DSpace at VNU: On the permanents of matrices with restricted entries over finite fields

DSpace at VNU: On the permanents of matrices with restricted entries over finite fields tài liệu, giáo án, bài giảng , l...

c 2012 Society for Industrial and Applied Mathematics Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php SIAM J DISCRETE MATH Vol 26, No 3, pp 997–1007 ON THE PERMANENTS OF MATRICES WITH RESTRICTED ENTRIES OVER FINITE FIELDS∗ LE ANH VINH† Abstract For a prime power q, we study the distribution of permanents of n × n matrices over a finite field Fq of q elements We show that if A is a sufficiently large subset of Fq , then the set of permanents of n × n matrices with entries in A covers all (or almost all) of F∗q When q = p is a prime, and A is a subinterval of [0, p − 1] of cardinality |A| p1/2 log p, we show that the number of matrices with entries in A having permanent t is asymptotically close to the expected value Key words permanents, matrices with restricted entries, finite fields AMS subject classifications 15A15, 15B33, 11L07 DOI 10.1137/110835050 Introduction Throughout this paper, let q = pr , where p is an odd prime and r is a positive integer Let Fq be a finite field of q elements The prime base field Fp of Fq may then be naturally identified with Zp = Z/pZ Let M be an n × n matrix Two basic parameters of M are its determinant n Det(M ) := sgn(σ) aiσ(i) i=1 σ∈Sn and its permanent n Per(M ) := aiσ(i) σ∈Sn i=1 The distribution of the determinant of matrices with entries in a finite field Fq has been studied by various researchers Suppose that the ground field Fq is fixed and M = Mn is a random n × n matrix with entries chosen independently from Fq If the entries are chosen uniformly from Fq , then it is well known that (1.1) (1 − q −i ) as n → ∞ Pr(Mn is nonsingular) → i It is interesting that (1.1) is quite robust Specifically, Kahn and Koml´os [9] proved a strong necessary and sufficient condition for (1.1) Theorem 1.1 (see [9]) Let Mn be a random n × n matrix with entries chosen according to some fixed nondegenerate probability distribution μ on Fq Then (1.1) holds if and only if the support of μ is not contained in any proper affine subfield of Fq ∗ Received by the editors May 23, 2011; accepted for publication (in revised form) April 17, 2012; published electronically July 19, 2012 This research was supported by Vietnam National University, Hanoi under the project Some problems on matrices over finite fields An extended abstract of this work appeared in Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb 2009), Electron Notes Discrete Math 34, Elsevier Sci B.V., Amsterdam, 2009, pp 519–523 [13] http://www.siam.org/journals/sidma/26-3/83505.html † University of Education, Vietnam National University, Hanoi, Vietnam (vinhla@vnu.edu.vn) 997 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 998 LE ANH VINH An extension of the uniform limit to random matrices with μ depending on n was considered by Kovalenko, Leviskaya, and Savchuk [10] They proved that the standard limit (1.1) holds under the condition that the entries mij of M are independent and Pr(mij = α) > (log n + ω(1))/n for all α ∈ Fq The behavior of the nullity of Mn for − μ(0) close to log n/n and μ(α) = (1 − μ(0))/(q − 1) for = was also studied by Blă omer, Karp, and Welzl [3] Another direction is to fix the dimension of matrices For an integer number n and a subset E ⊆ Fnq , let Mn (E) denote the set of n × n matrices with rows in E For any t ∈ Fq , let Dn (E; t) be the number of n × n matrices in Mn (E) having determinant t Ahmadi and Shparlinski [1] studied some natural classes of matrices over finite fields Fp of p elements with components in a given subinterval [−H, H] ⊆ [−(p − 1)/2, (p − 1)/2] They showed that (2H + 1)n Dn ([−H, H] ; t) = (1 + o(1)) p n (1.2) q 3/4 In the case n = 2, the lower bound can be improved to if t ∈ F∗q and H H q 1/2+ε for any constant ε > Covert et al [4] studied this problem in a more general setting A subset E ⊆ Fnq is called a product-like set if |Hd ∩ E| |E|d/n for any d-dimensional subspace Hd ⊂ Fnq Covert et al [4] showed that D3 (E; t) = (1 + o(1)) |E|3 q q 15/8 Using the if t ∈ F∗q and E ⊂ F3q is a product-like set of cardinality |E| geometry incidence machinery developed in [4], and some properties of nonsingular matrices, the author [12] obtained the following result for higher dimensional cases (n ≥ 4): |A|n Dn (A ; t) = (1 + o(1)) q n d q 2d−1 if t ∈ F∗q and A ⊆ Fq of cardinality |A| On the other hand, little has been known about the permanent The only known uniform limit similar to (1.1) for the permanent is due to Lyapkov and Sevast’yanov [11] They proved that the permanent of a random n × m matrix Mnm with elements from Fp and independent rows has the limit distribution of the form lim Pr(Per(Mnm ) = k) = ρm δk0 + (1 − ρm )/p, k ∈ Fp , n→∞ where δk0 is the Kronecker delta Here, as usual, the permanent Per(A) of an n × m matrix A is defined as the sum of all possible products of n coefficients of A chosen such that no two of the coefficients are taken from the same row nor from the same column The purpose of this paper is to study the distribution of the permanent when the dimension of matrices is fixed For any t ∈ Fq and E ⊂ Fdq , let Pn (E; t) be the number of n × n matrices with rows in E having determinant t We are also interested in the set of all permanents, Pn (E) = {Per(M ) : M ∈ Mn (E)} The first result of this paper is the following Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PERMANENTS OF MATRICES OVER FINITE FIELDS 999 Theorem 1.2 Suppose that q is an odd prime power and gcd(q, n) = n+1 (a) If E ∩ (F∗q )n = ∅, and |E| > cq , then F∗q ⊆ Pn (E) q 2/3 , then for each t ∈ F∗q , (b) If A ⊆ Fq of cardinality |A| P3 (A3 ; t) = (1 + o(1)) |A|9 q We have an immediate corollary of Theorem 1.2 Corollary 1.3 Suppose that q is an odd prime power and gcd(q, n) = (a) If E ⊂ Fnq of cardinality |E| > nq n−1 , then F∗q ⊆ Pn (E) 1 (b) If A ⊂ Fq of cardinality |A| q + 2n , then F∗q ⊆ Pn (An ) Note that the bound in the first part of Corollary 1.3 is tight up to a factor of n For example, |{x ∈ Fnq : x1 = 0}| = q n−1 and Pn ({x ∈ Fnq : x1 = 0}) = When E is a product-like set, we can get a positive proportion of the permanents under a weaker assumption Theorem 1.4 Suppose that q is an odd prime power and gcd(q, n) = If E ⊂ Fnq is a product-like set of cardinality |E| |Pn (E)| n2 q 2n−1 , then (1 + o(1))q In the special case E = A × · · · × A, we have the following corollary Corollary 1.5 Suppose that q is an odd prime power and gcd(q, n) = If 1 A ⊆ Fq of cardinality |A| q + 2n−1 , then |Pn (An )| (1 + o(1))q Furthermore, if we restrict our study to matrices over a finite field Fp of p elements (p is a prime) with components in a given interval, we obtain a stronger result Theorem 1.6 Suppose that q = p is a prime, and entries of M are chosen from a given interval I := [a + 1, a + b] ⊆ Fp , where p1/2 b → ∞, p → ∞; log p then Pn (I n ; t) = (1 + o(1)) bn p for any t ∈ Fp As we will see in the proof of Theorem 1.6, the method can be applied to a large variety of multihomogeneous forms In particular, we have a similar result for the determinant Theorem 1.7 Suppose that q = p is a prime and entries mij of M are chosen from a given interval I := [a + 1, a + b] ⊆ Fp with b → ∞, p → ∞; p1/2 log p then bn Dn (I ; t) = (1 + o(1)) p n for any t ∈ Fp Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1000 LE ANH VINH This result improves [1, Theorem 11] and [12, Theorem 1.5] The proof of Theorem 1.7 (and the statement for multihomogeneous forms) is left for interested readers Throughout the paper, the implied constants in the symbols o and may depend on integer parameter n We recall that the notation U V is equivalent to the assertion that the inequality U c|V | holds for some constant c > Some estimates 2.1 Incidence estimates Let f be a complex-valued function on Fnq The Fourier transform of f on Fnq with respect to a nontrivial principal additive character ψ on Fq is given by fˆ(m) = q −n f (x)ψ(−x · m) x∈Fn q One of our main tools is a two-set version of the geometric incidence theorem developed by Hart et al in [8] (see also [7] for an earlier version and [4] for a functional version of this theorem) Note that going from a one-set formulation in the proof of Theorem 2.1 in [8] to a two-set formulation is just a matter of inserting a different letter in a couple of places Theorem 2.1 (see [8, Theorem 2.1]) For any E, F ⊂ Fnq , let νt (E, F ) = E(x)F (y), x·y=t where here and throughout the paper, E(x) denotes the characteristic function of E (a) Then νt (E, F ) = |E||F |q −1 + Rt (E, F ), (2.1) where |Rt (E, F )|2 |E||F |q n−1 , if t = 0, and |R0 (E, F )|2 |E||F |q n (b) Moreover, if (0, , 0) ∈ E, then νt2 (E, F ) (2.2) |E|2 |F |2 q −1 + |E|q 2n−1 |Fˆ (k)|2 |E ∩ lk |, k=(0, ,0) t∈Fq where lk = {tk : t ∈ F∗q } 2.2 Arithmetic estimates The congruence x1 x2 + x3 x4 ≡ λ (mod p), (2.3) where p is a large prime, arises in many problems of number theory (see, for example, [2, 5] and the references therein) Let Li , Ni , i 4, be integers with Li < Li + Ni < p Denote by J(λ) the number of solutions of congruence (2.3) in the box (2.4) Li + xi L i + Ni (1 i 4) Ayyad, Cochrane, and Zheng [2] proved that (2.5) J(0) = N1 N2 N3 N4 + O( p N1 N2 N3 N4 log2 p) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1001 Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PERMANENTS OF MATRICES OVER FINITE FIELDS It has been shown by Garcia [6], in a personal communication, that a similar estimate holds for J(λ) when λ = Theorem 2.2 (see [6]) The following asymptotic formula holds for any λ ∈ Fq : (2.6) J(λ) = N1 N2 N3 N4 + O( p N1 N2 N3 N4 log2 p) Proof Writing J(λ) in terms of character sums, we have p−1 J(λ) = χ(x∗1 x∗2 (λ − x3 x4 )), χ x1 x2 x3 x4 where χ runs over the set of multiplicative characters modulo p, and the range for the variables in summations over x1 , x2 , x3 , and x4 is defined by (2.4) In addition, x∗ denotes the inverse multiplicative of x (mod p) Separating the principal character χ = χ0 , we have J(λ) = N1 N2 N3 N4 + R(λ) p By the Cauchy–Schwarz inequality, we have R(λ) p−1 χ=χ0 ⎛ ⎝ p−1 χ(x∗1 x∗2 ) x1 χ=χ0 (2.7) = J1 − N12 N22 p−1 x2 χ(λ − x3 x4 ) x3 ⎞1/2 ⎛ χ(x∗1 x∗2 ) ⎠ x1 x2 1/2 J2 − x4 N32 N42 p−1 ⎝ p−1 ⎞1/2 χ(λ − x3 x4 ) ⎠ χ=χ0 x3 x4 1/2 , where, J1 and J2 denote, respectively, the number of solutions of the congruence x1 x2 ≡ x1 x2 (mod p), L1 + λ − x3 x4 ≡ λ − x3 x4 (mod p), L3 + x1 , x1 x3 , x3 L + N1 , L + L + N3 , L + x2 , x2 x4 , x4 L + N2 ; L + N4 It follows from (2.5) that (2.8) (2.9) N12 N22 p−1 N32 N42 J2 − p−1 J1 − N1 N2 log2 p, N3 N4 log2 p The theorem follows immediately from (2.7), (2.8), and (2.9) Corollary 2.3 For any L1 , L2 , L3 , L4 , N1 , N2 , N3 , N4 with Ni → ∞, p → ∞, log p p1/2 let J(λ) be the number of solutions of x1 x4 + x2 x3 = λ, where xi ∈ [Li +1, Li +Ni ] (mod p) Then J(λ) = (1+o(1)) N1 N2qN3 N4 for any λ ∈ Fp Proof If [Li + 1, Li + Ni ] (mod p) is not a subinterval of [1, p − 1], we break it into two disjoint subintervals and an element Applying Theorem 2.2 for these subintervals, and adding up the results, we obtain the corollary Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1002 LE ANH VINH Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2) 2.3 Estimation of matrices with small per-rank Let Pn (An ) be the number of n × n matrices M with entries in A such that all (n − 1) × (n − 1) minors of M have permanent Lemma 2.4 For any fixed n and A ⊂ Fq , we have Pn(2) (An ) = O(|A|n −4 ) Proof The proof proceeds by induction The base case n = is trivial Suppose that the statement holds for n − We show that it also holds for n For any n × n matrix M = (xij )1 i,j n with entries in A, let Mst (2 s, t n) be the submatrix obtained from M by deleting the first row, the sth row, the first column, and the tth column Suppose that all (n − 1) × (n − 1) minors of M have permanent We have two cases (1) Suppose that Per(Mst ) = for some s, t n Without loss of generality, i, j 2, let M ij be the (n − 1) × (n − 1) we assume that Per(M22 ) = For minor of M obtained by deleting the ith column and the jth row Let i1 = − i and j1 = − j; then Per(M ij ) = xi1 j1 Per(M22 ) + α for some α does not depend on xi1 j1 Therefore, if we fix M22 with Per(M22 ) = 0, there is at most one choice of xi1 j1 such that Per(M ij ) = Hence, this case contributes at n2 −4 most n−1 = O(|A|n −4 ) matrices whose (n − 1) × (n − 1) minors have zero |A| permanent (2) Suppose that Per(Mst ) = for all s, t n From the induction hypoth2 (2) esis, this case contributes at most |A|2(n−1)+1 Pn−1 (An−1 ) = O(|A|2n−1+(n−1) −4 ) = O(|A|n −4 ) matrices whose (n − 1) × (n − 1) minors have zero permanent Putting these two cases together, we complete the proof of the lemma Proof of Theorem 1.2 (a) Fix an a = (a1 , , an ) ∈ E ∩ (F∗q )n For any x = (x1 , , xn ), and y = (y1 , , yn ) ∈ E, let M (a; x, y) denote the matrix whose rows are x, y and (n − 2) a’s Letting := (1, , 1), x/a := (x1 /a1 , , xn /an ), and y/a := (y1 /a1 , , yn /an ), we have n Per(M (a; x, y)) = Per(M (1; x/a, y/a)) i=1 n = n i=1 i=1 xi j=i yj aj Set (3.1) (3.2) E1 := {(xi /ai )ni=1 : (x1 , , xn ) ∈ E} , ⎧⎛ ⎫ ⎞n ⎨ ⎬ yi /ai ⎠ : (y1 , , yn ) ∈ E E2 := ⎝ ⎩ ⎭ j=i i=1 It is clear that |E1 | = |E2 | = |E| (as gcd(n, q) = 1) From (2.1), for any t ∈ F∗q , we have (3.3) νt (E1 , E2 ) = q −1 |E1 ||E2 | + O q d−1 |E1 ||E2 | = q −1 |E|2 + O(q (d−1)/2 |E|) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1003 Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PERMANENTS OF MATRICES OVER FINITE FIELDS This follows that F∗q ⊂ {Per(M (a; x, y)) : x, y ∈ E} if |E| > cq (d+1)/2 for some large constant c > (b) For any a, x, y ∈ A3 , let M (a, x, y) denote the × matrix whose rows are a, x, and y We have three cases (1) Suppose that a ∈ (A ∩ F∗q )3 Similarly as above, for a fixed a, we have |{Per(a, x, y) = t : x, y ∈ A3 }| = |A|6 |A|6 + O(q|A|3 ) = (1 + o(1)) q q Since |(A ∩ F∗q )3 | = (1 + o(1))|A|3 , (3.4) |{Per(a, x, y) = t : a ∈ (A ∩ F∗q )3 , x, y ∈ A3 }| = (1 + o(1)) |A|9 q (2) Suppose that a = (0, a2 , a3 ) for some a2 , a3 ∈ A We have Per(a, x, y) = x1 (a3 y2 + a2 y3 ) + y1 (a3 x2 + a2 x3 ) Therefore, for any choice of a2 , a3 , x2 , x3 , y1 , y2 , y3 , there is at most one possibility of x1 such that Per(a, x, y) = t (t ∈ F∗q ) This implies that |{Per(a, x, y) = t : a = (0, a2 , a3 ), x, y ∈ A3 }| = O(|A|7 ) (3.5) (3) Suppose that a = (0, 0, a3 ) for some a3 ∈ A We have Per(a, x, y) = a3 (x1 y3 + x2 y1 ) Therefore, for any choice of x1 , y3 , x2 , y1 , there is at most one possibility of a3 such that Per(a, x, y) = t (t ∈ F∗q ) This implies that |{Per(a, x, y) = t : a = (0, 0, a3 ), x, y ∈ A3 }| = O(|A|6 ) (3.6) Putting (3.4), (3.5), and (3.6) together, we have P3 (A3 ; t) = (1 + o(1)) if |A| |A|9 |A|9 + O(|A|7 ) = (1 + o(1)) q q q 2/3 This completes the proof of Theorem 1.2 Proof of Theorem 1.4 Since E is a product-like set, E ∩ (F∗q )n = ∅ Fix an a ∈ E ∩ (F∗q )n Define E1 and E2 as in (3.1) and (3.2), respectively Similarly to section 3, we have Pn (E; t) (4.1) |{x · y : x ∈ E1 , y ∈ E2 }| Removing (0, , 0) from E does not increase the lower bound of Pn (E; t), so we may assume that (0, , 0) ∈ E From (2.2), we have νt2 (E1 , E2 ) t∈Fq |E1 |2 |E2 |2 q −1 + |E1 |q 2n−1 |E2 (k)|2 |E1 ∩ lk | k=(0, ,0) Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1004 LE ANH VINH Since E is product-like, so is E1 (note that E1 is obtained from E by scaling) Hence, |E1 ∩ lk | |E1 |1/n It follows that νt2 (E1 , E2 ) |E1 |2 |E2 |2 q −1 + O |E1 |1+ n q 2n−1 |E2 (k)|2 k t∈Fq = |E1 |2 |E2 |2 q −1 + O |E1 |1+ n q n−1 |E2 (y)|2 y −1 = |E| q (4.2) + O(|E| 2+ n n−1 q ), where the second line follows from Plancherel’s theorem, and the last line follows from |E1 | = |E2 | = |E| Applying the Cauchy–Schwarz inequality, we have (4.3) ⎛ |E|4 = |E1 |2 |E2 |2 = ⎝ ⎞2 vt (E1 , E2 )⎠ νt2 (E1 , E2 ) |{x · y : x ∈ E1 , y ∈ E2 }| t∈Fq t∈Fq It follows from (4.2) and (4.3) that (4.4) |{x · y : x ∈ E1 , y ∈ E2 }| |E|4 |E|4 q −1 + O(|E|2+ n q n−1 ) = q 1+O qn |E|2− n Theorem 1.4 now follows immediately from (4.1) and (4.4) Proof of Theorem 1.6 The proof proceeds by induction The case n = follows from Theorem 2.2 Let M = (xij )1 i,j be a × matrix with entries in [a + 1, a + b] We have two cases (1) If x33 = 0, then we can write x13 x31 x33 x13 x31 x23 x32 −2 x33 x11 + Per(M ) = x33 x22 + x23 x32 x33 + x12 + x13 x32 x33 x21 + x23 x31 x33 Hence, Per(M ) = t if and only if x13 x31 x23 x32 x22 + x33 x33 tx33 − 2x13 x31 x23 x32 = x233 x11 + + x12 + x13 x32 x33 x21 + x23 x31 x33 Since xij + xi3 x3j xi3 x3j xi3 x3j ∈ a+ ,a + +b x33 x33 x33 (mod p), i, j 2, it follows from Corollary 2.3 that, for any fixed ai3 , a3j ∈ A, ≤ i, j ≤ 3, there are (1 + o(1))b4 /p choices of x11 , x12 , x21 , and x22 such that Per(M ) = t Therefore, this case contributes (1 + o(1))(b − I(0))b4 b4 /p = (1 + o(1))b9 /p matrices having permanent t, where I(0) = if ∈ I and I(0) = otherwise Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1005 PERMANENTS OF MATRICES OVER FINITE FIELDS Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (2) If x33 = 0, then Per(M ) = x32 (x11 x23 + x21 x13 ) + x31 (x22 x23 + x13 x22 ) If x11 = 0, there are at most b7 choices of the remaining entries such that Per(M ) = t If x11 = 0, there is at most one choice of x23 such that x11 x23 + x21 x13 = If x11 x23 + x21 x13 = 0, there is at most one choice of x32 such that Per(M ) = t Thus, this case contributes at most 3b7 matrices having permanent t p log2 p, putting these two cases together, we have Since b2 P3 (I ; t) = (1 + o(1))b9 /p Suppose that the statement holds for n − 2 We show that it also holds for n For any n × n matrix M = (xij )1 i,j n with entries in [a, a + b], let M be the submatrix obtained from M by deleting the first two rows and two columns, and let Mst (3 s, t n) be the submatrix obtained from M by deleting the first row, the second row, the sth row, the first column, the second column, and the tth column Choose entries of M randomly from the interval [a + 1, a + b] We have three cases i, (A) Suppose that per(M ) = For any choice of x1i , x2i , xj1 , xj2 , xij (3 j n), we can write Per(M ) = Per(M )(x11 x22 + x12 x21 ) + v11 x11 + v12 x12 + v21 x21 + v22 x22 + v for some fixed v11 , v12 , v21 , v22 , and v (depending on chosen entries of M ) Hence, Per(M ) = t if and only if x11 + v22 Per(M ) x22 + v11 Per(M ) v21 v21 x21 + Per(M ) Per(M ) v11 v22 + v21 v12 t−v − = Per(M ) (Per(M ))2 + x12 + Since xij + vij vij vij ∈ a+ ,a+ + b (mod p), Per(M ) Per(M ) Per(M ) it follows from Corollary 2.3 that there are (1 + o(1))b4 /p choices for x11 , x12 , x21 , x22 Therefore, this case contributes 2 (b(n−2) − Pn−2 (I n−2 ; 0))b4(n−2) (1 + o(1)) bn b4 = (1 + o(1)) p p −(1 + o(1))Pn−2 (I n−2 ; 0) b4n−4 p matrices having permanent t (B) Suppose that Per(M ) = and Per(Mst ) = for some s, t n Without loss of generality, we may assume that Per(M33 ) = For any choice of x1i , x2i , xi1 , xi1 (3 i n), we can write Per(M ) = v11 x11 + v12 x12 + v21 x21 + v22 x22 + v, where v11 , v12 , v21 , v22 , and v depend on xij , i, j ≥ By factoring Per(M ) by rows, we have (5.1) v11 = x23 (x32 Per(M33 ) + α) + β, Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1006 LE ANH VINH where α does not depend on x23 , x32 and β does not depend on x23 Now, suppose that Per(M ) = t For any fix x1i , xi1 , x2j , xj2 (i 3, j 4), there is at most one possibility of x32 such that x32 Per(M33 ) + α = Besides, if x32 Per(M33 ) + α = 0, there is at most one possibility of x23 such that v11 = Hence, there are at most 2b possibilities of (x23 , x32 ) such that v11 = From (5.1), if v11 = 0, there are at most b3 possibilities of (x11 , x12 , x21 , x22 ) such that Per(M ) = t Therefore, this case contributes at most (2) (Pn−2 (I n−2 ; 0) − Pn−2 (I n−2 ))(2b4(n−2)−1 b4 + b4(n−2) b3 ) (2) = 3b4n−5 (Pn−2 (I n−2 ; 0) − Pn−2 (I n−2 )) matrices having permanent t (C) Suppose that Per(Mst ) = for all s, t n It is trivial that there are at most b4(n−2)+4 choices of a1i , a2i , ai1 , ai2 , ≤ i ≤ n, such that Per(M ) = t Therefore, this cases contributes at most (2) b4n−4 Pn−2 (I n−2 ) matrices having permanent t Putting all three cases together, we have b4n−4 + 3b4n−5 p bn + O Pn−2 (I n−2 ; 0) Pn (I ; t) = (1 + o(1)) p n (5.2) (2) + O(b4n−4 Pn−2 (I n−2 )) From the induction hypothesis, (5.3) Pn−2 (I n−2 ; 0) = O(b(n−2) /p), and from Lemma 2.3, (5.4) (2) Pn−2 (I n−2 ) = O(b(n−2) −4 ) It follows immediately from (5.2), (5.3), and (5.4) that Pn (I n ; t) = (1 + o(1)) bn p whenever b/p1/2 log p → ∞, completing the proof of the theorem Acknowledgment The author would like to thank Victor Garcia for kindly granting his permission to present Theorem 2.2 here REFERENCES [1] O Ahmadi and I E Shparlinski, Distribution of matrices with restricted entries over finite fields, Indag Math (N.S.), 18 (2007), pp 327–337 [2] A Ayyad, T Cochrane, and Zh Zheng, The congruence x1 x2 ≡ x3 x4 (mod p), the equation x1 x2 = x3 x4 , and mean values of character sums, J Number Theory, 59 (1996), pp 398 413 ă mer, R Karp, and E Welzl, The rank of sparse random matrices over finite fields, [3] J Blo Random Structures Algorithms, 10 (1997), pp 407–419 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 01/01/13 to 150.135.135.70 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PERMANENTS OF MATRICES OVER FINITE FIELDS 1007 [4] D Covert, D Hart, A Iosevich, D Koh, and M Rudnev, Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields, European J Combin., 31 (2010), pp 306–319 [5] M Z Garaev and V C Garcia, The equation x1 x2 = x3 x4 + λ in fields of prime order and applications, J Number Theory, 128 (2008), pp 2520–2537 [6] V C Garcia, Personal communication, 2009 [7] D Hart and A Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in Radon Transforms, Geometry, and Wavelets, Contemp Math 464, Amer Math Soc., Providence, RI, 2008, pp 129–135 [8] D Hart, A Iosevich, D Koh, and M Rudnev, Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdă osFalconer distance conjecture, Trans Amer Math Soc., 363 (2011), pp 3255–3275 ´ s, Singularity probabilities for random matrices over finite fields, Com[9] J Kahn and J Komlo bin Probab Comput., 10 (2001), pp 137–157 [10] I N Kovalenko, A A Leviskaya, and M N Savchuk, Selected Problems in Probabilistic Combinatorics, Naukova Dumka, Kiev, Ukraine, 1986 (in Russian) [11] L A Lyapkov and B A Sevast’yanov, Limit distribution of the probabilities of the permanent of a random matrix in the field GF(p), Diskr Mat., (1996), pp 3–13 (in Russian) English translation in Discrete Math Appl., (1996), pp 107–116 [12] L A Vinh, On the distribution of determinants of matrices with restricted entries over finite fields, J Combin Number Theory, (2009), pp 203–212 [13] L A Vinh, On the distribution of permanents of matrices over finite fields, in Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb 2009), Electron Notes Discrete Math 34, Elsevier Sci B.V., Amsterdam, 2009, pp 519– 523 [14] L A Vinh, Singular matrices with restricted rows in vector spaces over finite fields, Discrete Math., 312 (2012), pp 413–418 Copyright © by SIAM Unauthorized reproduction of this article is prohibited  ... Vinh, On the distribution of determinants of matrices with restricted entries over finite fields, J Combin Number Theory, (2009), pp 203–212 [13] L A Vinh, On the distribution of permanents of matrices. .. Distribution of matrices with restricted entries over finite fields, Indag Math (N.S.), 18 (2007), pp 327–337 [2] A Ayyad, T Cochrane, and Zh Zheng, The congruence x1 x2 ≡ x3 x4 (mod p), the equation... column The purpose of this paper is to study the distribution of the permanent when the dimension of matrices is fixed For any t ∈ Fq and E ⊂ Fdq , let Pn (E; t) be the number of n × n matrices with