+x++xN 1 (2{2), OverZ2[x], P(x)=YdjMe(d) d(x)e(2d) 2d(x)e(2t+1d) 2t+1d(x): so 1+x++xN 1=1(x) 1YdjM2t f(d)=e(d)+t+1 d(x); =2tXi=12i 1e(2id) 2t 1ford=1. fordjm,d>1, (2 5)

5 Therefore,from(2{5)and2.3, Tt+1 2[P(x)]=YdjMf(d) =(xm 1)2t(x 1) 1: d(x)=1(x) 1YdjM2t d(x) Borwein lary2.7.toprove(2{7),weusedebruijn'sasymp- toticestimationforb(n)in[debruijn1948]: Proof.Formula(2{6)followsfrom(2{5)andCorol- and Choi: On Cyclotomic Polynomials with1 Coefficients 403 T2[P(x)]equals1+x++xN 1forallcyclotomic 2.6,whenNisodd(t=0),showsthat B(n)exp((logn)2=log4): ofcyclotomicpolynomialswithoddcoecients. (2{5),wethenhavethefollowingcharacterization polynomialswithoddcoecients.from(2{2)and N 1iscyclotomicifandonlyif Apolynomial,P(x),withoddcoecientsofdegree P(x)=YdjMe(d) 2.7.LetN=2tMwitht0andModd. mialofhigherdegreeisasfollows:ifp1(x)andp2(x) thecoecientsareall+1or 1. Wenowspecializethediscussiontothecasewhere OnenaturalwaytobuildupLittlewoodpolyno- d(x)e(2d) 2d(x)e(2t+1d) 2t+1d(x); P(x),withoddcoecientsofevendegreeN 1is cyclotomicifandonlyif andthee(d)satisfythecondition(2{5). istheonlywaytoproducecyclotomiclittlewood ofhigherdegree.inthissection,weshowthatthis arelittlewoodpolynomialsandp1(x)isofdegree Furthermore,ifNisodd,thenanypolynomial, polynomials,atleastforevendegree. N 1thenP1(x)P2(xN)isaLittlewoodpolynomial P(x)=YdjN d>1e(d) wherethee(d)arenonnegativeintegers. d(x); suchthatali+n=aliforall1ni 1and ProvingthisisequivalenttoshowingthatthecoecientsofP(x)are\periodic"inthesensethat ifp(x)=pn 1 n=0anxn,thenthereisa\period"i 0lN=i 1.ThisisourTheorem3.3below. B(n)bethenumberofpartitionsofnintoasum cyclotomicpolynomialswithoddcoecients.let oftermsofthesequencef1;1;2;4;8;16;:::g.then 2.7allowsustocomputethenumberof wehavexn 1P(1=x)=P(x).Thusitfollowsfrom nomialinlandletskbethesumofthek-thpowers ofalltherootsofp(x).sincep(x)iscyclotomic, SupposeP(x)=PN 1 n=0anxnisacyclotomicpoly- Newton'sidentitiesthat B(n)hasgeneratingfunction a1=1byreplacingp(x)by P(x)orP( x)if forkn 2.Wemayfurtherassumethata0= Sk+a1Sk 1++ak 1S1+kak=0 ThenumberofcyclotomicpolynomialswithoddcoecientsofdegreeN 1is Weclaimthat S1= a1= 1: 2.8.LetN=2tMwitht0andModd. F(x)=(1 x) 11Yk=0(1 x2k) 1: forsomeintegeri2.from(3{1),wehave necessary.wenowlet a0=a1==ai 1=1andai= 1; (3 1) whered(m)denotesthenumberofdivisorsofm. C(N)=B(2t)d(M) 1B(2t 1); Furthermore, SupposeS1==Sj= 1forj<i 1.Then s1=s2==si 1= 1; logc(n)(12t2log2)(d(m) 1)+(log(2t 1))2 from(3{1)again, log4 : SoS1==Si 1= 1.Similarly,from(3{1), Sj+1= a1sj ajs1 (j+1)aj+1 =j (j+1)= 1: (2 6) Si= a1si 1 ai 1S1 iai=2i 1: (2 7) Now(2{7)followsfromthisand(2{6). 3. CYCLOTOMIC LITTLEWOOD POLYNOMIALS Si=2i 1: (3 2)

6 alifor1ni 1and0lk 2.Then Xl=0ali S(k l)i+j+1 S(k l 1)i+j Let2kN 1 i 1andsupposeali+n= 404 Experimental Mathematics, Vol. 8 (1999), No. 4 +(ki+j+1)(aki+j+1 aki+j) for0ji 2. +i 1 Proof.Suppose0ji 2.From(3{1)and(3{2) wehave 0=k 1 Xl=0i 1 Xn=0a(k 1)i+n(Si+j n+1 Si+j n)=0(3 3) +j 1 Xn=0ali+nS(k l)i+j n =k 2 Xl=0alii 1 Xn=0aki+nSj n+(ki+j)aki+j Xn=0S(k l)i+j n+i 1 jxn=0aki+n+(ki+j+1)aki+j: Xn=0a(k 1)i+nSi+j n 0=k 2 Similarly, Xl=0alii 1 Xn=0S(k l)i+j n+1+i 1 Xn=0a(k 1)i+nSi+j n+1 (3 4) 0=k 2 Hence,onsubtracting(3{5)from(3{4),wehave Xl=0ali S(k l)i+j+1 S(k l 1)i+j+1 jxn=0aki+n+(ki+j+1)aki+j+1:(3 5) +(ki+j+1)(aki+j+1 aki+j) Thisproves(3{3). +i 1 Xn=0a(k 1)i+n(Si+j n+1 Si+j n): 3.2.Let0kN 1 i 1.Supposeali+n=ali for1ni 1and0lk. for1ni 1and0lk.Then Sli+n= 1 (3 6) Proof.Weprovethisbyinductiononk.Wehave istruefork 1.Then,forany0ji 2, provedthat(3{6)istruefork=0.suppose(3{6) 0=a0(Ski+j+1 S(k 1)i+j+1) =Ski+j+1+1+a(k 1)i(Si+j+1 Sj+1) +a(k 1)ii 1 Xn=0(Si+j n+1 Si+j n) by(3{3).henceski+j+1= 1for0ji 2. =Ski+j+1+1; 3.3.SupposeNisodd.Ifa0=a1== for1ni 1and0lNi 1. ai 1=1andai= 1,then Proof.WerstshowthatS2k= 1for1kN 1 ali+n=ali Theorem odd,from2.7, andhenceiisoddbecausesi=2i 1.SinceNis P(x)=YdjNe(d) d(x)e(2d) If1kN 1,then wheree(d)+e(2d)=1ifd>1ande(1)=e(2)=0. 2d(x) S2k=XdjN(e(d)Cd(2k)+e(2d)C2d(2k)) =XdjN(e(d)+e(2d))Cd(k) =XdjNCd(k) C1(k) rootsofqdjnd(x)=xn 1,whichisequaltozero hencepdjncd(k)isthesumofk-thpowersofthe k-thpowersoftheprimitived-throotsofunityand wheretheramanujansum,cd(k),isthesumofthe when1kn 1. Suppose Wecontinueourproofbyusinginductiononk. for1ni 1and0lk 1where1k N 1 i 1.Froms3.1and3.2wehave ali+n=ali Ski+j+1+1+(ki+j+1)(aki+j+1 aki+j)=0(3 8) = 1; (3 7)

7 andhencefrom(3{3)again 0=a0(S(k+1)i+j+1 Ski+j+1)+ai(Ski+j+1 S(k 1)i+j+1) +aki+j(si+1 Si)+aki+j+1(Si Si 1) clotomicinlofdegreen 1ifandonlyif 3.5.SupposeNisodd.ThenP(x)iscy- Borwein and Choi: On Cyclotomic Polynomials with1 Coefficients 405 =S(k+1)i+j+1+1+2((k+1)i+j+1)(aki+j+1 aki+j) =(S(k+1)i+j+1 2Ski+j+1 1)+2i(aki+j+1 aki+j) +((k+1)i+j+1)(a(k+1)i+j+1 a(k+1)i+j) where"=0or1,n0=1;nt=nandni 1isa P(x)=tYi=1xNi+( 1)"+i properdivisorofnifori=1;2;:::;t. xni 1+( 1)"+i Proof.Withoutlossofgenerality,wemayassume ifjiseven.sofrom(3{8)and(3{9),wehave (3{7),Ski+j+1= 1ifjisoddandS(k+1)i+j+1= 1 for0ji 2.Supposekiseven.Theninviewof P(x)iscyclotomicinLifandonlyif thatp(x)=1+x+a2x2+.fromtheorem3.4, P(x)=p1(x)p2(xp1)pr(xp1pr 1) aki+j+1=aki+j =p1(x)pn1(xp1pn1 1) (3 9) forj=1;3;:::;i 2and forj=0;2;:::;i 3.However,sincetheai'sare+1 pnt 1+1(( 1)t 1xp1pnt 1) pn1+1( xp1pn1)pn2( xp1pn2 1) forj=0;2;:::;i 3.Henceaki+n=akiforn= or 1,Equation(3{10)impliesthat aki+j+1=aki+j anda(k+1)i+j+1=a(k+1)i+j theprecedingequationbecomes wheren=p1pnt.sincep(x)=(xp 1)=(x 1), pnt(( 1)t 1xp1pnt 1); P(x)=tYi=1xNi+( 1)i 2(aki+j+1 aki+j)=a(k+1)i+j+1 a(k+1)i+j(3 10) nomial,p(x),ofdegreen 1iscyclotomicifand 1;2;:::;i 1.Thecaseofkoddcanbeprovedin xni 1+( 1)i; onlyif thesameway. 3.4.SupposeNisodd.ALittlewoodpoly- cyclotomiclittlewoodpolynomialsofgivenevendegree.foranypositiveintegersnandt,dene Thisprovesourcorollary. Using3.5,wecancountthenumberof wheren0=1andni=p1pnifori=1;:::;t. wheren=p1p2prandthepiareprimes,not Theorem Conversely,supposeP(x)isacyclotomicLittlewood thenp(x)isacyclotomiclittlewoodpolynomial. necessarilydistinct. andfori1,di(n):=xnjndi 1(n) r(n;t):=#(n1;n2;:::;nt):n1jn2jjnt; 1<N1<N2<<Nt=N ; polynomial.asbeforewemayassumethata0= Proof.ItisclearthatifP(x)isoftheform(3{11), whered0(n)=1. a1==ai 1=1andai= 1.WeproveourresultbyinductiononN.FromTheorem3.3,wehave 3.6.Forl;t0andpprime,wehave P(x)=P1(x)P2(xi),whereP1(x)=1+x++xi 1 dt(pl)=l+t t: (3 12) andp2(x)isacyclotomiclittlewoodpolynomialof P2(x)areoftheform(3{11)andhencesoisP(x) degreelessthann 1.Byinduction,P1(x)and becausethedegreeofp1(x)isi 1. suppose(3{13)istruefort 1wheret1.Then clearlytruefort=0becaused0(n)=1.wethen Proof.Weworkbyinductionont.Equality(3{13)is dt(pl)=xnjpldt 1(n)=lXi=0dt 1(pi)=lXi=0i+t 1 t 1: (3 13) P(x)=p1(x)p2(xp1)pr(xp1p2pr 1); (3 11)

8 (x+1)t 1+(x+1)t++(x+1)l+t 1 Sodt(pl)isthecoecientofxt 1in =(x+1)t 1(x+1)l+1 1 x 406 Experimental Mathematics, Vol. 8 (1999), No. 4 (x+1)t 1.Therefore,dt(pl)= l+t Hencedt(pl)isthecoecientofxtin(x+1)l+t =(x+1)l+t (x+1)t 1 t. x : have Sincedt(N)isamultiplicativefunctionofN,we aredistinctprimes,then 3.7.IfN=pr1 dt(n)=syi=1ri+t 1prs swhereri1andpi t: have r(n;t):=0 3.8.ForanypositiveintegersNandt,we Pti=1( 1)t i tidi 1(N)ifN>1. ifn=1, fromthedenitionthatr(1;t)=0andr(n;1)=1 Proof.Weagainprovebyinductionont.Itisclear (3{14)istruefort 1wheret2.Then foranyt;n1.wethensupposen>1and r(n;t)=xn1jn N1>1r(N=N1;t 1) (3 14) =XN1jN =XN1jNt 1 N>N1>1r(N=N1;t 1) t 1 Xi=1( 1)t i 1t 1 Xi=1( 1)t i 1t 1 idi 1(N=N1) =txi=2( 1)t it 1 i 1di 1(N) i(di 1(1)+di 1(N)) t 1 =txi=1( 1)t itidi 1(N) Xi=1( 1)t i 1t 1 idi 1(N)+( 1)t 1 from(3{12)andthefactthat t 1 i 1+ t 1 inlofdegreen 1,whereN=pr1 3.9.Thenumberofcyclotomicpolynomials i= ti. andthepiaredistinctoddprimes,is 4r1++rs Xi=1 ixj=1( 1)i jijsyk=1rk+j 1 1prs s;ri1 j 1: Proof.From3.5,thenumberofcyclotomic polynomialsinlofdegreen 1is Thecorollarynowfollowsfrom3.7and 4r1++rs Xi=1r(N;i): 3.8. forpolynomialsofodddegree. WeconjectureexplicitlythatTheorem3.4alsoholds degreen 1iscyclotomicifandonlyif 4.1.ALittlewoodpolynomial,P(x),of necessarilydistinct. wheren=p1p2prandthepiareprimes,not P(x)=p1(x)p2(xp1)pr(xp1p2pr 1); Conjecture putingallcyclotomicpolynomialswithoddcoe- cientsofagivendegreeandthencheckingwhich Wecomputeduptodegree210(exceptforthecase N 1=191).Thecomputationwasbasedoncom- matchedthesetgeneratedbytheconjecture.for wereactuallylittlewoodandseeingthatthisset arelittlewood.forn 1=191thereare cyclotomicpolynomialswithoddcoecients(which example,forn 1=143thereare cyclotomicpolynomialswithoddcoecientsofwhich416 wastoobigforourprogram). ecientsfrom2.7quiteeasilysothebulk height1.thesetintheconjecturecomputesvery oftheworkisinvolvedincheckingwhichoneshave Wecangenerateallthecyclotomicswithoddco- easilyrecursively. 4. CYCLOTOMIC LITTLEWOOD POLYNOMIALS OF ODD DEGREE (4 1)

9 proofisasfollows.from2.7,wehave MostnotablythecasewhereNisapowerof2.The Somespecialcasesalsosupporttheconjecture. P(x)=e(1) 1(x)e(2) 2(x)e(2t+1) 2t+1(x): Borwein and Choi: On Cyclotomic with1 Coefficients 407 x2 1and Again,weassumea0=a1=1.Since1(x)2(x)= [BorweinandErdelyi1995]P.BorweinandT.Erdelyi, L.Schumaker,VanderbiltUniversityPress,Nashville, TN,1998. ApproximationTheoryIX,editedbyC.Chuiand forl2,wehavee(2) e(1)=1andhence P(x)=2(x)Q(x2); 2l(x)=2(x2l 1) [debruijn1948]n.g.debruijn,\onmahler'spartition Polynomialsandpolynomialinequalities,Lect.Notes inmath.161,springer,newyork,1995. forsomecyclotomiclittlewoodpolynomialq(x). (1948),210{220. problem",nederl.akad.wetensch.,proc.51(1948), 659{669.AlsoappearsasIndagationesMath.10 TheauthorswishtothankProfessorD.Boydforhis adviceandsupportconcerningthispaper. [LidlandNiederreiter1983]R.LidlandH.Niederreiter, [Hungerford1974]T.W.Hungerford,Algebra,Holt, GraduateTextsinMath.73,Springer,Berlin,1980. RinehartandWinston,NewYork,1974.Reprintedas [Borwein1998]P.Borwein,\Someoldproblemson [Littlewood1968]J.E.Littlewood,Someproblems Finiteelds,Seconded.,Addison-Wesley,Reading, MA,1983.Secondedition,CambridgeUniv.Press, polynomialswithintegercoecients",pp.31{50in [Mahler1963]K.Mahler,\Ontwoextremumproperties inrealandcomplexanalysis,heathmathematical Monographs,D.C.Heath,Lexington,MA,1968. PeterBorwein,DepartmentofMathematicsandStatistics,SimonFraserUniversity,Burnaby,B.C.,Canada ofpolynomials",illinoisj.math.7(1963),681{701. REFERENCES Kwok-KwongStephenChoi,DepartmentofMathematics,UniversityofBritishColumbia,Vancouver,B.C.,Canada V6T1Z2andDepartmentofMathematicsandStatistics,SimonFraserUniversity,Burnaby,B.C.,Canada ReceivedJune3,1998;acceptedinrevisedformNovember28,1998 Therefore,byinduction,P(x)satises(4{1). ACKNOWLEDGMENT

minimal polyonomial Example

minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

Die ganzen zahlen hat Gott gemacht

Die ganzen zahlen hat Gott gemacht Die ganzen zahlen hat Gott gemacht Polynomials with integer values B.Sury A quote attributed to the famous mathematician L.Kronecker is Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Any Any Any natural number that that that greater greater than than than 1 1can can 1 be can be be factored into into into a a a product of of of prime prime numbers. For For For

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

5. Factoring by the QF method

5. Factoring by the QF method 5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Polynomial Factoring. Ramesh Hariharan

Polynomial Factoring. Ramesh Hariharan Polynomial Factoring Ramesh Hariharan The Problem Factoring Polynomials overs Integers Factorization is unique (why?) (x^2 + 5x +6) (x+2)(x+3) Time: Polynomial in degree A Related Problem Factoring Integers

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

Factorization Algorithms for Polynomials over Finite Fields

Factorization Algorithms for Polynomials over Finite Fields Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 2011-05-03 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is

More information

Monogenic Fields and Power Bases Michael Decker 12/07/07

Monogenic Fields and Power Bases Michael Decker 12/07/07 Monogenic Fields and Power Bases Michael Decker 12/07/07 1 Introduction Let K be a number field of degree k and O K its ring of integers Then considering O K as a Z-module, the nicest possible case is

More information

15. Symmetric polynomials

15. Symmetric polynomials 15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.

More information

Chapter 1 A Pri Characterization of T m e Pairs w in Proceedings NCUR V. (1991), Vol. I, pp. 362{366. Jerey F. Gold Department of Mathematics, Department of Physics University of Utah DonH.Tucker Department

More information

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

More information

Prime numbers and prime polynomials. Paul Pollack Dartmouth College

Prime numbers and prime polynomials. Paul Pollack Dartmouth College Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

LINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS

LINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS LINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS LESLIE D. GATES, JR.1 1. Introduction. The main result of this paper is an extension of the analysis of Laurent Schwartz [l ] concerning the primitive of

More information

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 [email protected] March 16, 2011 Abstract We give

More information

A SIMPLE PROCEDURE FOR EXTRACTING QUADRATICS FROM A GIVEN ALGEBRAIC POLYNOMIAL.

A SIMPLE PROCEDURE FOR EXTRACTING QUADRATICS FROM A GIVEN ALGEBRAIC POLYNOMIAL. A SIMPLE PROCEDURE FOR EXTRACTING QUADRATICS FROM A GIVEN ALGEBRAIC POLYNOMIAL. S.N.SIVANANDAM Professor and Head: Department of CSE PSG College of Technology Coimbatore, TamilNadu, India 641 004. [email protected],

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

More information

FACTORING SPARSE POLYNOMIALS

FACTORING SPARSE POLYNOMIALS FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

SOLVING POLYNOMIAL EQUATIONS BY RADICALS

SOLVING POLYNOMIAL EQUATIONS BY RADICALS SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang De-Qi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,

More information

Introduction to Finite Fields (cont.)

Introduction to Finite Fields (cont.) Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

More information

calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0,

calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0, Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials

More information

A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number

A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and

More information

3.2 The Factor Theorem and The Remainder Theorem

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

More information

Influences in low-degree polynomials

Influences in low-degree polynomials Influences in low-degree polynomials Artūrs Bačkurs December 12, 2012 1 Introduction In 3] it is conjectured that every bounded real polynomial has a highly influential variable The conjecture is known

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao [email protected]

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao [email protected] We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

More information

The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 2012 The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems

On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University

More information

2 Primality and Compositeness Tests

2 Primality and Compositeness Tests Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 1635-1642 On Factoring R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 http://www.math.ucalgary.ca/

More information

Principle of Data Reduction

Principle of Data Reduction Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

More information

3 Contour integrals and Cauchy s Theorem

3 Contour integrals and Cauchy s Theorem 3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

More information

The cyclotomic polynomials

The cyclotomic polynomials The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =

More information

STAT 830 Convergence in Distribution

STAT 830 Convergence in Distribution STAT 830 Convergence in Distribution Richard Lockhart Simon Fraser University STAT 830 Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 Fall 2011 1 / 31

More information

10 Splitting Fields. 2. The splitting field for x 3 2 over Q is Q( 3 2,ω), where ω is a primitive third root of 1 in C. Thus, since ω = 1+ 3

10 Splitting Fields. 2. The splitting field for x 3 2 over Q is Q( 3 2,ω), where ω is a primitive third root of 1 in C. Thus, since ω = 1+ 3 10 Splitting Fields We have seen how to construct a field K F such that K contains a root α of a given (irreducible) polynomial p(x) F [x], namely K = F [x]/(p(x)). We can extendthe procedure to build

More information

Solutions to Homework 6

Solutions to Homework 6 Solutions to Homework 6 Debasish Das EECS Department, Northwestern University [email protected] 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation. Lectures INF2320 p. 1/88 The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

8 Polynomials Worksheet

8 Polynomials Worksheet 8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1

Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1 Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms

More information

CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015

CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis. Linda Shapiro Winter 2015 CSE373: Data Structures and Algorithms Lecture 3: Math Review; Algorithm Analysis Linda Shapiro Today Registration should be done. Homework 1 due 11:59 pm next Wednesday, January 14 Review math essential

More information

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

More information

SECTION 10-2 Mathematical Induction

SECTION 10-2 Mathematical Induction 73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

More information

Why? A central concept in Computer Science. Algorithms are ubiquitous.

Why? A central concept in Computer Science. Algorithms are ubiquitous. Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

More information

The Greatest Common Factor; Factoring by Grouping

The Greatest Common Factor; Factoring by Grouping 296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

More information

Derive 5: The Easiest... Just Got Better!

Derive 5: The Easiest... Just Got Better! Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; [email protected] 1. Introduction Engineering

More information

OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem")

OPTIMAL SELECTION BASED ON RELATIVE RANK* (the Secretary Problem) OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem") BY Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT n rankable persons appear sequentially in random order. At the ith

More information

Erdős on polynomials

Erdős on polynomials Erdős on polynomials Vilmos Totik University of Szeged and University of South Florida [email protected] Vilmos Totik (SZTE and USF) Polynomials 1 / * Erdős on polynomials Vilmos Totik (SZTE and USF)

More information

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

More information

FACTORING. n = 2 25 + 1. fall in the arithmetic sequence

FACTORING. n = 2 25 + 1. fall in the arithmetic sequence FACTORING The claim that factorization is harder than primality testing (or primality certification) is not currently substantiated rigorously. As some sort of backward evidence that factoring is hard,

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the

More information

Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15. 9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

More information

Factorization in Polynomial Rings

Factorization in Polynomial Rings Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18 of Gallian s Contemporary Abstract Algebra. Most of the important

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 425, PRACTICE FINAL EXAM SOLUTIONS. MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either

More information

Analysis of Binary Search algorithm and Selection Sort algorithm

Analysis of Binary Search algorithm and Selection Sort algorithm Analysis of Binary Search algorithm and Selection Sort algorithm In this section we shall take up two representative problems in computer science, work out the algorithms based on the best strategy to

More information

AMBIGUOUS CLASSES IN QUADRATIC FIELDS

AMBIGUOUS CLASSES IN QUADRATIC FIELDS MATHEMATICS OF COMPUTATION VOLUME, NUMBER 0 JULY 99, PAGES -0 AMBIGUOUS CLASSES IN QUADRATIC FIELDS R. A. MOLLIN Dedicated to the memory ofd. H. Lehmer Abstract. We provide sufficient conditions for the

More information

Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem

Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem February 21, 214 In many problems, you are asked to show that something exists, but are not required to give a specific example or formula

More information

SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

More information

TOPPER Sample Paper - I. Class : XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100

TOPPER Sample Paper - I. Class : XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100 TOPPER Sample Paper - I Class : XI MATHEMATICS Questions Time Allowed : 3 Hrs Maximum Marks: 100 1. All questions are compulsory.. The question paper consist of 9 questions divided into three sections

More information

RESULTANT AND DISCRIMINANT OF POLYNOMIALS

RESULTANT AND DISCRIMINANT OF POLYNOMIALS RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results

More information

Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field

Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field 1. Throughout this section, F is a field and F [x] is the ring of polynomials with coefficients in F. We will

More information

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

More information

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

More information

Roots of Polynomials

Roots of Polynomials Roots of Polynomials (Com S 477/577 Notes) Yan-Bin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x

More information

2 Polynomials over a field

2 Polynomials over a field 2 Polynomials over a field A polynomial over a field F is a sequence (a 0, a 1, a 2,, a n, ) where a i F i with a i = 0 from some point on a i is called the i th coefficient of f We define three special

More information

i/io as compared to 4/10 for players 2 and 3. A "correct" way to play Certainly many interesting games are not solvable by this definition.

i/io as compared to 4/10 for players 2 and 3. A correct way to play Certainly many interesting games are not solvable by this definition. 496 MATHEMATICS: D. GALE PROC. N. A. S. A THEORY OF N-PERSON GAMES WITH PERFECT INFORMA TION* By DAVID GALE BROWN UNIVBRSITY Communicated by J. von Neumann, April 17, 1953 1. Introduction.-The theory of

More information

Factoring Algorithms

Factoring Algorithms Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

Ideal Class Group and Units

Ideal Class Group and Units Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many

More information

Some applications of LLL

Some applications of LLL Some applications of LLL a. Factorization of polynomials As the title Factoring polynomials with rational coefficients of the original paper in which the LLL algorithm was first published (Mathematische

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information