Certain diagonal equations over finite fields

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Certain diagonal equations over finite fields

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Title:
Certain diagonal equations over finite fields
Creator:
Sze, Christopher
Place of Publication:
[Tampa, Fla]
Publisher:
University of South Florida
Publication Date:
Language:
English

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Subjects / Keywords:
Irreducible polynomial
Gaussian sum
Planar function
Hasse-Weil bound
Elliptic curve
Dissertations, Academic -- Mathematics and Statistics -- Masters -- USF ( lcsh )
Genre:
non-fiction ( marcgt )

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Abstract:
ABSTRACT: Let F q to the t be the finite field with q to the t elements and let F q to the t star be its multiplicative group. We study the diagonal equation a times x to the (q minus 1) plus b times y to the (q minus 1) equals c, where a,b and c are elements of F q to the t star. This equation can be written as x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta, where alpha and beta are elements of F q to the t star. Let N sub t (alpha,beta) denote the number of solutions (x,y) in F q to the t star cross F q to the t star of the equation x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta and I(r;a,b) be the number of monic irreducible polynomials f with coefficients in F q of degree r with f(0) equals a and f(1) equals b. We show that N sub t (alpha,beta) can be expressed in terms of I(r;a,b), where r divides t and a,b are elements of F q star are related to alpha and beta. A recursive formula for I(r;a,b) will be given and we illustrate this by computing I(r;a,b) for r greater than or equal to 2 but less than or equal to 4. We also show that N sub 3 (alpha,beta) can be expressed in terms of the number of monic irreducible cubic polynomials over F q with prescribed trace and norm. Connsequently, N sub 3 (alpha,beta) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a,b elements of F q star and integer r greater than or equal to 3, there always exists a monic irreducible polynomial f with coefficients in F q of degree r such that f(0) equals a and f(1) equals b. We also use the result on N sub 2 (alpha,beta) to construct a new family of planar functions.
Thesis:
Thesis (M.A.)--University of South Florida, 2009.
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Includes bibliographical references.
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Title from PDF of title page.
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Document formatted into pages; contains 45 pages.
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by Christopher Sze.

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002063929 ( ALEPH )
558652129 ( OCLC )
E14-SFE0003018 ( USFLDC DOI )
e14.3018 ( USFLDC Handle )

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Certain diagonal equations over finite fields
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ABSTRACT: Let F q to the t be the finite field with q to the t elements and let F q to the t star be its multiplicative group. We study the diagonal equation a times x to the (q minus 1) plus b times y to the (q minus 1) equals c, where a,b and c are elements of F q to the t star. This equation can be written as x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta, where alpha and beta are elements of F q to the t star. Let N sub t (alpha,beta) denote the number of solutions (x,y) in F q to the t star cross F q to the t star of the equation x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta and I(r;a,b) be the number of monic irreducible polynomials f with coefficients in F q of degree r with f(0) equals a and f(1) equals b. We show that N sub t (alpha,beta) can be expressed in terms of I(r;a,b), where r divides t and a,b are elements of F q star are related to alpha and beta. A recursive formula for I(r;a,b) will be given and we illustrate this by computing I(r;a,b) for r greater than or equal to 2 but less than or equal to 4. We also show that N sub 3 (alpha,beta) can be expressed in terms of the number of monic irreducible cubic polynomials over F q with prescribed trace and norm. Connsequently, N sub 3 (alpha,beta) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a,b elements of F q star and integer r greater than or equal to 3, there always exists a monic irreducible polynomial f with coefficients in F q of degree r such that f(0) equals a and f(1) equals b. We also use the result on N sub 2 (alpha,beta) to construct a new family of planar functions.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
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Advisor: Xiang-Dong Hou, Ph.D.
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Irreducible polynomial
Gaussian sum
Planar function
Hasse-Weil bound
Elliptic curve
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CertainDiagonalEquationsoverFiniteFields by ChristopherSze Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematicsandStatistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:Xiang-DongHou,Ph.D. BrianCurtin,Ph.D. StephenSuen,Ph.D. DateofApproval: May29,2009 Keywords:irreduciblepolynomial,Gaussiansum,planarfunction,Hasse-Weil bound,ellipticcurve c Copyright2009,ChristopherSze

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Dedication Tothememoryofmymother VirginiaSze

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TableofContents Abstract.......................................ii 1Introduction...................................1 2PreliminaryResults...............................4 2.1Characters...............................4 2.2GaussianSums............................6 2.3MobiusInversion...........................8 3DiagonalEquations...............................10 4TheMainProblem...............................13 5NumberofIrreduciblePolynomialswithPrescribedValues.........17 5.1ARecursiveFormulafor I r ; a;b ..................17 5.2TheCase r =2............................21 5.3TheCase r =3............................23 5.4TheCase r =4............................26 6 N 3 ; andEllipticCurves..........................30 7Positivityof I t ; a;b ;t 3...........................36 8ApplicationstoPlanarFunctions.......................40 References......................................43 i

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CertainDiagonalEquationsoverFiniteFields ChristopherSze Abstract Let F q t betheniteeldwith q t elementsandlet F q t beitsmultiplicativegroup. Westudythediagonalequation ax q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + by q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = c where a;b;c 2 F q t .Thisequation canbewrittenas x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = ,where ; 2 F q t .Let N t ; denotethenumber ofsolutions x;y 2 F q t F q t of x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = and I r ; a;b bethenumberof monicirreduciblepolynomials f 2 F q [ x ]ofdegree r with f = a and f = b .We showthat N t ; canbeexpressedintermsof I r ; a;b ,where r j t and a;b 2 F q are relatedto and .Arecursiveformulafor I r ; a;b willbegivenandweillustratethis bycomputing I r ; a;b for2 r 4.Wealsoshowthat N 3 ; canbeexpressedin termsofthenumberofmonicirreduciblecubicpolynomialsover F q withprescribed traceandnorm.Consequently, N 3 ; canbeexpressedintermsofthenumberof rationalpointsonacertainellipticcurve.Wegiveaproofthatgivenany a;b 2 F q andinteger r 3,therealwaysexistsamonicirreduciblepolynomial f 2 F q [ x ]of degree r suchthat f = a and f = b .Wealsousetheresulton N 2 ; to constructanewfamilyofplanarfunctions. ii

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1Introduction Let F q betheniteeldwith q elementsandlet t beapositiveinteger.Themultiplicativegroupof F q t isdenotedby F q t .Thepurposeofthisthesisistostudythe numberofsolutions x;y 2 F q t F q t oftheequation ax q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + by q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = c; .1 where a;b;c 2 F q t .Equation.1isequivalentto x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = ; .2 where = b a = c a 2 F q t .Let N t ; denotethenumberofsolutions x;y 2 F q t F q t of.2.Thenumber N t ; isrelatedtothenumberofrationalpoints ontheprojectiveFermatcurve C : x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(z q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =0.3 over F q t .Thenumberofrationalpointson C isgivenby jC F q t j = N t ; + k q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; .4 where k isthenumberofelementsinthemultiset f)]TJ/F27 11.9552 Tf 15.276 0 Td [(;;= g whichare q )]TJ/F19 11.9552 Tf 12.171 0 Td [(1st powersin F q t .Equation.4wasstatedin[23]. Equation.2isaspecialdiagonalequation.Ingeneral,thenumberofsolutions ofadiagonalequationcanbeexpressedintermsofGaussiansumsandestimatesfor 1

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thenumberofsolutionscanbeobtainedthereafter.However,theexactnumberof solutionsofadiagonalequationisnotknownexceptinsomespecialcases.Wolfmann [29]determinedthenumberofsolutionsof a 1 x d 1 + + a s x d s = b over F p 2 m where d isaspecial"divisorof p 2 m )]TJ/F19 11.9552 Tf 12.017 0 Td [(1,meaningthat d j p r +1forsome r j m .Assume q t = p 2 m i.e., t j 2 m and q = p 2 m t .Then p r +1 ;q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1= 8 > > > > > > < > > > > > > : p r; 2 m t +1if 2 2 m t > 2 r ; 2if 2 2 m t 2 r and p> 2 ; 1if 2 2 m t 2 r and p =2 ; where 2 isthe2-adicorder;see[5,Lemma2.6]and[17,Lemma5.3].Thus, q )]TJ/F19 11.9552 Tf 12.377 0 Td [(1 isnotaspecialdivisorof p 2 m )]TJ/F19 11.9552 Tf 12.713 0 Td [(1exceptwhen q =2 ; 2 2 or3.Hence,ingeneral, equation.2isnotcoveredtheresultof[29]. Thefocusofthisthesisisthenumber N t ; .Let I r ; a;b denotethenumberof monicirreduciblepolynomials f 2 F q [ x ]ofdegree r suchthat f = a and f = b Weshallseethat N t ; canbeexpressedintermsof I r ; a;b where r j t and a;b 2 F q arerelatedto and .Thisreducesourproblemtonding I r ; a;b .The problemofcountingmonicirreduciblepolynomialswithprescribedvaluesresembles thatofcountingmonicirreduciblepolynomialswithprescribedcoecients;the latterisawellstudiedtopicinniteelds,seeforexample[4,12,13,25,28,30], buttheformer,toourknowledge,hasnotattractedmuchattention.Herearisesa naturalquestion:Is I r ; a;b alwayspositive?Namely,given r> 0and a;b 2 F q doestherealwaysexistamonicirreduciblepolynomial f 2 F q [ x ]suchthat f = a and f = b ?Theanswerisobviouslynegativefor r =1 ; 2,andisobviouslypositive for r =3.Weareabletoprovethat I r ; a;b > 0forall r 4and a;b 2 F q InChapter2wewillpresentpreliminaryresultsonGaussiansumsandMobius 2

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Inversion;thesearethebasictoolsofourinvestigation.InChapter3welookatthe diagonalequationsingeneralandgivethenumberofsolutionsintermsofGaussian sums.InChapter4,weconsiderourmainproblem,andweshallexpress N t ; in termsof I r ; a;b .Wewillgivearecursiveformulafor I r ; a;b andcomputations of I r ; a;b forsmallvaluesof r inChapter5.InChapter6,wewillderiveanother formulafor N 3 ; usingadierentperspective.Thisnewformulaallowsusto relate N 3 ; tothenumberofirreduciblecubicsover F q withprescribedtraceand normandfurtherallowsustorelate N 3 ; toacertainellipticcurve.InChapter 7,wegiveaproofthatassertsthepositivityof I r ; a;b ,for t 3.Inthelastchapter wewilldiscusssomeapplicationtoplanarfunctionswhicharealsoknownasperfect linearfunctions.Wewillusetheresulton N 2 ; toconstructanewfamilyof planarfunctions. 3

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2PreliminaryResults 2.1Characters Let G beaniteabeliangroupwrittenmultiplicatively.A character of G isamap from G intothemultiplicativegroupofcomplexnumbersofabsolutevalue1such that g 1 g 2 = g 1 g 2 forall g 1 ;g 2 2 G: Equivalently,acharacterofaniteabeliangroup G isahomomorphism : G C If1 G istheidentityelementin G ,then G =1.If g 2 G ,then g isa j G j throot ofunityand g )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = g )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = g ,wherethebardenotescomplexconjugation. Foranyniteabeliangroup G ,wehavethe trivial character 0 denedby 0 g = 1forall g 2 G: Foreachcharacter of G; thereisassociatedthe conjugatecharacter denedby g = g forall g 2 G .Giventhecharacters 1 ;:::; n ,wedene theproduct 1 n by 1 n g = 1 g n g : Theset G ^ ofcharactersof G formsanabeliangroupundermultiplicationofcharactersand j G j = j G ^ j .Infact, G = G ^ althoughtheisomoprhismisnotcanonical. Let F q betheniteeldwith q elements.Then F q and F q areniteabeliangroups underadditionandmultiplication,respectively.Considerrsttheadditivegroupof F q .Let q = p n ,where p isaprime.LetTr F q = F p : F q F p betheabsolutetrace functionfrom F q to F p .Thenthe canonicaladditivecharacter of F q ,denotedby 1 isgivenby 1 c = e 2 i Tr F q = F p c =p forall c 2 F q : .1 4

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Foreach b 2 F q ,thefunction b denedby b c = 1 bc forall c 2 F q isalsoanadditivecharacterof F q andalladditivecharactersof F q arefoundinthis manner.Now,letusconsiderthemultiplicativegroup F q of F q .Thecharactersof F q arecalledthe multiplicativecharacters of F q .Let g beaxedprimitiveelementof F q .Thenforeach j =0 ; 1 ;:::;q )]TJ/F19 11.9552 Tf 11.955 0 Td [(2,thefunction j denedby j g k = e 2 ijk= q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 for k =0 ; 1 ;:::;q )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 isamultiplicativecharacterof F q andallmultiplicativecharactersof F q areobtained inthisway.Furthermore,thesetofallmultiplicativecharactersof F q formsacyclic groupoforder q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1. Let q beoddand bethefunctionon F q denedby c = 8 > < > : 1if c isasquarein F q ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1otherwise. Then isamultiplicativecharacterof F q calledthe quadraticcharacter of F q .For convenience,wedene =0 : Wehavethefollowingidentitiesinvolvingtheadditiveandmultiplicativecharactersof F q .If a and b areadditivecharactersof F q wehave X c 2 F q a c b c = 8 > < > : 0if a 6 = b; q if a = b: Inparticular, X c 2 F q a c =0for a 6 =0 : 5

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Moreover,if c;d 2 F q then X b 2 F q b c b d = 8 > < > : 0if c 6 = d; q if c = d: .2 Formultiplicativecharacters and of F q wehave X c 2 F q c c = 8 > < > : 0if 6 = ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if = : Inparticular, X c 2 F q c =0for 6 = 0 : .3 Furthermore,if c;d 2 F q then X c d = 8 > < > : 0if c 6 = d; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if c = d; .4 wherethesumisoverallmultiplicativecharacters of F q Charactersareusedtondexpressionsforthenumberofsolutionsofequations inaniteabeliangroup G .Let f x 1 ;:::;x n = b beanequationin n indeterminates over G .Let N b bethenumberof x 1 ;:::;x n 2 G n suchthat f x 1 ;:::;x n = b Then N b = 1 j G j X x 1 2 G ::: X x n 2 G X 2 G ^ f x 1 ;:::;x n b : .5 2.2GaussianSums Let beamultiplicativeand beanadditivecharacterof F q .The Gaussiansum G ; isdenedby G ; = X c 2 F q c c : 6

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Let 0 and 0 bethetrivialadditiveandmultiplicativecharactersof F q respectively. TheGaussiansum G ; satises G ; = 8 > > > > < > > > > : q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if = 0 ; = 0 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1if = 0 ; 6 = 0 ; 0if 6 = 0 ; = 0 ; .6 and j G ; j = q 1 = 2 if 6 = 0 ; 6 = 0 : .7 TheGaussiansumsfortheniteeld F q alsohavethefollowingproperties: i G ; ab = a G ; b for a 2 F q ;b 2 F q ; ii G ; = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 G ; ; iii G ; = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 G ; ; iv G ; G ; = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 q for 6 = 0 and 6 = 0 ; v G p ; b = G ; b for b 2 F q ,where p isthecharacteristicof F q and b = b p : Let beamultiplicativecharacterof F q .By.2wehave,forany c 2 F q c = 1 q X d 2 F q d X b 2 F q b c b d = 1 q X b 2 F q b c X d 2 F q d b d = 1 q X G ; c ; wherethelastsumisextendedoveralladditivecharacters of F q .Similarly,if is 7

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anadditivecharacterof F q ,thenby.4,weget,forany c 2 F q c = 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 X d 2 F q d X c d = 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 X c X d 2 F q d d = 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 X G ; c ; .8 wherethesumisextendedoverallmultiplicativecharacters of F q 2.3MobiusInversion A partiallyorderedset S; isanorderedpairconsistingofaset S andabinary relation on S thatisreexive,transitiveandanti-symmetric.An interval ofa partiallyorderedset S; isgivenby[ x;y ]= f z 2 S : x z y g : Wesaythata partiallyorderedsetis locallynite ifeveryintervalhasanitenumberofelements. Let S; bealocallynitepartiallyorderedset.The Mobiusfunction of S; isanintegervaluedfunctionoftwovariableson S denedby x;y =0if x y; andby X z 2 [ x;y ] x;z = x;y if x y; where istheKroneckerdeltafunction. Theorem2.1 MobiusInversionFormula[1] Let S; bealocallynitepartially orderedsetwithMobiusfunction .Let A beanabeliangroupand N = : S A bea function.Let l;m 2 S bexedandfor x 2 S dene N x = X y 2 [ x;m ] N = y 8

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and N x = X y 2 [ l;x ] N = y : Then N = x = X y 2 [ x;m ] x;y N y forall x 2 S with x m and N = x = X y 2 [ l;x ] y;x N y forall x 2 S with x l: Example2.2. [ ClassicalMobiusfunction ]Let Z + bethesetofallpositiveintegers. Then Z + ; j isalocallynitepartiallyorderedset,where x j y means x divides y TheMobiusfunctionisgivenby x;y = y x = 8 > > > > > < > > > > > : 1if y x =1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 k if y x isaproductof k distinctprimes ; 0if y x isdivisiblebyasquareofaprime. Example2.3. [ Partitionsofaset [1]]Let S n beanitesetconsistingof n elements. Let f 1 ; 2 ;::: g beapartitionof S n intosubsetsof S n .Thesets i arecalled blocks ofthepartition.Let P bethesetofallpartitionsof S n andlet ; 2P : Wewrite tomeanthat isarenementof .Then P ; isalocallynitepartially orderedset.ThentheMobiusfunctionisgivenby ; = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 r )]TJ/F28 7.9701 Tf 6.586 0 Td [(r r Y i =1 n i )]TJ/F19 11.9552 Tf 11.955 0 Td [(1!.9 where r denotesthenumberofblocksof andthe i thblockof forsomexed orderistheunionofexactly n i blocksof 9

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3DiagonalEquations A diagonalequation over F q isanequationoftheform a 1 x k 1 1 + ::: + a n x k n n = b; .1 where k 1 ;:::;k n arepositiveintegers, a 1 ;:::;a n 2 F q and b 2 F q .Inthischapter,we willuseGaussiansumstoexpressthenumberofsolutionsofdiagonalequations. Let N bethenumberofsolutionsof.1in F n q : By.5wehave N = 1 q X c 1 ;:::;c n 2 F q X a 1 c k 1 1 + + a n c k n n b ; where runsthroughalltheadditivecharacterof F q .Rearrangingandseparating thetrivialcharacter 0 ,weget N = 1 q X s 2 F q s b X c 1 ;:::;c n 2 F q s a 1 c k 1 1 s a n c k n n = 1 q q n + 1 q X s 2 F q s b X c 1 ;:::;c n 2 F q s a 1 c k 1 1 s a n c k n n = q n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + 1 q X s 2 F q s b 0 @ X c 1 2 F q a 1 s c k 1 1 1 A 0 @ X c n 2 F q a n s c k n n 1 A : Welookatthesum X c i 2 F q a i s c k i i .By2.8, a i s c k i i = 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 X G ; a i s c k i i ; 10

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wherethesumisoverallmultiplicativecharacters of F q .Wehave X c i 2 F q a i s c k i i =1+ X c i 2 F q a i s c k i i =1+ 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 X c i 2 F q X G ; a i s c k i i =1+ 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 X G ; a i s X c i 2 F q k i c i : By.3, X c i 2 F q k i c i = 8 > < > : q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if k i = 0 ; 0if k i 6 = 0 ; where 0 isthetrivialmultiplicativecharacterof F q .Nowlet d i =gcd k i ;q )]TJ/F19 11.9552 Tf 12.614 0 Td [(1 : Then k i istrivialifandonlyif o j d i ,where o istheorderof .Let i bea multiplicativecharacteroforder d i .Since i isoforder d i ,thenthecharacterswhose orderdivides d i areexactlygivenby i j i ,for j i =0 ; 1 ;:::;d i )]TJ/F19 11.9552 Tf 11.956 0 Td [(1.Hence, X c i 2 F q a i s c k i i =1+ 1 q )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 d i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j i =0 G j i i ; a i s X c i 2 F q i j i c i =1+ d i )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X j i =0 G j i i ; a i s Finally,by.6andpropertyiofGaussiansumsweget X c i 2 F q a i s c k i i = d i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j i =1 G j i i ; a i s = d i )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j i =1 i j i a i G j i i ; s 11

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Therefore, N = q n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + 1 q X s 2 F q s b d 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j 1 =1 1 j 1 a 1 G j 1 1 ; s d n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j n =1 n j n a n G j n n ; s = q n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + 1 q d 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j 1 =1 d n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X j n =1 1 j 1 a 1 ::: n j n a n X s 2 F q s b G j 1 1 ; s :::G j n n ; s : Fortheinnersum,wehave X s 2 F q s b G j 1 1 ; s :::G j n n ; s = G j 1 1 ; 1 G j n n ; 1 X s 2 F q b a 1 j 1 a n j n a = G j 1 1 ; 1 G j n n ; 1 G 1 j 1 n j n ; b : Thus, N = q n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + 1 q d 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j 1 =1 d n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X j n =1 1 j 1 a 1 G j 1 1 ; 1 n j n a n G j n n ; 1 G 1 j 1 n j n ; b = q n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + 1 q d 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 X j 1 =1 d n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 X j n =1 G j 1 1 ; a 1 G j n n ; a n G 1 j 1 n j n ; b : .2 12

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4TheMainProblem Let F q betheniteeldwith q elementsandlet t beapositiveinteger.Considerthe equation x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = ; .1 where ; 2 F q t .Wewanttoknowthenumberofsolutions x;y 2 F q t F q t of .1.Let N t ; = f x;y 2 F q t F q t : x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = g andfor a;b 2 F q and r 1,let I r ; a;b = jf f 2 F q [ x ]: f monic,irr.deg f = r;f = a;f = b gj : Wegiveaformulafor N t ; intermsof I r ; a;b where r j t and a;b 2 F q are relatedto and .Foranyinteger s ,let F s q t bethegroupdenedby F s q t = f x s : x 2 F q t g : Wedenotethenormfunctionfrom F q t to F q byN F q t = F q Theorem4.1. For ; 2 F q t N t ; = q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 X r j t ; 2 F q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 ;t=r q t r X a;b 2 F q a t=r =N F q t = F q b t=r =N F q t = F q I r ; a;b : 13

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Proof. Put X = f x;y 2 F q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q t F q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q t : x + y = g .Thenwehave N t ; = q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 jXj : .2 Let U = f u 2 F q t :N F q t = F q u =N F q t = F q ; N F q t = F q u +1=N F q t = F q g : Weclaimthatthemapping : X)167(!U x;y 7)167(! y x isabijection.Let x + y = : Then N F q t = F q y x = y x q t )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = q t )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 =N F q t = F q : Similarly, N F q t = F q y x +1 =N F q t = F q x =N F q t = F q : Thisshowsthat iswell-dened.Let x 1 + y 1 = = x 2 + y 2 and y 1 x 1 = y 2 x 2 ; then clearly x 1 ;y 1 = x 2 ;y 2 : Thus isone-to-one.Toshowthat isonto,let u 2U Then 1+ u ; u + u 2 F q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q t since 1+ u q t )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = N F q t = F q N F q t = F q + u =1 and u + u q t )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = N F q t = F q N F q t = F q u N F q t = F q N F q t = F q + u =1 : Furthermore, 1+ u + u + u = : 14

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Thereforewehaveshownthat isabijectionwithinverse )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 : U)167(!X u 7)167(! 1+ u 1 ; u : Hence, jXj = jUj : .3 Let U r = f u 2U :[ F q u : F q ]= r g : Then jUj = P r j t jU r j .Let u 2 F q t suchthat [ F q u : F q ]= r andlet f 2 F q [ x ]betheminimalpolynomialof u over F q .Wehave N F q t = F q u =N F q r = F q N F q t = F q r u =N F q r = F q u t=r =[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 r f ] t=r : Similarly,since f x )]TJ/F19 11.9552 Tf 11.955 0 Td [(1istheminimalpolynomialof u +1over F q wehave N F q t = F q u =[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 r f )]TJ/F19 11.9552 Tf 9.299 0 Td [(1] t=r : Let I r = f f 2 F q [ x ]: f monic,irr.deg f = r;f t=r =N F q t = F q ;f t=r =N F q t = F q g : .4 Thenitisclearthatthemapping U r )167(!I r u 7)167(! )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 r f )]TJ/F27 11.9552 Tf 9.299 0 Td [(x ; where f istheminimalpolynomialof u over F q ,isontoand r -to-1.So jU r j = r jI r j From.4weseethat I r = ; unlessN F q t = F q ; N F q t = F q 2 F t=r q Werstclaimthat F t=r q = F q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;t=r q ,where q )]TJ/F19 11.9552 Tf 10.14 0 Td [(1 ;t=r =gcd q )]TJ/F19 11.9552 Tf 10.14 0 Td [(1 ;t=r .Clearly, F t=r q F q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ;t=r q .If 2 F q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ;t=r q ,then = x a q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1+ b t=r forsome x 2 F q and 15

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integers a;b .Andso = x b t=r 2 F t=r q .Next,weshowthatN F q t = F q 2 F t=r q if andonlyif 2 F q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;t=r q t : N F q t = F q 2 F t=r q q t )]TJ/F26 5.9776 Tf 5.757 0 Td [(1 q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 2 F t=r q = F q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;t=r q q t )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 t=r;q )]TJ/F26 5.9776 Tf 5.757 0 Td [(1 =1 q t )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 t=r;q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 =1 2 F q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ;t=r q t : Therefore, I r = ; unless ; 2 F q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ;t=r q t .Andsoweget jUj = X r j t jU r j = X r j t ; 2 F q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 ;t=r q t r jI r j = X r j t ; 2 F q )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 ;t=r q t r X a;b 2 F q a t=r =N F q t = F q b t=r =N F q t = F q I r ; a;b : .5 Theconclusionfollowsfrom.2,.3,and.5. 16

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5NumberofIrreduciblePolynomialswithPrescribedValues Inthischapter,wegivearecursiveformulafor I r ; a;b .Wealsogiveexplicitformulas for I r ; a;b for r =2 ; 3 ; 4. 5.1ARecursiveFormulafor I r ; a;b Forinteger r> 0and a;b 2 F q ,let I r ; a;b = f f 2 F q [ x ]: f monic,irr.deg f = r;f = a;f = b g : So I r ; a;b = jI r ; a;b j : If f 2I ; a;b ; then f = b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a x + a .So I ; a;b = 8 > < > : 1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a =1 ; 0otherwise. .1 Forinteger i> 0 ; 0and a = a;b 2 F q F q ,let I i ; a = f f 1 f : f 1 ;:::;f 2 F q [ x ]monic,irr.ofdeg i; f 1 f = a; f 1 f = b g 17

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andlet I i ; a = jI i ; a j : Wedene I 0 i ; a =1,andwrite I 1 i ; a = I i ; a .We have q r )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 = f 2 F q [ x ]: f monicofdeg r f = a f = b = X 1 1 + + r r = r X a 1 ;:::; a r 2 F q F q a 1 a r = a;b r Y i =1 I i i ; a i : Andso I r ; a;b = q r )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 )]TJ/F32 11.9552 Tf 45.405 11.357 Td [(X 1 1 + + r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 r )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 = r X a 1 ;:::; a r )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 2 F q F q a 1 a r )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 = a;b r )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Y i =1 I i i ; a i : .2 InthefollowingLemma,wewillexpress I i ; a intermsof I i ; a 0 where a 0 2 F q F q A partitionofaninteger 0isasequenceofintegers = 1 ;:::; k suchthat 1 k 1and 1 + + k = .Wewrite ` tomeanthat isapartition of .For = 1 ;:::; k ` ,let n s = jf j : j = s gj ; 1 s : Lemma5.1. For i> 0 0 and a 2 F q F q ,wehave I i ; a = X = 1 ;:::; k ` 1 n 1 n X a 1 ;:::; a k 2 F q F q distinct a 1 1 a k k = a k Y j =1 I i ; a j + j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j : .3 Proof. Sincetheelementsof I i ; a areproductsof irreduciblepolynomials,we partition I i ; a bylookingattheimagesof0and1undereachirreduciblefactorand grouptheelementsof I i ; a havingthesamesetofimages,countingmultiplicities. 18

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Soforeach = 1 ;:::; k ` ,let I i ; a = f 1 f : 9 a 1 ;:::; a k 2 F q F q distinctsuchthat a 1 1 a k k = a and f s 2I i ; a j for 1 + + j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1
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WiththisLemma,.2becomesarecursiveformulafor I r ; a;b .However,inthe innersumof.3,therequirementthat a 1 ;:::; a k bedistinctisdiculttoimplement inactualcomputation.WeshalluseaMobiusinversiontowaivethisrequirement. Fix = 1 ;:::; k ` andlet A = f a 1 ;:::; a k : a 1 ;:::; a k 2 F q F q ; a 1 1 a k k = a g : Foreach a 1 ;:::; a k 2A wedeneanequivalencerelationontheset f 1 ;:::;k g as follows: i j ifandonlyif a i = a j .Wedenotetheinducedpartitionon f 1 ;:::;k g by a 1 ;:::; a k .Let P k bethesetofallpartitionsof f 1 ;:::;k g .For ; 2P k ,we write tomeanthat isarenementof .Then P k ; isapartiallyordered setwhosesmallestelementis 0 = f 1 g ;:::; f k g .For 2P k ,put A = f a 1 ;:::; a k 2A : a 1 ;:::; a k = g : Let 1 ;:::; l betheblocksof 2P k .Wehave X X a 1 ;:::; a k 2A jJ a 1 ;:::; a k j = X a 1 ;:::; a k 2A a 1 ;:::; a k jJ a 1 ;:::; a k j = X a 1 ;:::; a l 2 F q F q a P j 1 2 1 j 1 1 a P j l 2 l j l l = a l Y s =1 Y j 2 s I i ; a s + j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j : BytheMobiusinversionformula, X a 1 ;:::; a k 2A 0 jJ a 1 ;:::; a k j = X = f 1 ;:::; l g2P k X a 1 ;:::; a l 2 F q F q a P j 1 2 1 j 1 1 a P j l 2 l j l l = a l Y s =1 Y j 2 s I i ; a s + j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j ; 20

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where = 0 ; and istheMobiusfunctionof P k ; : By.9, = 0 ; = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 k )]TJ/F28 7.9701 Tf 6.587 0 Td [(l l Y s =1 j s j)]TJ/F19 11.9552 Tf 17.933 0 Td [(1! : But A 0 = f a 1 ;:::; a k : a 1 ;:::; a k 2 F q F q distinct ; a 1 1 a k k = a g : Andsoby.5, X a 1 ;:::; a k 2A 0 jJ a 1 ;:::; a k j = X a 1 ;:::; a k 2 F q F q distinct a 1 1 a k k = a k Y j =1 I i ; a j + j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j : Hence,wecanwrite.3as I i ; a = X = 1 ;:::; k ` 1 n 1 n X = f 1 ;:::; l g2P k X a 1 ;:::; a l 2 F q F q a P j 1 2 1 j 1 1 a P j l 2 l j l l = a l Y s =1 Y j 2 s I i ; a s + j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j : .7 5.2TheCase r =2 Inthissectionwewillndanexpressionfor I ; a;b andconsequently,for N 2 ; Let f 2 F q [ x ]bemonicwithdeg f =2 ;f = a;f = b .Then f isoftheform f = x 2 + b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 x + a: Wendconditionssuchthat f isirreducible. Werstassumethat q isodd.Then f isirreducibleifandonlyif b )]TJ/F27 11.9552 Tf 11.058 0 Td [(a )]TJ/F19 11.9552 Tf 11.057 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.057 0 Td [(4 a isanonsquarein F q .Let bethequadraticcharacterof F q .Notethatwedene 21

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=0.Then b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a = 8 > < > : 1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a isasquarein F q ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a isanonsquarein F q : Thus, I ; a;b = 8 > > < > > : 1 2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F19 11.9552 Tf 5.48 -9.683 Td [( b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a if b )]TJ/F27 11.9552 Tf 11.956 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a 6 =0 ; 0if b )]TJ/F27 11.9552 Tf 11.956 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a =0 : .8 If q iseven,then f = x 2 + b )]TJ/F27 11.9552 Tf 12.621 0 Td [(a )]TJ/F19 11.9552 Tf 12.621 0 Td [(1 x + a 2 F q [ x ]isreducibleifandonlyif b )]TJ/F27 11.9552 Tf 12.582 0 Td [(a )]TJ/F19 11.9552 Tf 12.582 0 Td [(1=0orthereexists 2 F q suchthat 2 + b )]TJ/F27 11.9552 Tf 12.582 0 Td [(a )]TJ/F19 11.9552 Tf 12.582 0 Td [(1 + a =0.When b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 6 =0, 2 + b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + a =0 )]TJ/F28 7.9701 Tf 17.023 -4.428 Td [( b )]TJ/F28 7.9701 Tf 6.587 0 Td [(a )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 + )]TJ/F28 7.9701 Tf 17.023 -4.428 Td [( b )]TJ/F28 7.9701 Tf 6.587 0 Td [(a )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = a b )]TJ/F28 7.9701 Tf 6.587 0 Td [(a )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 Tr F q = F 2 a b )]TJ/F28 7.9701 Tf 6.587 0 Td [(a )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 =0 ; by[22,Theorem2.25].Therefore f isirreducibleifandonlyif b )]TJ/F27 11.9552 Tf 12.517 0 Td [(a )]TJ/F19 11.9552 Tf 12.517 0 Td [(1 6 =0and Tr F q = F 2 a b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 =1.Nowlet 1 bethecanonicaladditivecharacterof F q Thenby.1, 1 a b )]TJ/F27 11.9552 Tf 11.956 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 Tr F q = F 2 )]TJ/F28 7.9701 Tf 22.559 -4.977 Td [(a b )]TJ/F28 7.9701 Tf 6.586 0 Td [(a )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 = 8 > < > : 1ifTr F q = F 2 a b )]TJ/F28 7.9701 Tf 6.586 0 Td [(a )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 =0 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1ifTr F q = F 2 a b )]TJ/F28 7.9701 Tf 6.586 0 Td [(a )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 =1 : Hence, I ; a;b = 8 > > > < > > > : 1 2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 1 a b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 6 =0 ; 0if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1=0 : .9 WeshalluseTheorem4.1togetherwith.8and.9todetermine N 2 ; 22

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Let a =N F q 2 = F q and b =N F q 2 = F q .ByTheorem4.1, N 2 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 =2 I ; a;b + X a 1 ;b 1 2 F q a 2 1 = a;b 2 1 = b I ; a 1 ;b 1 : Using.1, X a 1 ;b 1 2 F q a 2 1 = a;b 2 1 = b I ; a 1 ;b 1 = a 1 2 F q : a 2 1 = a; a 1 +1 2 = b = 8 > < > : 1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a =0 ; 0otherwise : .10 Combining.10with.8and.9,weget,if q isodd, N 2 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 =1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F19 11.9552 Tf 5.48 -9.683 Td [( b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a : .11 If q iseven, N 2 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = 8 > > > < > > > : 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 1 a b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 6 =0 ; 1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0 : 5.3TheCase r =3 Let f 2 F q [ x ]bemonicwithdeg f =3 ;f = a;f = b .Then f = x 3 + cx 2 + b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F27 11.9552 Tf 11.955 0 Td [(c )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 x + a; 23

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forsome c 2 F q .Now f isirreducibleifandonlyif f x 6 =0forall x 2 F q nf 0 ; 1 g Let V a;b = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 x 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(x 3 + b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 x + a : x 2 F q nf 0 ; 1 g = )]TJ/F27 11.9552 Tf 9.299 0 Td [(x + a x )]TJ/F27 11.9552 Tf 24.221 8.088 Td [(b x )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1: x 2 F q nf 0 ; 1 g : .12 Then f isirreducibleifandonlyif c= 2 V a;b .Therefore I ; a;b = q )-222(j V a;b j : .13 Todetermine N 3 ; ,let a =N F q 3 = F q and b =N F q 3 = F q .ByTheorem4.1 and.13, N 3 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 =3 I ; a;b + X a 1 ;b 1 2 F q a 3 1 = a;b 3 1 = b I ; a 1 ;b 1 =3 )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(q )-222(j V a;b j + a 1 2 F q : a 3 1 = a; a 1 +1 3 = b : .14 Wedetermine a 1 2 F q : a 3 1 = a; a 1 +1 3 = b in.14inthenextlemma. Lemma5.2. Let a;b 2 F q iWhen p 6 =3 a 1 2 F q : a 3 1 = a; a 1 +1 3 = b = 8 > > > > < > > > > : 2 if a =1 b = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 and 3 j q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; 1 if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2 6 =0 and 3 a + b )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 a +2 b +1= b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2 2 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; 0 otherwise : 24

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iiWhen p =3 a 1 2 F q : a 3 1 = a; a 1 +1 3 = b = 8 > < > : 1 if b = a +1 ; 0 if b 6 = a +1 : Proof. iLet a;b 2 F q and A = a 1 2 F q : a 3 1 = a; a 1 +1 3 = b .If a 1 2 A ,then 3 a 2 1 +3 a 1 +1= b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a .So a 2 1 + a 1 +1= 1 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2. If b )]TJ/F27 11.9552 Tf 10.675 0 Td [(a +2=0,then a 3 1 =1.So a =1and b = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1.Itfollowsthat a 1 isaprimitive cuberootofunity.But F q hasaprimitivecuberootofunityifandonlyif3 j q )]TJ/F19 11.9552 Tf 11.964 0 Td [(1. Andsoif a =1 ;b = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1and3 j q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1then j A j =2. If b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2 6 =0,wehave a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1= a 3 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 a 2 1 + a 1 +1 = 3 a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2 : So a 1 = 2 a + b )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2 : If a 1 = 2 a + b )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 b )]TJ/F28 7.9701 Tf 6.586 0 Td [(a +2 ,theequation a 2 1 + a 1 +1= 1 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2becomesequivalentto 3 a + b )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 a +2 b +1= b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a +2 2 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 : .15 Thus,if.15issatisedthen j A j =1. iiObvious. 25

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5.4TheCase r =4 Inthissectionwewillpresentanexplicitformulafor I ; a;b .Using.2wehave I ; a;b = q 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(I 4 ; a;b )]TJ/F32 11.9552 Tf 29.342 11.357 Td [(X a 1 ; a 2 2 F q F q a 1 a 2 = a;b I 2 ; a 1 I ; a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(I 2 ; a;b )]TJ/F32 11.9552 Tf 29.342 11.358 Td [(X a 1 ; a 2 2 F q F q a 1 a 2 = a;b I ; a 1 I ; a 2 : .16 In.16,weshalluse.7tocompute I 2 ; a ;I 2 ; a;b and I 4 ; a;b .Note thatfor i =1 ; 2,then I i ; a =0or1andsothebinomialcoecientin.7is simpliedto I i ; a s + j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j = I i ; a s : Performingthecomputations,weget I 4 ; a;b = 1 24 X a 1 a 4 = a;b I ; a 1 I ; a 2 I ; a 3 I ; a 4 + 1 4 X a 2 1 a 2 a 3 = a;b I ; a 1 I ; a 2 I ; a 3 + 1 8 X a 2 1 a 2 2 = a;b I ; a 1 I ; a 2 + 1 3 X a 3 1 a 2 = a;b I ; a 1 I ; a 2 + 1 4 X a 4 1 = a;b I ; a 1 : .17 Eachsumin.17representsthenumberofsolutionsofarationalequationor somepolynomialequations.Andso.17canbewrittenas I 4 ; a;b = 1 24 n a 1 ;a 2 ;a 3 : a i 2 F q ; + a 1 + a 2 + a 3 1+ a a 1 a 2 a 3 = b o + 1 4 n a 1 ;a 2 2 F q F q :+ a 1 2 + a 2 1+ a a 2 1 a 2 = b o + 1 8 a 1 ;a 2 2 F q F q : a 2 1 a 2 2 = a; + a 1 2 + a 2 2 = b + 1 3 n a 1 2 F q :+ a 1 3 1+ a a 3 1 = b o + 1 4 a 1 2 F q : a 4 1 = a; + a 1 4 = b : .18 26

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Next,for i =1 ; 2 ; a 2 F q F q wehave I 2 i ; a = 1 2 X a 1 a 2 = a I i ; a 1 I i ; a 2 + 1 2 X a 2 1 = a I i ; a 1 : .19 Let i =1and a = a;b in.19.Thenby.10, X a 2 1 = a I ; a 1 = 8 > < > : 1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a =0 ; 0otherwise. Now X a 1 a 2 = a I ; a 1 I ; a 2 = n a 1 2 F q :+ a 1 1+ a a 1 = b o = a 1 2 F q : a 2 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 a 1 + a =0 = 8 > > > > > < > > > > > : 1+ )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [( b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a if q isodd ; 1+ 1 a b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 if q isevenand b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 6 =0 ; 1if q isevenand b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0 : Hence,if q isodd,then I 2 ; a = 8 > < > : 1if )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [( b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a =0or1 ; 0if )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [( b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; andso X a 1 ; a 2 2 F q F q a 1 a 2 = a;b I 2 ; a 1 I ; a 2 = a 1 ;b 1 2 F q F q : )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [( b 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a 1 =0or1; b b 1 )]TJ/F28 7.9701 Tf 15.226 4.707 Td [(a a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a a 1 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 : .20 27

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Nowif q iseven,then I 2 ; a = 8 > > < > > : 0if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 6 =0and 1 a b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1otherwise : andsowehave X a 1 ; a 2 2 F q F q a 1 a 2 = a;b I 2 ; a 1 I ; a 2 = n a 1 ;b 1 2 F q F q : b 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0or 1 a 1 b 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(a 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 =1; 1 a a 1 b b 1 )]TJ/F28 7.9701 Tf 15.227 4.707 Td [(a a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 o : .21 Nowlet i =2and a = a;b in.19.Then I 2 ; a;b = 1 2 X a 1 a 2 = a;b I ; a 1 I ; a 2 + 1 2 X a 2 1 = a;b I ; a 1 : When q isodd,wehave I 2 ; a;b = 1 2 n a 1 ;b 1 2 F q F q : )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [( b 1 )]TJ/F27 11.9552 Tf 9.963 0 Td [(a 1 )]TJ/F19 11.9552 Tf 9.963 0 Td [(1 2 )]TJ/F19 11.9552 Tf 9.962 0 Td [(4 a 1 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; b b 1 )]TJ/F28 7.9701 Tf 13.233 4.707 Td [(a a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a a 1 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 o + 1 2 a 1 ;b 1 2 F q F q : a 2 1 = a;b 2 1 = b; )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [( b 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 a 1 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 : .22 28

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When q iseven,wehave I 2 ; a;b = 1 2 n a 1 ;b 1 2 F q F q : 1 a 1 b 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1 a a 1 b b 1 )]TJ/F28 7.9701 Tf 15.227 4.707 Td [(a a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 o + 8 > > < > > : 1 2 if 1 a 1 b 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 0otherwise : .23 Thelastsumin.16isgivenby X a 1 ; a 2 2 F q F q a 1 a 2 = a;b I ; a 1 I ; a 2 = X a 1 2 F q ;a 1 6 = )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F32 11.9552 Tf 11.955 13.749 Td [( V a a 1 ; b a 1 +1 ; .24 where V a a 1 ; b a 1 +1 isdenedin.12. Now,Equation.16combinedwith.18and.20{.24,isthemostexplicit formulafor I ; a;b thatthismethodcanoer. 29

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6 N 3 ; andEllipticCurves Inarecentpaper[23],Moisiofoundaformulafor N 3 ; intermsofthenumberof rationalpointsonaprojectivecubiccurve.Let a =N F q 3 = F q and b =N F q 3 = F q Let A betheanecubiccurvedenedby A : ax 2 y + axy 2 + x 2 + ay 2 + a +1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(b xy + x + y =0 andlet A betheprojectiveclosureof A .Moisio[23,Theorem2]provedthat N 3 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = j A F q j ; where A F q denotesthesetofrationalpointson A over F q .Inthischapter,wewill deriveanotherformulafor N 3 ; intermsofthenumberofrationalpointsona dierentandsimplerprojectivecubic. Werstdeneafewterms.For a;b 2 F q a 0 2 F q andinteger r 1,let S r a;b = f u 2 F q r :N F q r = F q u = a; N F q r = F q u +1= b g ; T r a 0 ;b = f u 2 F q r :Tr F q r = F q u = a 0 ; N F q r = F q u = b g ; J r ; a 0 ;b = f x r )]TJ/F27 11.9552 Tf 11.956 0 Td [(a 0 x r )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + + )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 r b 2 F q [ x ]isirreducible g : Remark. Let beaprimitiveelementof F q r andassumethat a =N F q r = F q i b =N F q r = F q j .Then j S r a;b j isthe cyclotomicnumber i;j q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 over F q r ;see[22, p.247].CyclotomicnumbersarecloselyrelatedtoJacobisums;see[2, x 11.6]. 30

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Throughoutthischapter,let a =N F q t = F q b =N F q t = F q .By.2and.3, N 3 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = j S 3 a;b j : .1 Ourmethodinthischapterconsistsoftwosteps:Firstweproveapeculiarconnection between S 3 a;b and T 3 b )]TJ/F27 11.9552 Tf 9.392 0 Td [(a )]TJ/F19 11.9552 Tf 9.392 0 Td [(1 ;ab Theorem6.1.ThenweusearesultofMoisio[24] toexpress T 3 b )]TJ/F27 11.9552 Tf 11.851 0 Td [(a )]TJ/F19 11.9552 Tf 11.851 0 Td [(1 ;ab intermsofthenumberofrationalpointsonaprojective cubic. Theorem6.1. Let a;b 2 F q .Themapping : S 3 a;b )167(! T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 ;ab u 7)167(! u + u 1+ q isonto.Moreprecisely,foreach v 2 T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 v j = 8 > < > : q +1 if a =1 and v = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1 otherwise : Proof. Firstweshowthatif u 2 S 3 a;b ,then u + u 1+ q indeedbelongsto T 3 b )]TJ/F27 11.9552 Tf 11.776 0 Td [(a )]TJ/F19 11.9552 Tf -422.702 -20.922 Td [(1 ;ab .Clearly, N F q 3 = F q u + u 1+ q =N F q 3 = F q )]TJ/F27 11.9552 Tf 5.479 -9.683 Td [(u + u q =N F q 3 = F q u N F q 3 = F q + u = ab: Wealsohave b =N F q 3 = F q u +1= u +1 1+ q + q 2 = u 1+ q + q 2 + u 1+ q + u q + q 2 + u q 2 +1 + u 1 + u q + u q 2 +1 =N F q 3 = F q u +1+Tr F q 3 = F q u + u 1+ q = a +1+Tr F q 3 = F q u + u 1+ q : SoTr F q 3 = F q u + u 1+ q = b )]TJ/F27 11.9552 Tf 11.956 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1. 31

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Nowlet 2 F q 3 suchthatN F q 3 = F q = a .Then u 2 F q 3 satisesN F q 3 = F q u = a ifandonlyif u = x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 forsome x 2 F q 3 For v 2 T 3 b )]TJ/F27 11.9552 Tf 12.047 0 Td [(a )]TJ/F19 11.9552 Tf 12.046 0 Td [(1 ;ab ,let u 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 v suchthatN F q 3 = F q u = a .Hencewecan write u = x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ,forsome x 2 F q 3 .Weclaimthat u = x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 v ifandonlyif x 2 F q 3 isasolutionof 1+ q x q 2 + x q )]TJ/F27 11.9552 Tf 11.955 0 Td [(vx =0 : .2 Firstassume x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 v .Then x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 1+ q = v andso x q + 1+ q x q 2 = vx: Next,weassume x 2 F q 3 isasolutionof.2.Thenwehave x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 1+ q = v: Itremainstoshowthat u 2 S 3 a;b .WeonlyneedtoshowthatN F q 3 = F q u +1= b Wehave N F q 3 = F q u +1=N F q 3 = F q u +Tr F q 3 = F q u + u 1+ q +1 = a + b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1+1 = b: Thisprovestheclaim. Thenumberofsolutions x 2 F q 3 of.2is q 3 )]TJ/F25 7.9701 Tf 6.587 0 Td [(rank A ,where A = 2 6 6 6 4 v )]TJ/F27 11.9552 Tf 9.298 0 Td [( )]TJ/F27 11.9552 Tf 9.298 0 Td [( 1+ q )]TJ/F27 11.9552 Tf 9.299 0 Td [( q + q 2 v q )]TJ/F27 11.9552 Tf 9.299 0 Td [( q )]TJ/F27 11.9552 Tf 9.298 0 Td [( q 2 )]TJ/F27 11.9552 Tf 9.299 0 Td [( q 2 +1 v q 2 3 7 7 7 5 ; 32

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see[16,Proposition2.1].Wehave det A = v 1+ q + q 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 1+ q + q 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2+ q + q 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 1+ q + q 2 v 1 + v q + v q 2 =N F q 3 = F q v )]TJ/F19 11.9552 Tf 11.956 0 Td [(N F q 3 = F q )]TJ/F19 11.9552 Tf 11.955 0 Td [(N F q 3 = F q 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(N F q 3 = F q Tr F q 3 = F q v = ab )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 =0 : Sorank A =1or2.Itiseasytoseethat rank A = 8 > < > : 1if a =1and v = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 2otherwise : Hencethenumberof x 2 F q 3 of.2is 8 > < > : q 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1if a =1and v = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1otherwise : .3 Nowsuppose x 2 F q 3 isasolutionof.2.Thenfor 2 F q x isalsoasolution since 1+ q x q 2 + x q )]TJ/F27 11.9552 Tf 11.955 0 Td [(v x = 1+ q x q 2 + x q )]TJ/F27 11.9552 Tf 11.955 0 Td [(vx : Therefore,for v 2 T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 ;ab ,thenumberof u = x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 v is 8 > < > : q +1if a =1and v = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1otherwise : If a =1and v = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 2 T 3 b )]TJ/F27 11.9552 Tf 9.482 0 Td [(a )]TJ/F19 11.9552 Tf 9.482 0 Td [(1 ;ab ,thenTr F q 3 = F q )]TJ/F19 11.9552 Tf 9.299 0 Td [(1= b )]TJ/F19 11.9552 Tf 9.483 0 Td [(2andN F q 3 = F q )]TJ/F19 11.9552 Tf 9.298 0 Td [(1= b ,whichimply b = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1.Wehavethefollowingcorollary. 33

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Corollary6.2. Let a;b 2 F q .Then j S 3 a;b j = 8 > < > : j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j + q if a =1 and b = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j otherwise : .4 Combining.4and.1,wearriveatanewformulafor N 3 ; : N 3 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = 8 > < > : j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j + q if a =1and b = )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j otherwise ; .5 where a =N F q 3 = F q and b =N F q 3 = F q Moisio[24]studiedthenumberofirreduciblepolynomialsoverniteeldswith prescribedtraceandnorm.Thenumber j T 3 b )]TJ/F27 11.9552 Tf 10.302 0 Td [(a )]TJ/F19 11.9552 Tf 10.302 0 Td [(1 ;ab j in.5issubjecttofurther interpretationsbytheresultsof[24]. For c 2 F q ,let B c betheanecubiccurvedenedby B c : y 2 + cy + xy = x 3 andlet B c denotetheprojectiveclosureof B c .ByTheorems3.2and5.1of[24],we have j T 3 b )]TJ/F27 11.9552 Tf 11.956 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j = 8 > > > > < > > > > : B c F q ; where c = ab b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 3 ; if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 6 =0 ; q +1+ 1 q X x 2 F q 3 e x ;q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0 ; .6 where e isthecanonicaladditivecharacterof F q 3 34

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Wecanalsowrite j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j = T 3 b )]TJ/F27 11.9552 Tf 11.956 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab F q 3 n F q + T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab F q =3 J ; b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab + f v 2 F q :3 v = b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;v 3 = ab g = 8 > < > : 3 J ; b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab +1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 3 =27 ab; 3 J ; b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab otherwise : .7 If b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 3 =27 ab andchar F q 6 =3,byCorollary5.2of[24], J ; b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab = 1 3 q +1 ; so j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j =3 1 3 q +1 +1 : Thisisalsotruewhenchar F q =3sincein.6,wehave P x 2 F q 3 e x =0and 3 1 3 q +1 +1= q +1.Thus.7canbemadealittlemoreexplicit: j T 3 b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab j = 8 > < > : 3 b 1 3 q +1 c +1if b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 3 =27 ab; 3 J ; b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ;ab otherwise : .8 By.5and.8weobtainthefollowingformulafor N 3 ; : N 3 ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 = 8 > > > > < > > > > : 3 b 1 3 q +1 c + q +1if a;b =1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 3 b 1 3 q +1 c +1if a;b 6 = ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1and b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 3 =27 ab; 3 J ; b )]TJ/F27 11.9552 Tf 11.955 0 Td [(a )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 ;ab otherwise, where a =N F q 3 = F q and b =N F q 3 = F q 35

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7Positivityof I t ; a;b ;t 3 Inthischapter,wewanttodetermineif I t ; a;b ispositivewith a;b 2 F q andinteger t> 0.Namely,given a b and t ,doesthereexistamonicirreduciblepolynomial f 2 F q [ x ]ofdegree t suchthat f = a and f = b ?By.1,.8and.9,we seethatfor t =1 ; 2,wehave I t ; a;b =0or1dependingoncertainconditionson a and b .When t =3,thenby.13, I ; a;b 2since j V a;b j q )]TJ/F19 11.9552 Tf 10.713 0 Td [(2 : Wewillprove that I t ; a;b > 0for t 4.Ourproofisbasedontherelationbetween I t ; a;b and anestimatefor N t ; Insomesense,thepositivityof I t ; a;b t 3iscomparablewiththeHansenMullenconjectureforirreduciblepolynomials[13]provedbyWan[28]andHamand Mullen[12]whichpostulatesthataprescribeddegreeandoneprescribedcoecient canalwaysbeachievedbyamonicirreduciblepolynomialin F q [ x ]excludingtwo obviousnonattainablecases. Werstlookat N t ; = n x;y 2 F q t F q t : x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 = o ,where ; 2 F q t .Thenumberofsolutionsofadiagonalequationcanbeexpressedintermsof Gaussiansumsandisgivenin.2.Butnotethaninthisexpression,weconsidered allsolutionsin F n q .Fornonzerosolutions,thecomputationissimilar. Wenowconsiderthesolutions x;y 2 F q t F q t ofthediagonalequation x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = .Let 1 bethecanonicaladditivecharacterof F q t andlet beamulti36

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plicativecharacteroforder q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1of F q t .Then N t ; = 1 q t X x;y 2 F q t X s 2 F q t s x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 s = q t )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 q t + 1 q t X s 2 F q t s X x;y 2 F q t s x q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 s y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = q t )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 q t + 1 q t q )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 X j =0 q )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 X k =0 G j ; 1 G k ; G )]TJ/F28 7.9701 Tf 6.586 0 Td [(j )]TJ/F28 7.9701 Tf 6.587 0 Td [(k ; = q t )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 q t + 1 q t q )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 X j =0 q )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 X k =0 j )]TJ/F27 11.9552 Tf 9.299 0 Td [( k )]TJ/F27 11.9552 Tf 10.62 8.088 Td [( G j ; 1 G k ; 1 G )]TJ/F28 7.9701 Tf 6.587 0 Td [(j )]TJ/F28 7.9701 Tf 6.586 0 Td [(k ; 1 : .1 Observethatby.6and.7, j G j ; 1 G k ; 1 G )]TJ/F28 7.9701 Tf 6.586 0 Td [(j )]TJ/F28 7.9701 Tf 6.586 0 Td [(k ; 1 j = 8 > > > > < > > > > : 1if j;k; )]TJ/F27 11.9552 Tf 9.299 0 Td [(j )]TJ/F27 11.9552 Tf 11.955 0 Td [(k areall 0mod q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; q t ifexactlyoneof j;k; )]TJ/F27 11.9552 Tf 9.298 0 Td [(j )]TJ/F27 11.9552 Tf 11.955 0 Td [(k is 0mod q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; q 3 2 t ifnoneof j;k; )]TJ/F27 11.9552 Tf 9.299 0 Td [(j )]TJ/F27 11.9552 Tf 11.955 0 Td [(k is 0mod q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 : Thus N t ; q t )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 2 q t )]TJ/F19 11.9552 Tf 14.812 8.087 Td [(1 q t 1+3 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 q t + q )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 q 3 2 t = q t +4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 q )]TJ/F19 11.9552 Tf 11.955 0 Td [( q 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(5 q +6 q t 2 : Andsowehavethefollowinglemma. Lemma7.1. Let ; 2 F q t .Thenwehave N t ; q t +4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 q )]TJ/F19 11.9552 Tf 11.955 0 Td [( q 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(5 q +6 q t 2 : Remark. Lemma7.1alsofollowsfromtheHasse-Weilbound[27,TheoremV.2.3]. Sincethegenusof C in.3is 1 2 q )]TJ/F19 11.9552 Tf 12.358 0 Td [(2 q )]TJ/F19 11.9552 Tf 12.357 0 Td [(3[11,p.199],theHasse-Weilbound 37

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gives jC F q t j q t +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( q )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 q t 2 .Thusby.4, N t ; jC F q t j)]TJ/F19 11.9552 Tf 17.933 0 Td [(3 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 q t +4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 q )]TJ/F19 11.9552 Tf 11.955 0 Td [( q 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(5 q +6 q t 2 : Wenowprovethepositivityof I t ; a;b for t 4inthenexttheorem. Theorem7.2. For a;b 2 F q and t 4 wehave I t ; a;b > 0 Proof. If q =2,then a = b =1.Everyirreduciblepolynomial f 2 F 2 [ x ]with deg f> 1musthave f =1and f =1.Thus, I t ;1 ; 1 > 0for t 2. Henceforthweassume q 3. Let ; 2 F q t suchthat a =N F q t = F q and b =N F q t = F q .ByTheorem4.1, tI t ; a;b = N t ; q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F32 11.9552 Tf 16.787 11.358 Td [(X r j t;r > < > > : @A @q = t 2 q t 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 q +5 ; @A @t = 1 2 q t 2 ln q )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 t: 38

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Wehave A ; 4=3, A ; 8=17and @A @q > 0 ; @A @t > 0for q 5, t 4or q 3, t 8 : Sowhen q 5, t 4or q 3, t 8,wehave A q;t 3andconsequently tI t ; a;b 1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 q t 2 3+4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 q > 0 : For3 q< 5and4 t< 8,thepositivityof I t ; a;b ischeckeddirectlyusinga computer. 39

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8ApplicationstoPlanarFunctions Afunction f : F q )167(! F q iscalled planar ifforevery u 2 F q x 7)167(! f x + u )]TJ/F27 11.9552 Tf 11.956 0 Td [(f x isapermutationof F q .PlanarfunctionswereintroducedbyDembowskiandOstrom [7]todescribecertainaneplanes.Forfurtherresultsonplanarfunctionsand relatedtopics,see[6],[14],[18],[21].Recently,planarfunctionshavefoundimportant applicationsincryptographywheretheyarecalled perfectnonlinearfunctions ;see [26].Constructionsofperfectnonlinearfunctionsandtheircloserelatives almost perfectnonlinearfunctions havebeenattractingmuchattentionforthepastdecade, see[3],[8],[9],[10],[15],[19]. Observethatplanarfunctionsexistonlywhen q isodd. Lemma8.1. Let p beanoddprimeand n beapositiveinteger.Let f x = x p m +1 + x 2 2 F p n [ x ] ; where m> 0 and 2 F p n : Let t = n m;n and q = p m;n = p n t so q t = p n .Then f isaplanarfunctionon F q t ifandonlyif N t ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 =0 ,i.e.,ifandonlyif x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + y q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 hasnosolution x;y 2 F q t F q t Proof. Let u 2 F q t .Since f u isconstant,then f x + u )]TJ/F27 11.9552 Tf 11.973 0 Td [(f x isapermutationof 40

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F q t ifandonlyif f x + u )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(f u isapermutationof F q t .Wehave f x + u )]TJ/F27 11.9552 Tf 11.955 0 Td [(f x )]TJ/F27 11.9552 Tf 11.955 0 Td [(f u = ux p m + u p m x +2 ux = ux x p m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + u p m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 +2 : Now f x + u )]TJ/F27 11.9552 Tf 12.917 0 Td [(f x )]TJ/F27 11.9552 Tf 12.917 0 Td [(f u isa p -polynomial.By[22,Theorem7.9],itisa permutationof F q t ifandonlyif x =0isitsonlyroot,i.e.,ifandonlyif x p m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + u p m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 6 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 forall x;u 2 F q t : But F p m )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q t = F p m )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ;q t )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 q t = F p m;n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q t .Andso f x + u )]TJ/F27 11.9552 Tf 12.536 0 Td [(f x )]TJ/F27 11.9552 Tf 12.535 0 Td [(f u isa permutationof F q t ifandonlyif x q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 + u q )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 6 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 forall x;u 2 F q t : Therefore, f isaplanarfunctionon F q t ifandonlyif N t ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 =0. InLemma8.1,if t =1,then f x = +1 x 2 on F q ,whichisnotinteresting;if t 3,weknowfromchapter7that N t ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 > 0,soLemma8.1doesnotproduce anyplanarfunction.TheonlyinterestingcaseinthisLemmaiswhen t =2.Let t =2andlet b =N F q 2 = F q .ThenN F q 2 = F q )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 =4 b .By.11wehave N 2 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 =1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F19 11.9552 Tf 5.479 -9.683 Td [( b )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(4 =1 )]TJ/F27 11.9552 Tf 11.955 0 Td [( )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [(b b )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 : CombiningtheaboveequationandLemma8.1,wehavethefollowingproposition. Proposition8.2. Let p beanoddprimeandlet n;m bepositiveintegerssuchthat m;n = n 2 .Put q = p n 2 so p n = q 2 .Let f x = x p m +1 + x 2 2 F q 2 [ x ] ,where 2 F q 2 .Then f isaplanarfunctionon F q 2 ifandonlyif )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [(b b )]TJ/F19 11.9552 Tf 12.409 0 Td [(1 =1 ,where b = N F q 2 = F q 41

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Suppose f satisfytheassumptionsinProposition8.2.Weknowthat f isaplanar functionon F q 2 ifandonlyif N 2 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 =0.Wecountthenumberof sothat f isplanar.Write H = F q )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 q 2 .Then f 0 2 F q 2 : N 2 ; 0 > 0 g = F q 2 H + H = H + H nf)]TJ/F19 11.9552 Tf 21.254 0 Td [(1 g : Let q beoddand x;y 2 H nf)]TJ/F19 11.9552 Tf 21.254 0 Td [(1 g .Wehave 1+ y 1+ x 2 H 1+ y 1+ x q +1 =1 + y q +1 =+ x q +1 + y + y q =1+ x + x q 2+ y + y q =2+ x + x q y + y )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = x + x )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 y )]TJ/F27 11.9552 Tf 11.955 0 Td [(x xy )]TJ/F19 11.9552 Tf 11.955 0 Td [(1=0 y = x or y = x )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : Therefore,if x 6 =1,thentherearepreciselytwo y 2 H y = x or x )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 suchthat H + x = H + y .If x =1,thenthereisexactlyone y 2 H y = x suchthat H + x = H + y .Thus, j H + H nf)]TJ/F19 11.9552 Tf 21.253 0 Td [(1 g j = j H j 1 2 j H j)]TJ/F19 11.9552 Tf 17.933 0 Td [(2+1 = 1 2 j H j 2 = 1 2 q +1 2 : Hencethenumberof inProposition8.2sothat f isaplanarfunctionis q 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 13.15 8.088 Td [(1 2 q +1 2 = 1 2 q +1 q )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 : 42

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References [1]E.A.BenderandJ.R.Goldman, OntheapplicationsofMobiusinversionincombinatorial analysis ,Amer.Math.Monthly 82 ,789{803. [2]B.C.Berndt,R.J.Evans,K.S.Williams, GaussandJacobiSums ,JohnWiley,NewYork, 1998. [3]L.Budaghyan,C.Carlet,G.Leander, TwoclassesofquadraticAPNbinomialsinequivalentto powerfunctions ,IEEETrans.Inform.Theory 54 ,4218{4229. [4]S.D.Cohen, Primitivepolynomialswithaprescribedcoecient ,FiniteFieldsAppl. 12 425{491. [5]R.S.Coulter, ExplicitevaluationsofsomeWeilsums ,ActaArith. 83 ,241{251. [6]R.S.Coulter,R.W.Matthews, PlanarfunctionsandplanesofLenz-BarlotticlassII ,Des. CodesCryptogr. 10 ,167{184. [7]P.DembowskiandT.G.Ostrom, Planesoforder n withcollineationgroupsoforder n 2 ,Math. Z. 103 ,239{258. [8]H.Dobbertin, Almostperfectnonlinearpowerfunctionson GF n :theWelshcase ,IEEE Trans.Infor.Theory 45 ,1271-1275. [9]H.Dobbertin, Almostperfectnonlinearpowerfunctionson GF n :anewcasefor n divisible by5 ,FiniteFieldsandApplications,Springer,Berlin,2001,113-121. [10]T.Helleseth,C.Rong,D.Sandberg, Newfamiliesofalmostperfectnonlinearpowermappings IEEETrans.Inform.Theory 45 ,474-485. [11]W.Fulton, AlgebraicCurves ,Addison-Wesley,ReadingMA,1989. [12]K.H.HamandG.L.Mullen, Distributionofirreduciblepolynomialsofsmalldegreesover niteelds ,Math.Comp. 67 ,337{341. 43

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