Pertanika 11(1), 125-131 (1988)

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

KAMEL ARIFF1N MOHD. ATANMathematics Department,

Faculty of Science and Environmental Studies,Universiti Pertanian Malaysia

43400 Serdang, Selangor, Malaysia

Key words: Congruence equations; n-tuple polynomials; cardinality; p-adic common zeros.

ABSTRAK

Kekardinalan set penyelesaian kepada sesuatu sistem persamaan kongruen modulo kuasa perdana,dianggarkan melalui penggunaan kaedah Newton polihedron. Anggaran kepada nilai ini didapatkan bagin-rangkap polinomial^ = (fj, . . . . , fn) dalam koordinat x = fxp . • . ,xn) dengan pekali dalam Zp. Per-bincangan adalah mengenai anggaran yang berkaitan dengan polinomial f yang linear dalam £ dan sepasangpolinomial yang kuadratik tertentu dalam ZJx, y]

ABSTRACT

The 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 = (fj, > • - > fn) tri coordinates x. = (Xj, . . . , xn) with coefficients in Z .The discussion is on the estimates corresponding to the polynomials f that are linear inland a specificpair of quadratics in ZJx, y].

INTRODUCTION

For each prime p and Z the ring of p-adic integerslet£,= (f | . . . , f ) be an n-tuple of polynomialsin Z fx] where x = ( x , , . . . , xM). We will consider

p *•** "w i n

the set

V(f ; p a ) = { u m o d p a : f(u) =£ O mod p a }

where a > 0. Let N(f ; pa) • card V(f ; pa).

For a polynomial f(x) defined over the ring ofintegers Z Sandor showed that

N(f ;, mp 1 / 2 O r dpD

where D =£ 0, ot > ord D and D is the discrimi-nant of f.

Let K be the algebraic number field generatedby the roots if, 1 < i < m qf 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 numbers

i > 1 where ei is the multiplicity of the root f.;

Loxton and Smith (1982) showed that

a a - (a - 5 ) / eN(f ; pa) < mp

KAMEL ARIFFIN MOHD. ATAN

where 5 = ord D(f), With this suitably definedglobal different of 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 Z .

Chalk and Smith (1982) obtained a result ofsimilar form with 5 = max, ord f. where f. is the

Taylor coefficient at the

with integer coefficients. Suppose

where the £.'s are distinct algebraic numbers with

respective multiplicities e.. Let 5(f) = ord D(f)

as before and e(f) = max ord_ —1J

distinct roots £.. The proof used a version ofHensel's Lemma.

Forj; • ( f j , . . . , f ) an n-tuple of polyno-mials in Z[£] define the discriminant D.(f) o Mas follows. If the resultant of f and the Jacobianof f vanishes set D(f) = 0 otherwise let D(f) be thesmallest positive integer in the ideal in Zfx] gene-rated by the Jacobian o f j and the components ofL Loxton and Smith (1982) showed that

N(f,pa)_nc*p for a < 26(Degf)pn d for a > 25

where Deg^rneans the product of the degrees ofall the components of f.

In this paper we will arrive at an estimate ofNC£ * Pa) f o r c e r t a i n polynomials^ by using theNewton polyhedral method as described by MohdAtan (1986). With p denoting a prime, we definethe valuation on Q the field of p-adic numbersas usual. That is

-ord xP

i 0 ifx = 0W p ' I P V i f x # 0

Then the following theorem gives an estimate forN(f ; Pa). 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.

Lemma 2.1

Let p be a prime and f(x) a polynomial with coef-ficients in the complete field £2 . If a segment ofthe Newton polygon of f has slope X, and horizon-tal length N(i.e. it extends from (i, ord ap to (i +N, XN + ord then f has precisely N root c*| in

dN

12 with ord a. = - X (counting multiplicities).P P i

Theorem 2.1

Let p be a prime and f a polynomial with integercoefficients which does not vanish identicallymodulo p. Set e = c(f), 6 = 5(f) and let m be thenumber of distinct zeros of f. Then

Proof:

As above we write

f(x) = aO

a 1x1

n " 1 EL= a o j = 1

where ord x denotes the highest power of p divi-p

ding x and ord x = °° if x = 0. The valuation ex-

tends uniquely from Q to Q the algebraic

closure of Q and to Q , and £2 is complete and

algebraically closed.

2. AN ESTIMATE FOR N(f ; p a ) WITH f(x) INZ[x]

In this section we will consider a polynomial f(x)

where the £-'s are distinct with multiplicities e-.We may suppose that £2 contains the numberfield K generated by the roots £.. Let V.(f; ptt)denote the set of points in V(f ; p a) which are p-adically closest to £-, that is

V.(f ; pa) ={x in V(f ; p a): ordp(x - ?.)

• max ord (x - & ) A

1 < i < m

126 PERTANIKA VOL. 11 NO. t, 1988

A METHOD FOR DETERMINING THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATIONS

T n e n m and for any k > 0card V(f ; p a) < 2 card V.(f ; pa).

i=l J

To estimate the terms on the right, we introducethe set

: ordp(x - t) - max ordp(x -- {x in

andord f(x)>6]P

and define

7j(fl) • i n f o r d p ( x - $ ) .

xinD.(0)

Since V.(f ; P ) is a subset of D.(<*), we have

ord.k!

This shows that the first edge of the Newton poly-gon of f(x + r\) goes from the point (0, ord f(r?))

as required byto the point (e., ord

Lemma 2.1. We can find 77 so that ord f(r?) = 0

and /JL- • 7(0) and for this choice of 17 we have

card V.(f ; p a) < card { x mod p a : ord (x - £.) Therefore 7-(0) is continuous, increasing and con-/ . P cave away from the origin. Further if 6 is suf-yi

< P J

We now require a lower bound for 7-(of). For this

choose 17 in D.(0) and consider the Newton

polygon of the polynomial f(x + 17). Let ju-

• ord (77 - J.) and let e be the total multiplicity

of all the roots £• with

ficiently large, £. is the unique closest root to 7}

and so e- = e- ana

X. = ordJ P c!

J

= 6. (say).

By considering the graph of 7.(8) we see that

ord (17 - £.) = ju. and set X. = ordP J J J P €\

5.)e. - 5 ) / e

We have Finally,

e.!J

card V(f ; p a) < max card Vj(f ; pa)Kj<m

where ord (17 - £.) <M for all i,

and the dots indicate terms with larger p-adic ordersthan the main term. Thus

for a > 5.

This proves the theorem since the requiredestimate is trivial when Ot > 5.

ord.. = X..J

In the same way,

ord f(T?) = X. +e./i.P J J J

3 ESTIMATE FOR N(f ; pa) WITH^(x) IN ZpTx]

In this section we will consider the set

V(f ; Pa) ={x mod p a : f(\) = 0 mod

PERTANIKA VOL. 11 NO. 1, 1988 127

KAMEL ARIFFIN MOHD. ATAN

where f = (f., . . . , fn) is an n-tuple of polyno-

mials in the coordinates £ " . ( * } » . . . , x n ) with

coefficients in Z . We will consider first polyno-

mials f, i = 1, 2, . . . , n that are linear in ( x p

. . . , x ) as in the following theorem.

Theorem 3.1

Let p be a prime and f = ff,, . . . , fR) be an n-tupleof non-constant linear polynomials in Z [x]

where x » ( x » xn). Suppose r is the rank ofmatrix A representing^. Let 6 be the minimum ofthe p-adic orders of r x r non-singular submatricesof A. If a > 0 then

On multiplying both sides of the congruence (4)by the adjoint of B, we obtain

(det B)x' = -(adj B)C,x" mod p a (5)

For a given x" in (5) the number of solutionsfor x' mod p a is either o or p since (5) deter-mines x' mod p a ~ . Thus there are p^ n ~ r ' a

choices for x" mod pa. It follows that the numberof solutions x mod p a to (2) and hence (1) is

p ( n - r ) a + r 6 a s a s s e r t e d

In Theorem 3.1 if n = 2, a > 5 and rank A = 2

N(i;P*)< p(,-r)C.+r6 l f Q > 5

Proof:

The assertion is trivial for Of < 5 . Suppose a > 5.Consider the set

-° mod

The equation

f(x) = 0 mod pa

is equivalent to

A x = O mod p a

(1)

(2)

where A is the matrix representing f. Now A isequivalent to a matrix A' of the form

B

O

C

O

where 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

A'x = O m o d p a (3)

Write x = (x\ x'V where x' comprises the first rcomponents of x and x" the remainder, and (a, b)denotes the transpose of (a, b). Then (3) becomes

then

where 8 is the p-adic order of the Jacobian of fjand fy We will give an alternative proof of thisassertion using the Newton polyhedral method.First we have the following lemma.

Lemma 3.1

Let p be a prime and f, g linear functions in thecoordinates x = (x, y) defined over ZM. Let J =

** P

fx8y — f z be their Jacobian. Suppose ^xQ in

S72 statisfies ord f(xQ) > a and ord g(xQ) > a.

If a > ord J then f and g have a common zero

£ in SI2 with ordp(i* - x ) > a - ord J.

Proof-

Lot X = fX, Y) -XQ and write

g o + g x X + g y Y

^x' = - C V ' mod p a (4)

where fo = f^xo), gQ{xQ).

Consider the indicator diagrams (as definedby Mohd Atan (1986)) of f(X) + xQ) and g(X +x0)-If no edges in these diagrams coincide, then byMohd Atan (1986) there exists a zero common tof and g statisfying ord X>a - o r d J. If some

edges coincide but with say ord f /f < ordgo/gx 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 the

128 PERTANIKA VOL. 11 NO. I, 1988

A METHOD FOR DETERMINING THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATIONS

1•

11

11i

g

r

Y h

iii

i

same 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 andthe result is therefore clear. Possible stages in theproof are shown in the following diagrams.

o r d p g ( x , y ) > x |

for any real number X. Define

?(X) - inf ord (x - t)P ~ ~

x e H(X)

Note:

The above equations can be obtained at once bysolving the simultaneous equations f(X + xQ) =g(X +,xo) • 0 for X. It is to avoid solving the equa-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 > 6 and rank of matrixrepresenting f j , ?2 *s e Q ^ t 0 2.

Theorem 3.2

Let f and g be linear polynomials in Zp[x, y ] .

Let Jf be the Jacobian of f and g and

5 = ord Jf , Let a > 0. Then

N(f ; g ; p

ifa<5

ifa>5

Proof: The result is trivial for a < 5 . We assumenext a > 5. As before let

V(f ;g ;p a) = {(x ,y) mod p a : f(x,y),

g(x,y) = O m o d p a j .

Consider the set

H(X) = { (x, y) in £2* : ordpf(x, y),

where x = (x, y) and % is the common zero of fandg.

V(f;g;pa)CH(a)

for each a > 1, it follows that

card V(f ;g; pa) < card|x mod p a: ord (x - $)

P 0)

w

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 + xQ) and g(X +JCQ) for^xQ in H(X). By ourhypothesis and Lemma 3.1,

7(a) > a - 5

It follows by (1) that

cardV(f ;g ;p a )<p 2 6

Since ord Jf < °°, f and g have a unique com-

mon zero. Hence

N( f ;g ;p a )<p 2 6

as required.

Next, we will consider a pair of non-linearpolynomials f and g in coordinates (x, y) defined

PERTANIKA VOL. 11 NO. 1, 1988 129

KAMEL ARIFFIN MOHD. ATAN

over Z of the form

f(x, y) • 3ax2 + by2 + c

g(x,y) = 2bxy + d.

First however we prove the following lemmawhich gives the estimate to the p-adic order ofzeros common to f and g.

Lemma 32

Let f(x, y) • 3ax2 + by2 + c, g(x, y) = 2bxy + dbe polynomials with coefficients in Z and with

p > 2. Let 5 = max< ord 3a, - ord b \ Supposet P 2 P *

(x o ,y o ) is inJ2 pwith

where uQ = \/3a xQ

FQ =

y o , u 0 =\/3a"xo ->/b~y

and GQ = fQ - g Q .1

hypothesis

ord F , ord G > a + min s ord Vb, ord x/3a-

We therefore see from the Newton polygon of Fthat F has a zero satisfying

F j f

> - a + min-{0,ord V3a/t. /u 0 V P

ord U > - ord

A similar result holds for G. These estimates leadto a zero (X, Y) of f and g satisfying the requiredinequality.

- . ,, . A ,y , r^2 ... a. x T n e following theorem gives the estimate for N(fThen, there is a point (£, rj) in il with f(§, rj) = ttv .fSfX* V) = 0 and ord (£ - xQ), ord (T? - yQ) >

2

Proof:

Write X = (X, Y) = x - x^ and set

f(X,Y) = 3aX2 +bY2 + 6axQ:

g(X,Y) = 2bXY + 2byoX^

wheref o = 3 a x o + b y o

2 + c

= 2 b V o+ d

With the change of variable

U = \/3a X + >/b"V

V

g; p ) where f and g are quadratics in the aboveform.

Theorem 33

Let f(x, y) = 3ax2 + by2 + c and

g(x, y) = 2bxy + b be polynomials with coefficieni

andcients in Z where p is a prime > 2. Let a > (

Then5 = max [ ord 3a, - ord b] .

P 2 P

p2a i f a < 5

Proof:

We consider the case when a > 6. The result istrivial when a < 5. As before let

V(f;g;pa) ={(x, y) mod p a : f(x, y), g(x, y).= 0 me

we find that

F(U,V)= Vbf(X,Y)+V3lg(X,Y)

= Vb U2 + 2Vb"u U + F° a

andG(U,V)= Vbf(X,Y)+V3ag(X,Y)

130 PERTANIKA VOL. II NO. 1, 1988

Following the method of Loxton and Smith(1982) we take Vi(f;g: pa) to indicate the set ofpoints in V(f; g; pa) that are p-adically close to acommon zeroj^ = ( ^ j , (.j) of f and g. That is

): ordp(x -^{)

max ordp (x

A METHOD FOR DKTERMINING THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATIONS

where x = (x,y). Then

N(f;g;pa)<S cardV(f;g;pa)i 1

Consider the set

H.(A) = {xeo ; ord (x-J=

• max ord (x — £.) and

ordpf(x),ordpg(x)>x}

For any real number X. Define

yfk) = inf ord (x - |.)xeH.(X)

for all L Now, for every a > 1.

coordinates x •, f ) is an n-tuple polynomials in the

^ j , . . . , x ) and coefficients in(1) Z We have considered both linear polynomials

and 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, p ) where f =(f f ) are n-tuple of non-linear poly-nomials of a more general form will be the subjectof our next discussion.

ACKNOWLEDGEMENT

The author would like to express his gratitude toProfessor J.H. Loxton of the University of Mac-quarie of Australia for suggesting the problem andfor enlightening discussions leading to the results.

V^g iP iCHjfa ) .

It follows that

card V-(f;g;pa)< card{x mod p a : ordJx -1.)

p 2 a - 2 7 i ( a )

(2)

where a > 7-(a) for all i.

We find the lower bound for the functionT| : R -> R by examining the combination of theindicator diagrams for f(X + xQ) and g(X + £o)with (x , y ) in H.(\). By hypothesis and Lemma

3.2

Hence, by (2)

ctrdVi(f;g;pa)< p a + ( (3)

for all i. By a theorem of Bezout (see for exampleHartshorne (1977) or Shafarevich (1977) the num-ber of common zeros of f(X + x ) and g(X + x_)does not exceed the product of the degrees of fand g. Hence by (1) and (3) the above assertionholds.

REFERENCES

KOBLITZ, N. (1977): 4tp-adic power series" p-adic Num-bjers, 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 HensePsLemma C.R.Math. Rep. Acad. Set Canada 4(1):

HARTSHORNK, 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. /. London *Math Soc26(2): 15-20.

LOXTON, J.H. and R.A. SMITH (1982): On Hua's esti-Exponential Sums. 7. Australian Math. Soc. 33:125-134.

MOHD ATAN, K.A. (1986): Newton Polyhedra and p-adic Estimates of Zeros of Polynomials u > ^ p [ x , y ] "Pertaniaka 9(1): 51-56 . Universiti Pertaniah Malay-sia.

MOHD ATAN, K.A. (1986): Newton Polyhedral Methodof Determining p-adic Orders of Zeros Commonto Two Polynomials in Q [x, y ) . Pertanika 9(3).375-380. Universiti Pertanmn Malaysia.

SHAFAREVICH, I.R. (1977): Basic Algebraic GeometrySpringer Verlag, Berlin, New York, 198.

CONCLUSION

In this paper we give estimates for N(f, pa) where

PERTANIKA VOL. 11 NO. 1, 1988

(Received 30 November, 1987)

131