International Mathematical Forum, Vol. 8, 2013, no. 8, 357-367 Small Zeros of Quadratic Forms Mod P m Ali H. Hakami Deartment of Mathematics, Faculty of Science, Jazan University P.O. Box 277, Jazan, Postal Code: 45142, Saudi Arabia aalhakami@jazanu.edu.sa Abstract. Let Q(x) Q(x 1,x 2,..., x n ) be a quadratic form over Z, be an odd rime, and Δ ( ( 1) n/2 det A Q / ). A solution of the congruence Q(x) 0 (mod m ) is said to be a rimitive solution if x i for some i. We rove that if this congruence has a rimitive solution then it has a rimitive solution with x max{6 1/n m[(1/2)+(1/n)], 2 2(n+1)/(n 2) 3 2/(n 2) }. Mathematics Subject Classification: 11D79, 11E08, 11H50, 11H55 Keywords: Quadratic forms, congruences, small solutions 1 Introduction Let Q(x) Q(x 1,x 2,..., x n ) 16i6j6n a ijx i x j be a quadratic form with integer coefficients and be an odd rime. Set x max x i. When n is even we let Δ (Q) ( ( 1) n/2 det A Q / ) if det A Q and Δ (Q) 0if det A Q, where ( /) denotes the Legendre-Jacobi symbol and A Q is the n n defining matrix for Q(x). Q(x) is called nonsingular (mod ) if det A Q. Consider the congruence Q(x) 0 (mod M), (1) where M is a ositive integer. The roblem of finding a small solution of (1) means finding a nonzero integral solution x such that x M δ for some ositive constant δ<1. The constant δ may deend on n, but not on M. Our interest in this aer is in finding a small rimitive solution of (1) in the case where M m, m 2, a solution x with gcd(x 1,..., x n,m)1. A rimitive solution is sought to rule out solutions of the tye y where y satisfies Q(y) 0 (mod ). For the quadratic form Q(x) x 2 1 + + x 2 n,it is clear that any nonzero solution x of (1) must satisfy, max x i 1 n M 1/2. Thus δ 1/2 is the best ossible exonent for a small solution in general.
358 Ali H. Hakami Schinzel, Schlickewi and Schmidt [15] roved that (1) has a nonzero solution with x <M (1/2)+1/2(n 1) for n 3. Thus for any ε>0 we get a nonzero solution of (1) with x <M (1/2)+ε rovided n is sufficiently large. We note that the solution obtained by their method is not necessarily a rimitive solution. Indeed, when M m, m 2 they use a solution of the tye y with Q(y) 0 (mod ). Dealing with M, an odd rime, Heath-Brown [13] obtained a nonzero solution of (1) with x 1/2 log for n 4. His result was an imrovement on the result of [15] in this case. Wang Yuan [16],[17] and [18] generalized Heath-Brown s work to all finite fields. Cochrane, in a sequence of aers [1], [2] and [3] imroved this to x < max{2 19 1/2, 2 22 10 6 }. An imrovement on Cochrane s result is given by Hakami [7, Theorem 1.3] and [10, Theorem 1] who obtained x < min{ 2/3, 2 19 1/2 }. Hakami [8, Theorem 1] and [9, Theorem 1] resectively, generalized Cochrane s method to find a rimitive solution of (1) with x for n 4 when M 2 and with x 3/2, for n 6 when M 3 where Q(y) is nonsingular (mod ). Final Cochrane and Hakami [6, Theorem 1] roved that x for n>1. For M q a roduct of two distinct rimes, we find Heath-Brown [11] obtained x M (1/2)+ε, for n>4 and ε>0. Later Cochrane [5] sharened this result to x M 1/2, for n>4. Our urose in this aer is to generalize (mod) methods for obtaining a small rimitive solution of Q(x) 0 (mod m ). (2) and more generally of obtaining a rimitive solution in a box B with B sufficiently large. Our main result is the following theorem. Theorem 1.1 For any quadratic form Q(x) with n even, n 4 and any odd rime ower m, m 2, there exists a rimitive solution of (2) with x max{6 1/n m[(1/2)+(1/n)], 2 2(n+1)/(n 2) 3 2/(n 2) }. Theorem 1.1 is an immediate consequence of Corollary 3.1. To rove Theorem 1.1 we shall use finite Fourier series over Z m, the ring of integers (mod m ). 2 Basic Identities and Lemmas Henceforth we shall assume that n is even, is an odd rime, and that Q(x) is a nonsingular quadratic form (mod ). Let e m(α) e 2πiα/m. Let V m
Quadratic forms 359 V m(q) be the set of zeros of Q contained in Z n and let m Q (y) be the quadratic form associated with inverse of the matrix for Q (mod ). For y Z n set m e m(x y) for y 0 φ(v m, y) V m m(n 1) for y 0 We abbreviate comlete sums over Z n m and Zn in the manner m x x mod m x 1 1 m x n1,. x mod x 1 1 x n1 The following lemma gives us a formula for φ(v,y). Lemma 2.1 Let n be is even ositive integer. For y Z n, ut y j y in case y. Then where φ(v,y) (mn/2) m j y i foralli δ j jn/2 ω j (y ), δ j { 1 if m j is even, Δ if m j is odd, (3) and 1, Q (y ), ω j (y ) 1, 1 Q (y ), 0, 1 Q (y ). The roof of Lemma 2.1 is given (with some work) in Carlitz [12], and in comlete detail in [11, Theorem 1]. Let α(x) be a comlex valued function defined on Z n with Fourier exansion α(x) y a(y)e m(x y) where a(y) mn x α(x)e m( x y). Then m by definition of φ(v,y),
360 Ali H. Hakami a(y) e m(y x) m α(x) a(0) V m + y 0 m a(0) V m + a(y) φ(v,y) y 0 a(0) [ φ(v,0)+ m(n 1)] + a(y) φ(v,y) y 0 a(0) m(n 1) + y a(y) φ(v,y) mn x α(x) m(n 1) + y a(y) φ(v,y). Thus by Lemma 2.1, m α(x) m x m x α(x)+ (mn/2) m y α(x)+ (mn/2) m a(y) δ j jn/2 j y i for all i y j y i for all i δ j jn/2 ω j (y ) a(y) ω j (y ). But by noticing that the inner sum y a(j y ) ω j (y ) can be written y j y i for all i y Q (y ) y Q (y ) a(y) ω j (y ) y (mod ) a( j y ) ( 1) a( j y ) ω j (y ) a( j y ) 1 y 1 Q (y ) y 1 Q (y ) a( j y ) 1 a( j y ), we obtain, Lemma 2.2 (The fundamental identity) For any comlex valued α(x)
Quadratic forms 361 as given above α(x) m m + mn/2 jn/2 δ j x (mod m ) j α(x) y i 1 Q (y ) a( j y ) j 1 y i 1 1 Q (y ) a( j y ), where δ j as we defined in (3): δ j 1when m j is even and δ j Δif m j is odd. 3 Proof of Theorem 1.1 when Δ + 1 Let B be the box of oints in Z n given by B {x Z n a i x i <a i + B i, 1 i n}, (4) where a i, B i Z and 1 B i m, 1 i n. Then B n i1 B i, the cardinality of B. View the box B in (4) as a subset of Z n and let α(x) χ B χ m B with Fourier exansion α(y) y a(y)e m(x y). Thus for any y Zn m, a( y) mn n sin 2 πb i y i / m sin 2 πy i / m where the term in the roduct is taken to be m i if y i 0. In articular n { ( ) } a(y) mn min Bi 2 m 2,. 2y i Consider the congruence (2): i1 i1 Q(x) 0 (mod m ). For later reference, we construct the following lemma which hel us on estimating the sum y i 1 a(j y). Lemma 3.1 Let B be any box of tye (4) and α(x) χ B χ B (x). Then we have a( j y) B 2B i jn. y i 1 B i >,
362 Ali H. Hakami Proof. First, y i 1 a( j y) y m x i 1 m x i 1 m x i 1 1 m α(x)e m( x j y) 1 mn α(x) y i e m( x j y) 1 α(x) mn m x i 1 x 0 (mod ) m jn y i 1 x i 1 x 0 (mod ) jn 1 jn u B v B u+v 0(mod ) n ] i1 B i ([ Bi e ( x y) mn α(x) ()n α(x) ) +1. (5) To verify the last inequality in (5) we count the number of solutions of the congruence u + v 0 (mod ), with u, v B. In fact for each choice of v, there are at most n i1 ([B i/ ]+ 1) ch-oices u. Thus the total number of solutions is less than or equal to n i1 ([B i/ ] + 1). Hence it follows from (5), 2 y i 1 a(y) jn n i1 We may slit the roduct in (6) such that n ([ ] ) Bi B i +1 i1 Then by this equality we therefore have y i 1 a( j y) B jn B i < B i ([ ] ) Bi B i +1. (6) B i > B i > B i 2B i. ( 2Bi ).
Quadratic forms 363 Our roof is comlete. Now we can roceed to estimate the error term in the fundamental identity which is given by Error mn/2 jn/2 δ j j y i 1 Q (y ) a( j y ) j 1 y i 1 1 Q (y ) a( j y ). By Lemma 3.1 we get Error mn/2 jn/2 ϕ(y) a( j y ) y i 1 (7) where Continuing from (7), ϕ(y) { j j 1 if Q (y), j 1 if 1 Q (y). Error mn/2 (jn/2) j B jn mn/2 1 B (jn/2)+j B i > B i > 2B i 2B i. (8) We now restrict our attention to the case where B is cube, that is, B i B, 1 i n. Say j 0 B< j 0+1 for some j 0 N, 1 j 0 <m. Then B m j j 0 j m j 0. Continuing from (8), we get Error mn/2 B 2n B 2 mn/2 j 0 2n 0 1 B (jn/2)+j + 1 nm nj mn/2 B (jn/2)+j j 0 (n/2)j j + mn/2 B 0 1 < 2n B 2 mn/2 [(n/2) 1]() 2+ mn/2 B 2 1 (jn/2)+j
364 Ali H. Hakami 2 n+1 B 2 m (n/2)+1 +2 mn/2 B 2n+1 B 2 m+(n/2) 1 +2mn/2 B. Proosition 3.1 Suose that m 2, n 4, neven. Then for any cube B centered at the origin, where α(x) B 2 m Error, Error 2n+1 B 2 +2 mn/2 B. } m+(n/2) 1 }{{}{{} Error 2 Error 1 We comare Error 1 and Error 2 in Proosition 3.1 to the main term B 2 / m. In order to make the left-hand side ositive, we make each error term less than 1/3 of the main term. For the error term Error 1, we need 2 n+1 B 2 < 1 B 2 (n/2) 1 > 3.2 n+1 m+(n/2) 1 3 m For the error term Error 2, >2 2(n+1)/(n 2) 3 2/(n 2). (9) 2 mn/2 B < 1 B 2 B > 3.2 (mn/2)+m 3 m B>6 1/n (m/2)+(m/n). (10) Collecting together the two criteria (9) and (10), we obtain Theorem 3.2 Suose that m 2, n 4, neven. Then for any cube B centered at the origin, if (9), (10) hold, then α(x) 1 B 2 3. m In articular V (B + B) B 3 m.
Quadratic forms 365 It remains to rove under the hyothesis of Theorem 3.2, the existence of rimitive solutions of the congruence (1). Recall x is called rimitive if gcd(x 1,..., x n,) 1. We shall write x for imrimitive oints. Thus we have to rove α(x) > α(x). x Corollary 3.1 Suose m 2, n>m, n even, >2 2(n+1)/(n 2) 3 2/(n 2) and B is a cube with B>6 1/n (m/2)+(m/n). Then B + B contains a rimitive solution of (2). Proof. As in the roof of Lemma 3.1 with j m 1, we have x α(x) α(x) < x i, 16i6n n ([ ] Bi B i i1 B n ([ Bi ] +1 < B2n n (1+ B B2n n (1 + ε)n, 1 u B v B u+v 0(mod) ) n ) +1 where ε< [(m/2)+(m/n)]+1. However according to Theorem 3.2 we have, ) n α(x) B 2 3 m. Hence it follows that α(x) -x B 2 3 m x α(x) B 2 3 m B2n n (1 + ε)n > 0, by our hyotheses on the size of. This comletes the roof. References [1] T. Cochrane,Small zeros of quadratic forms modulo, J. Number Theory, 33, no.3 (1989), 286-292.
366 Ali H. Hakami [2] T. Cochrane, Small zeros of quadratic forms modulo, II, Proceedings of the Illinois Number Theory Conference, (1989), Birkhauser, Boston (1990), 91-94 [3] T. Cochrane, Small zeros of quadratic forms modulo, III, J. Number Theory, 33, no. 1 (1991), 92-99. [4] T. Cochrane, Small zeros of quadratic congruences modulo q, Mathematika, 37 (1990), no. 2, 261-272. [5] T. Cochrane, Small zeros of quadratic congruences modulo q, II, J. Number Theory 50 (1995), no. 2, 299-308. [6] T. Cochrane and A. Hakami, Small zeros of quadratic congruences modulo 2, II, Proceedings of the American Mathematical Society 140 (2012), no.12, 4041-4052. [7] A. Hakami, Small zeros of quadratic congruences to a rime ower modulus, PhD thesis, Kansas State University, 2009. [8] A. Hakami,Small zeros of quadratic forms modulo 2, JP J. Algera, Number Theory and Alications, 17,no. 2, (2011),141-162. [9] A. Hakami, Small zeros of quadratic forms modulo 3, J. Advanvces and Alications in Mathematical Sciences 9 (2011), no. 1, 47-69. [10] A. Hakami, On Cochrane s estimate for small zeros of quadratic forms modulo, Far East J. Math. Sciences,50 (2011), no. 2, 151-157. [11] A. Hakami, Weighted quadratic artitions (mod m ), A new formula and new demonstration, Tamaking J. Math., 43, (2012), 11-19. [12] L. Carlitz, Weighted quadratic artitions (mod r ), Math Zeitschr. Bd, 59, (1953), 40-46. [13] D.R. Heath-Brown, Small solutions of quadratic congruences, Glasgow Math. J, 27 (1985), 87-93. [14] D.R. Heath-Brown, Small solutions of quadratic congruences II, Mathematika, 38 (1991), no. 2, 264-284. [15] A. Schinzel, H.P. Schlickewei and W.M. Schmidt, Small solutions of quadratic congruences and small fractional arts of quadratic forms, Acta Arithmetica, 37 (1980), 241-248. [16] Y. Wang, On small zeros of quadratic forms over finite fields, Algebraic structures and number theory (Hong Kong, 1988), 269 274, World Sci. Publ., Teaneck, NJ, 1990.
Quadratic forms 367 [17] Y. Wang, On small zeros of quadratic forms over finite fields, J. Number Theory, 31 (1989) 272-284. [18] Y. Wang, On small zeros of quadratic forms over finite fields II, A Chinese summary aears in Acta Math. Sinica 37 No.5 (1994), 719-720. Acta Math. Sinica (N.S.) 9 no.4 (1993), 382-389. Received: October, 2012