Distribution of Matrices with Restricted Entries over Finite Fields

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1 Distribution of Matrices with Restricted Entries over Finite Fields Omran Ahmadi Deartment of Electrical and Comuter Engineering University of Toronto, Toronto, ON M5S 3G4, Canada Igor E. Sharlinski Deartment of Comuting Macquarie University, Sydney, NSW 209, Australia January 0, 2007 Abstract For a rime, we consider some natural classes of matrices over a finite field F of elements, such as matrices of given rank or with characteristic olynomial having irreducible divisors of rescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with comonents in a given subinterval [ H, H] [ ( )/2,( )/2] is asymtotically close to the exected value. Introduction For integer numbers m and n we use M m,n (F ) to denote the set of m n matrices over the field F of elements where is a rime. We always assume that F is reresented by the elements of the set {0, ±,..., ±( )/2}. Accordingly given a ositive integer H ( )/2

2 we use M m,n (H; F ) to denote the set of (2H +) n2 matrices X = (x ij ) m n M m,n (F ), with x ij H for i m and j n. For square matrices we also ut M n (F ) = M n,n (F ) and M n (H; F ) = M n,n (H; F ). Let T n be the set of vectors t = (t,...,t n ) with nonnegative integer comonents such that n t j j = n. j= We say that a olynomial f of degree n over F is of factorisation attern t T n if it has exactly t j irreducible factors of degree j, j =,...,n. For examle, t = (0,...,0, ) corresonds to irreducible olynomials and t = (n, 0,..., 0) corresonds to olynomials which slit in F. Motivated by a work of I. Rivin [7], we study various questions about the distribution of matrices X M m,n (H; F ). Another motivation for our work comes from the results of W. Duke, Z. Rudnick and P. Sarnak [6] which yield an asymtotic formula for the number of matrices in SL n (Z) of restricted Euclidean norm, as well as from a recent extension of these results (in the case n = 2) to algebraic number fields by C. Roettger [8]. More recisely, we obtain asymtotic formulas for the number R m,n (k, H; F ) of matrices X M m,n (H; F ) of rank at most rkx k, for the number F n,t (H; F ) of matrices X M n (H; F ) whose characteristic olynomial f has a rescribed factorisation attern t T n, and for the number U n (H; F ) of matrices X M n (H; F ) with det X =, that is, the number of matrices X M n (H; F ) SL n (F ). As we have mentioned these questions are F analogues of the results of similar sirit for matrices over Z and algebraic number fields, see [6, 7, 8] and references therein. We use these roblems to demonstrate several techniques which can be alied to many other similar questions and allow us to show that the number of matrices in certain classes and also with comonents in a given subinterval 2

3 [ H, H] [ ( )/2, ( )/2] is asymtotically close to the exected value. Throughout the aer, the imlied constants in the symbols O, and may deend on integer arameters k and d. We recall that the notations U = O(V ) and U V are all equivalent to the assertion that the inequality U cv holds for some constant c > 0. 2 Prearations 2. Determinant Varieties Let X = (x ij ) m n be the m n matrix in variables x,...,x mn, and let I C[x,..., x mn ] be the ideal generated by (k + ) (k + ) minors of X. Now let the affine set V k (I) be the set containing zeros of I in C mn. It is easy to see that the algebraic set V k (I) can be identified with M m n (C; k) where M m n (C; k) denotes the set of m n matrices over C of rank at most k. We say that an algebraic variety is not contained in a hyerlane if it is not contained in the zero set of an ideal in C[x,..., x mn ] generated by a nontrivial linear form in x,...,x mn. We need the following well-known result (for a simle roof see []) which is crucial in what follows. Lemma. The set V k (I) is an irreducible variety of dimension k(m+n k) in C mn and it is not contained in a hyerlane. Let U n be the affine set in C[x,...,x nn ] associated with SL n (C) matrices, that is the zero set of the equation det X = where X = (x ij ) n n. We have the following analogue of Lemma, see [2, Chater I, Section.6] or [6, Chater 3, Examle 2.2], which in fact can easily be derived from Lemma by examining degrees of ossible factors of the olynomial det X. Lemma 2. The set U n is an irreducible variety of dimension n 2 in C n2 and it is not contained in a hyerlane. 2.2 Distribution of Points on Varieties Now let F = {F, F 2,..., F r } be a family of r olynomials over Z in s variables. The set of solutions over C or F to the system of equations F j (a,...,a s ) = 0, 3 j =,...,r,

4 is called the zero set of F over C or F, resectively. Let Z F (H; F ) be the set of vectors (a, a 2,...,a s ) F s with a i H, i =, 2,..., s which are in the zero set of F over F. We also ut Z F (F ) = Z F (( )/2; F ). We need the following slight modification of a result of Fouvry [7]. Lemma 3. Suose that the affine zero-set of F = {F, F 2,...,F r } in C s is an irreducible variety of dimension d and is not contained in a hyerlane of C s. Then ( ) s 2H + #Z F (H; F ) = #Z F (F ) +O ( d/2 (log ) s + H d /2 (log ) s d+). For an s-dimensional vector a = (a,...,a s ) F s, we use T s(a, H; ) to denote the number of λ F for which a j λ b j (mod ), with b j H, j =,...,s. The following result is a secial case of several more general results which are essentially due to N. M. Korobov [9], which can also be found in many other works, see, for examle, [4, 5]. We resent it in a form which immediately follows from [5, Theorems 5.6 and 5.0]. Lemma 4. We have T (2H + )s s(a, H; ) ( ) s s (log ) s. a F s Let r m,n (k; F ) be the total number of m n matrices of rank k over F. The following exlicit formula for r m,n (k; F ) is well known, see, for examle, [3] for this and many other related formulas. Lemma 5. For any k 0, we have, r m,n (k; F ) = k i=0 (m i ) k i=0 (n i ) k. i=0 (k i ) 4

5 For a monic olynomial f F [T] of degree n, we denote by G n (f; F ) the set of matrices X M n (F ) whose characteristic olynomial is equal to f. By a result of Chavdarov [3, Theorem 3.9], if f(0) 0, then ( 3) n2 n #G n (f; F ) ( + 3) n2 n. Therefore we obtain the following estimate. Lemma 6. Let f F [T] be a monic olynomial of degree n, and let f(0) 0. Then #G n (f; F ) = n2 n + O( n2 n ). Notice that olynomials in the above lemma corresond to matrices in GL n (F ), the general linear grou over F. Finally, for t T n we denote by F n (t; F ) the set of monic olynomials f F [T] with a factorisation attern t. It is well-known (see, for examle, [4, 5, 9, 20]) that simle counting arguments imly the following asymtotic formula for the cardinality of F n (t; F ). Lemma 7. For every t = (t,..., t n ) T n, we have #F n (t; F ) = n n j= 2.3 Distribution of Products t j!j t j + O(n ). Let N a (H, ) denote the number of solutions to the congruence xy a (mod ), x, y H. The following bound on the average deviation between N a (H, ) and its exected value taken over a ( )/2 is a secial case of a more general estimate from [2] (and also the trivial estimate N a (H, ) = O(H)). Lemma 8. We have, ( )/2 a= ( )/2 N a(h, ) (2H + )2 2 H 2 o(). 5

6 3 Results 3. Ranks of Matrices of Bounded Height Theorem 9. For H ( )/2 and k min{m, n}, we have R m,n (k, H; F ) = (2H + ) mn (m k)(n k) +O ( k(m+n k)/2 (log ) mn + H k(m+n k) /2 (log ) (m k)(n k)+). Proof. From Lemma and Lemma 3, alied with ( )( ) m n r =, s = mn, d = k(m + n k), k k we infer that ( 2H + R m,n (k, H; F ) = R m,n (k; F ) ) mn +O ( k(m+n k)/2 (log ) mn + H k(m+n k) /2 (log ) (m k)(n k)+). By Lemma 5 we see that R m,n (k; F ) = k r m,n (l; F ) = k(m+n k) + O ( k(m+n k) ) l=0 which imlies the desired result. where One can easily see that Theorem 9 is nontrivial whenever γ k,m,n = max { /2 + H γ k,m,n+ε (m k)(n k), mn } 2(m k)(n k) + 2 for some fixed ε > 0 and sufficiently large. Secially when m = n and k = n (the case of singular matrices), then the result is nontrivial whenever H 3/4+ε. 6

7 3.2 Factors of Characteristic Polynomials of Matrices of Bounded Height Theorem 0. For H ( )/2 and t T n, we have F n,t (H; F ) = (2H + ) n2 n j= ( t j!j + O n2 (log ) n2). t j Proof. Clearly, if f(t) F [T] is the characteristic olynomial of X M n (F ), then for every λ F, the characteristic olynomial of λx M n (F ) is λ n f(tλ ) and thus has the same factorisation attern. Therefore F n,t (H; F ) = = (2H + )n2 n2 + f F n(t;f ) X G n(f;f ) λ F λx M n(h;f ) f F n(t;f ) X G n(f;f ) f F n(t;f ) X G n(f;f ) Using Lemmas 6 and 7, we derive f F n(t;f ) X G n(f;f ) = = f F n(t;f ) X G n(f;f ) f(0) 0 f F n(t;f ) f(0) 0 = n2 n = n2 n j= λ F λx M n(h;f ) + O ( n2 n + O( n2 n ) f F n(t;f ) f(0) 0 ( ) + O n2 t j!j t j + O(n2 ), X M n(f ) X singular (2H + )n2 n2 ) + O. ( ) n2 7

8 since obviously f F n(t;f ) f(0) 0 = f F n(t;f ) + O( n ). On the other hand, using Lemma 4, we estimate (2H + )n2 n2 f F n(t;f ) X G n(f;f ) λ F λx M n(h;f ) (2H + )n2 f F n(t;f ) X G n(f;f ) n2 λ F λx M n(h;f ) (2H + )n2 ( ) X M n(f ) n2 λ F n2 (log ) n2, λx M n(h;f ) which concludes the roof. One can easily see that Theorem 0 is nontrivial whenever H /n2 +ε for some fixed ε > 0 and sufficiently large. 3.3 Matrices of Bounded Height in SL n (F ) Following the same arguments as in the roof of Theorem 9 and using Lemma 2 instead of Lemma and recalling that #SL n (F ) = #GL n(f P ) = n ( n i ) = n2 + O( n2 2 ), we immediately obtain: i=0 8

9 Theorem. For H ( )/2, we have U n (H; F ) = (2H + )n2 ) + O (H n2 2 /2 (log ) 2. The bound of Theorem is nontrivial if H 3/4+ε. for any fixed ε > 0 and sufficiently large. However for n = 2 a different argument leads to a stronger result. Theorem 2. For H ( )/2, we have Proof. Let us define U 2 (H; F ) = (2H + )4 + O ( H 2 o()). a (H, ) = N a (H, ) (2H + )2 and note that ( )/2 a= ( )/2 a (H, ) = ( )/2 a= ( )/2 a+ (H, ) = 0. Then we have U 2 (H; F ) = = = ( )/2 a= ( )/2 ( )/2 a= ( )/2 (2H + )4 N a (H, )N a+ (H, ) ( (2H + ) 2 + ( )/2 a= ( )/2 ) ( ) (2H + ) 2 + a (H, ) + a+ (H, ) a (H, ) a+ (H, ). 9

10 Bu the Cauchy inequalty ( )/2 a= ( )/2 a (H, ) a+ (H, ) = ( )/2 a= ( )/2 ( )/2 a= ( )/2 a (H, ) 2 a (H, ) 2. ( )/2 a= ( )/2 a (H, ) 2 Now an alication of Lemma 8 concludes the roof. Clearly Theorem 2 is nontrivial if H /2+ε. for any fixed ε > 0 and sufficiently large. 4 Comments Analogues of Theorem 9 can be roven about the symmetric matrices over F. More recisely, suose that Y = (y ij ) n n is the n n symmetric matrix in variables y ij = y ji for i j n, and let I C[y,...,y nn ] be the ideal generated by (k + ) (k + ) minors of Y. Also suose that W k (I) is the set containing zeros of I in C (n+ 2 ). Notice that Wk (I) can be identified with the set of symmetric n n matrices over C of rank at most k. It follows that (see [0]) W k (I) is an irreducible variety in C (n+ 2 ) and is not contained in a hyerlane. Thus alying Lemma 3 one can get similar results as Theorem 9 for symmetric matrices over F. The determinant variety is not smooth, so the results about the distribution of oints on such varieties, see [8,, 2, 22, 23] and references therein, do not aly. Acknowledgements The authors wish to thank Igor Rivin for many stimulating discussions. 0

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