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1、May5,200915:25WSPC/203-IJNT00224InternationalJournalofNumberTheoryVol.5,No.3(2009)541–554cWorldScientificPublishingCompanyAMETHODOFESTIMATINGTHEp-ADICSIZESOFCOMMONZEROSOFPARTIALDERIVATIVEPOLYNOMIALSASSOCIATEDWITHAQUINTICFORMS.H.SAPAR∗andK.A.MOHD.ATAN†DepartmentofMathematics,Fa
2、cultyofScienceandLaboratoryofTheoreticalStudiesInstituteforMathematicalResearchUniversitiPutraMalaysia,43400UPMSerdangSelangor,Malaysia∗sitihas@putra.upm.edu.my†kamel@putra.upm.edu.myReceived13December2006Accepted19July2007P2πifItisknownthatthevalueoftheexponentialsumS(f;q)=ex
3、p()canbexmodqqderivedfromtheestimateofthecardinality
4、V
5、,thenumberofelementscontainedinthesetV={xmodq
6、f≡0modq}wherefisthepartialderivativesoffwithrespectxxtox.ThecardinalityofVinturncanbederivedfromthep-adicsizesofcommonzerosofthepartialderivativesf.Thispaperpresentsamethodofde
7、terminingthexp-adicsizesofthecomponentsof(ξ,η)acommonrootofpartialderivativepolynomialsoff(x,y)inZp[x,y]ofdegreefivebasedonthep-adicNewtonpolyhedrontechniqueassociatedwiththepolynomial.Thedegreefivepolynomialisoftheformf(x,y)=ax5+bx4y+cx3y2+sx+ty+k.Theestimateobtainedisintermsof
8、thep-adicsizesofthecoefficientsofthedominanttermsinf.Keywords:Exponentialsums;cardinality;p-adicsizes;Newtonpolyhedron.MathematicsSubjectClassification2000:11D45,11L071.IntroductionInourdiscussion,weusenotationsZp,Ωpandordpxtodenote,respectively,theringofp-adicintegers,completion
9、ofthealgebraicclosureofQpthefieldofrationalp-adicnumbersandthehighestpowerofpwhichdividesx.Foreachprimep,letf=(f1,f2,...,fn)beann-tupleofpolynomialsinZp[x]wherex=(x1,x2,...,xn).LetV={xmodq
10、f≡0modq}wherefarethepartialderivativesoffxxwithrespecttox.Theestimationofthecardinalityof
11、Vhasbeenthesub-jectofmuchresearchinnumbertheoryoneapplicationofwhichisinthequesttofindthebestpossibleestimatesformultipleexponentialsumsoftheform541May5,200915:25WSPC/203-IJNT00224542S.H.Sapar&K.A.Mohd.Atan2πifS(f;q)=exp()wheref(x)isapolynomialinZ[x]andthesumtakenxmodqqoveraco
12、mpletesetofresiduesxmoduloapositiveintegerq.LoxtonandVaughan[
