The cardinality of the set of solutions to a system of congruence equations module a prime

power is estimated by applying the Newton polyhedral method. Estimates to this value are obtained

for an n-tuple of polynomials f = _ (f1, ... ,fn) in coordinates -f = (xl' ... ,xn) with coefficients in Zp.

The discussion is 011 the estimates corresponding to the polynomials f that are linear in x and a specific

pair of quadratics in Zp(x,y

Mohd Atan, Kamel Ariffin

Universiti Putra Malaysia Institutional Repository

Pertanika 11(1),125-131 (1988)A Method for Determining the Cardinalityof the Set of Solutions to Congruence EquationsKAMEL ARIFFIN MOHD. ATANMathematics Department,Faculty ofScience and Environmental Studies,Universiti Pertanian Malaysia43400 Serdang, Selangor, MalaysiaKey words: Congruence equations; n-tuple polynomials; cardinality; p-adic common zeros.ABSTRAKKekardinalan set penyelesaian kepada sesuatu sistem persamaan kongruen modulo kuasa perdana,dianggarkall melalui penggunaan kaedah Newton polihedron. Anggaran kepada nilai ini didapatkan bagin-rangkap polinomialj = (f1, .... , f n) dalam koordinat,3 =(x l' ... , x n) dengan pekali dalam Zp' Per-bincangan adalah mengenai anggaran yang berkaitan dengan polinomial f yang linear dalam.x dan sepasangpolinomial yang kuadratik tertentu dalam Z,Jx, yj - -ABSTRACTThe cardinality of the set of solutions to a system of congruence equations modulo a primepower is estimated by applying the Newton polyhedral method. Estimates to this value are obtainedfor an n-tuple of polynomials f = (f1, ... ,fn) in coordinates x = (xl' ... ,x ) with coefficients in Z ._ - n pThe discussion is 011 the estimates corresponding to the polynomials f that are linear in x and a specificpairofquadraticsinZpfx,yj. - '"INTRODUCTIONFor each prime p and Zp the ring of p-adic integersleU>= (fl ... ,fn) be an n-tuple of polynomialsin Z [x J where x = (x I' ... , xn). We will considerp - -the setV([; pQ) ={ u mod pQ : feu) ~ 0 mod pQ}where Q> 0. Let (f; pQ) = card V(f ; pQ)."" -For a polynomial f(x) defined over the ring ofin tegers Z Sandor showed thatwhere D * 0, a > ordpD and D is the discrimi-nant of f.rLet K be the algebraic number field generatedby the roots ~i' I ~ i ~ m 01 the polynomial f(x)with m distinct zeros. Let D(f) denote the dif--ferent of f the intersection of the fractional idealsof K generated by the numberse.!1i;;;' I where ei is the multiplicity of the root ~i'Loxton and Smith (1982) showed thatQ Q-(Q-D)/e\!,; p ) ~ mpKAMELARIFFIN MOHO. ATANIn this section we will consider a polynomial f(x)where Deg~eans the product of the degrees ofall the components ofl:Lemma 2.1Let l' be a prime and f(x) a polynomial with coef-ficients in the complete field Up' If a segment ofthe Newton polygon of f has slope A, and horizon-tal length NO.e. it extends from 0, ordl' ai) to 0 +N, AN + ord a.)) then f has precisely N root a l· inl' IU with ord a. = - A (counting multiplicities).l' l' ITheorem 2.1Let l' be a prime and f a polynomial with integercoefficients which does not vanish identicallymodulo p. Set e = e(f), [j = [j(f) and let m be thenumber of distinct zeros of (ThenAs above we writeN(f ; pa) ~ mpa - (a - o)/eProof'where the ~j'S are distinct with multiplicities ej .We may suppose that U contains the numberl' 0 afield K generated by the roo~s ~j' Let Vj(f; l' )denote the set of points in V(f ; pa) which are p-adically closest to ~j' that isVo(f ; pa) ={x in V(f ; pal: ord (x - nJ l' J= max ordp(x - ~j) .}1~'i~m1 m e.f(x)=a xn+alx n- + ... +a =a .rr. (x-n Jo n oJ=1 JThen the following theorem gives an estimate forN(f ; pQ). The proof is a modification of that byLoxton and Smith (1982) illustrating the use ofthe Newton polygon of f whose special propertyis stated in Koblitz (1977) which we rewrite asfollows.where the ~/s are distinct algebraic numbers withrespective multiplicities ej' Let o(f) = ordpD(f)rCe.) (~j)as before and e(f) = max ord ~.Ll-----=--j l' e.!Jwith integer coefficients. Supposeat theif x * 0pna for a ~ 2[j(DegD pn[j for a > 2[jJp-ordpx\lo ifx=Owhere ord x denotes the highest power of l' divi-ding x and ord x = 00 if x = O. The valuation ex-l' - b .tends uniquely from Qp to Qp the alge ralCclosure of Q and to U , and U is complete andl' l' l'algebraically closed.Taylor coefficientIn this paper we will arrive at an estimate ofN[ ; pa) for certain polynomials L by using theNewton polyhedral method as described by MohdAtan (1986). With l' denoting a prime, we definethe valuation on Q the field of p-adic numbers. l'as usual. That IS2. AN ESTIMATE FOR N(f ; pa) WITH f(x) INZ[x]where [j = ord D(f). With this suitably definedglobal differenlof f(x) Loxton and Smith thUsimproved on the result of Sandor's. Both resultsare stated for polynomials defined over Z. They,however, can be adapted to work over Zp'Chalk and Smith (1982) obtained a result ofsimilar form with [j = maxI·ord f. where f. is thel' I If(ei) (~i)ei !distinct roots ~i' The proof used a version ofHensel's Lemma.For1 = (fI' ... , fn) an n-tuple of polyno-mials in Z [;Sl define the discriminant DoW of1as follows. If the resultant of! and the Jacobianof f vanishes set D(O = 0 otherwise let D(f) be thesm';;ilest positive in~ger in the ideal in Z r~l gene-rated by the Jacobian of-l and the components of1. Loxton and Smith (1982) showed that126 PERTANIKA YOLo 11 NO.1, 1988A METHOD FOR DETERMINING THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATIONSand for any k ~ 0This shows that the first edge of the Newton poly-gon of f(x + 1/) goes from the point (0, ord f(1/»pf(k)(1/)ordp ~ A. - (i - E.)J.1..k! J J Jas required by(E. )f J (1/)to the point (E., ordJ p E.!JLemma 2.1. We can find 1/ so that ordp f(1/) = eand J.1. = r·(O) and for this choice of 1/ we haveJ Jr}O) = (0 - Aj)/EyTherefore r·(e) is continuous, increasing and con-cave away (rom the origin. Further if e is suf-ficiently large, ~. is the unique closest root to 1/and so Ej = ej andand ord f(x) ~ 0 Jpand defineThen mcard V(f ; pet) ~ :E card V.(f; p(\j=1 JTo estimate the terms on the right, we introducethe setD.(O) ={x in n : ordp(x - n = max ord (x - ~i)J p J PI ~i~minf ord (x - O.P Jx in D.(O)JSince V.(f ; pet) is a subset of D.(et), we haveJ Jcard V.(f ; pet) ~ card {x mod pet: ord (x _ ~.)J P 1et-r.(et)~r.(et)J ~ p JJWe now require a lower bound for r}et). For thischoose 1/ in D.(O) and consider the ewtonpolygon of the J polynomial f(x + 1/). Let J.1j= ordp(1/ - ~j) and let Ej be the total multiplicityof all the roots ~j withfore ~o.By considering the graph of r.(O) we see thatJr.(e)~(O -o.)e.~(O -o)/eJ J Jord (1/ - 0 = J.1. and set A. = ordp J J J P(E.)f J (0JA.Jordp(e.)f J (~j)e.!. J&.J(say).We have Finally,where ord (1/ - 0 < J.1. for all i,p 1 J(E.)f J (1/)E.!Je.a 1T(1/ - 0 1 + ...o Icard V(f ; pet) ~ max card V/f ; pet)I~~m~ m et-(et - o)/e""" pfor et ~o.3 ESTIMATE FOR N(f ; pet) WITH!~) IN Zp~]This proves the theorem since the requiredestimate is trivial when et >o.A..JE.!Jand the dots indicate terms with larger p-adic ordersthan the main term. Thus(E.)f J (1/)In the same way, In this section we will consider the setord f(1/) = A. + E.J.1. ~ 0p J J JPERTANIKA VOL. 11 NO.1, 1988 127KAMEL ARIFFIN MOHO. A TANwhere 1. = (f l' ... , fn) is an n-tuple of polyno-mials in the coordinates 2S = (xl' ... , xn) withcoefficients in Zp' We will consider first polyno-mials f., i= i, 2, ... , n that are linear in (xl'1 . h... , xn) as in the followmg t eorem.Theorem 3.1Let p be a prime and f = ~l' ... , fn) be an n-tupleof non-constant lia.:ar polynomials in Zp[~lwhere x = (x l' ... , x ). Suppose r is the rank of~ nmatrix A representing!. Let Dbe the minimum ofthe p-adic orders of r x r non-singular submatricesof A. If 0: > 0 thenOn multiplying both sides of the congruence (4)by the adjoint of B, we obtain(det B13' =-(adj B)c.l' mod pO: (5)For a given x" in (5) the number of solutionsfor x' mod pO: i; either 0 or prD since (5) deter-min~ x' mod pO:-D. Thus there are p(n-r)o:choicesfor x" mod pO:. It follows that the numberof solution;-~ mod pO: to (2) and hence (1) isp(n-r)Q+rD as asserted.In Theorem 3.1 if n = 2, 0: > Dand rank A = 2if 0: ~Dp(n-r)o:+rD if 0: > DthenProof:The assertion is trivial for 0: ~ D. Suppose 0: > D.Consider the setwhere {) is the p-adic order of the Jacobian of f land f2' We will ~ive an alternative proof of thisassertion using the Newton polyhedral method.First we have the following lemma.v~; pO:) ={';LmOd pO: :!(.l9 ==,0 mod pO: }The equationf(x) ='0 mod pO:-- ~is equivalent to(1)(2)Lemma 3.1Let p be a prime and f, g linear functions in thecoordinates x = (x, y) defined over Z . Let J =~ Pfxg y - fy&x be their Jacobian. Suppose ~o inrl? statisfies ord f(x) ~ 0: and ordp g(x ) ~ 0:.P P ~o ~oIf 0: > ord J then f and g have a common zerop~ in n 2 with ordp(~ - xo) ~ 0: - ord J.~ p ..,.. ~ pwhere A is the matrix representing f. Now A isequivalent to a matrix A' of the formwhere B is an r x r non-singular matrix and C and rx(n-r) matrix both with rational entries. There-fore (2) is equivalent to Consider the indicator diagrams (as definedby Mohd Atan (1986» of f(X) + x ) and g(X + x ).-., _0 ,...., _0If no edges in these diagrams coincide, then byMohd Atan (1986) there exists a zero common tof and g statisfying ord X ~ 0: - ord J. If somep pedges coincide but with say .ord f If ~ ord. pox pgolgx we replace g by g - (gy/fy)f to eliminate Y.This transformation does not change J .and thehypothesis of the lemma are satisfied with theProof:Let X = (X, Y) = x - x and write~ -"-0f (X + x ) = f + f X + f Y...., ~o 0 x yg(X + x ) = g + g X + g Y~ ~o 0 x y(4)(3)" 0:- -C,2S mod pI ")t I • h f' tWrite x = (x , x where x compnses t e us r~ -., -." -., tcomponents of x and x" the remainder, and (a, b)denotes the tra;spose~of (a, b). Then (3) becomes128 PERTANlKA YOLo 1 I NO.1, 1988A METHOD FOR DETERMINING THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATIONSVbIIIr -----/g~ ////IV"'hIIIIsame a as before. If no edges of the indicator dia-grams coincide we can apply the same result byMohd Atan (1986) above to get the desired con-clusion. Otherwise we replace f by f - (fx/gx)gto eliminate X. Again this does not change J and~he result is therefore clear. Possible stages in theproof are shown in the following diagrams.Note:The above equations can be obtained at once bysolving the simultaneous equations f(3 +2'0) =g(X + x ) = 0 for X. It is to avoid solving the equa-,....,., _0 ""tions and to illustrate the use of Newton poly-hedron that we consider the above result.The following theorem gives an alternativeproof using Newton polyhedral method toTheorem 3.1 when n = 2, a> 0 and rank of matrixrepresenting f I' f2 is equal to 2.Theorem 3.2Let f and g be linear polynomials in Zp[x, y].Let Jf,g be the Jacobian of f and g and0= ordp Jf,g' Let a> O. Thenfor any real number X. Define"Y(X) = inf ord (x - np - -~ € H(X)where x = (x, y) and ~ is the common zero of fandg. '" -V(f; g; pa) ~ H(a)for each a ~ I, it follows thatwhere a ~ "Y(a).The lower bound for the function "Y : R -+R can be found by examining the indicator dia-grams associated with the Newton polynomialsof f(X +~o) and g(! t~o) for..3-oin H(X). By ourhypothesis and Lemma 3.1,ifa~oifa>o It follows by (1) thatProof: The result is trivial for a ~ O. We assumenext a> O. As before letV(f ;g ; pa) = { (x, y) mod pa : f(x, y),g(x, y) == 0 mod pa}-Consider the setH(X) = { (x, y) in n~ : ordpf(x, y),Since ordp Jf,g < 00, f and g have a unique com-mon zero. Henceas required.Next, we will consider a pair of non-linearpolynomials f and g in coordinates (x, y) definedPERTANIKA VOL. 11 NO.1, 1988 129KAMEL ARIFFIN MOHD. ATANord f(x ,y ),ord g(x ,y ) ~ a > O.p 0 0 p 0 0First however we prove the following lemmawhich gives the estimate to the p-adic order ofzeros common to f and g.over Zp of the formf(x, y) = 3ax2 + by2 + Cg(x, y) = 2bxy + d.ordp F ,ord G ~ a + min {ord Vb, ord v'3a:}o pop p= Vb V 2 0+ 2ybv V + Go 0o= max [ord 3a, lord b1.Then p 2 Pp2a if lX ~oN(f;g;plX)~4plX + 0 if lX > 0A similar result holds for G. These estimates leadto a zero (X, Y) of f and g satisfying the requiredinequality.The following theorem gives the estimate for N(f;g; p~) where f and g are quadratics in the aboveform.Theorem 3.3Let f(x, y) = 3ax2 + by2 + c andg(x, y) = 2bxy + b be polynomials with coeffi-cients in Zp where p is a prime> 2. Let a> 0andWe therefore see from the Newton polygon of F,that F has a zero satisfyingFord U ~ 1. ord ~ ~ 1. a + min{O ord Y3a/b}p 2 Py'b 2' Pwhere u o = v'3il Xo +y'b Yo' uo =y'3a Xo - vIiJYo'F0 = Vb f0 -i; V3a go and Go = Yb f0 - v'3il go . Byhypothesis2bXY + 2byoX + 2bxoY + go3aX2 + by2 + 6ax X + 2by Y + fo 0 0(X, Y) = x - x and set~ ~f(X, Y)g(X, Y)whereThen, there is a point (~, 1'/) in n~ with.f(t 1'/) =g(t 1'/) = 0 and ordp (~ - xo), ordp(1'/ - Yo) ~1. (a - 0)2Proof:Write~Lemma 3.2Let f(x, y) = 3ax2 + by2 + C, g(x, y) = 2bxy + dbe polynomials with coefficients in Zp and withp > 2. Let 0 = max{ordp3a, ~ ordpb} Suppose( X y) is in n2 with0' 0 PWith the change of variableProof:We consider the case when lX > O. The result istrivial when a ~ o. As before letUVy3a X +ybYv'3il X -VbYV(f; g; pa) ={(x, y) mod plx: f(x, y), g(x, y)= 0 me.J~} .we find thatF(U, V) = Vb f(X, Y) +y3a g(X, Y)= Vb u2 + 20u U + Fo 0andG(U, V) = Vb f(X, Y) + Y3a g(X, Y)Following the method of Loxton and Smith(1982) we take Vi(f;g; plx) to indicate the set ofpoints in V(f; g; pa) that are p-adically close to acommon zeroJi = (~il' ~i2) of f and g. That isVi(f; g; plx) ={xEV(f; g; plx): ord (x - 0p__1= max ord (x - E.)}J P ~ ~J130 PERTANIKA YOLo II NO.1, 1988A METHOD FOR DETERMINING THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATIONSwhere x = (x, y). ThenN(f;g;pO')';;;;l; card V.(f;g;pO')i 1Consider the set, { 2H.(A) = x € n ; ord (x-l:.)1 "" P p~-3i= max ordp(x - nand~ ~Jord f(x),ord g(x) ~ A}p p "'"For any real number A. Define'Y1.(A) = inf ord (x - 0p ~ ~1X € H.(A)~ 1for all i. Now, for every 0' ~ I.(I)L = (f I' ... , fn) is an n-tuple polynomials in thecoordinates,2S = (x I' ... , xn) and coefficients inZp . We have considered both linear polynomialsand a pair of non-linear polynomials of a specificform. Our discussion is centred in the use ofNewton polyhedral method to arrive at theseestimates. The extension to a more general methodto arrive at the estimates for N(f, pO') where f =(f I' . . .. , fn) are n-tuple of non-linear p;\y-nomials .of a more general form will be the subjectof our next discussion.ACKNOWLEDGEMENTThe author would like to express his gratitude toProfessor J.H. Loxton of the University of Mac-quarie of A ustralia for suggesting the pro blem andfor enlightening discussions leading to the results.REFERENCESIt follows thatI'Y.(O') ~ - (0' - 0).1 2card Vi(f; g; pO') ,;;;; card {'5-mod pO' : ordp-~ -;-li),;::: 20' - 2'Y .(0') ~ 'Y1'(0')}""" p 1(2)where 0' ~ 'Y.(O') for all i.1We find the lower bound for the function'Yi : R --+- R by examining the combination of theindicator diagrams for f(~ + ..30 ) and g(~ +~o)with (x y) in H.(A). By hypothesis and Lemma0' 0 13.2Hence, by (2)(3)for all i. By a theorem of Bezout (see for exampleHartshorne (1977) or Shafarevich (1977) the num-ber of common zeros of f(~ + ~o) and g(X +,!o)does not exceed the product of the degrees of fand g. Hence by (1) and (3) the above assertionholds.CONCLUSIONIn this paper we give estimates for N~, pO') whereKOBLITZ, N. (1977): "p-adic power series" p-adic Num-b.ers, p-adic Analysis and Zeta- Functions, Springer-Verlag New York, Berlin, 1977.CHALK, J .H.H. and R.A. SMITH (1982): Sandor's Theo-rem On Polynomial Congruences and Hensel'sLemma C.R.Math. Rep. Acad. Sci Canada 4(1);HARTSHORNE, R. (1977): Algebraic Geometry SpringerVerlag, New York, Berlin, 53-54.LOXTON, J.H. and R.A. SMITH, (1982): On Hua's esti-mate for exponential sums. J. London'Math. Soc.26(2): 15-20.LOXTON, J.H. and R.A. SMITH (1982): On Hua's csti-Exponential Sums. J. Australian Math. Soc. 33:125 -134.MOHO ATAN, K.A. (1986): Newton Polyhedra and p-adic Estimates of Zeros of Polynomials in S1p l x, y J"Pertamaka 9(1); 51-56. Universiti Pertaniafl Malay-sia.MOHO ATAN, K.A. (1986): Newton Polyhedral Methodof Determining p-adic Orders of Zeros Commonto Two Polynomials in Q [x, y J. Pertanika 9(3).375 - 380. Universiti Pertan&n Malaysia.SHAFAREYICH, I.R. (1977): Basic Algebraic GeometrySpringer Verlag, Berlin, New York, 198.(Received 30 November, 1987)PERTANlKA YOL.ll NO.I,1988 131

A Method for Determining the Cardinality

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A Method for Determining the Cardinality of the Set of Solutions to Congruence Equations

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