- Email: [email protected]
On the distribution of permanents of matrices over finite fields
Electronic Notes in Discrete Mathematics 34 (2009) 519–523 www.elsevier.com/locate/endm
On the distribution of permanents of matrices over finite fields Le Anh Vinh Mathematics Department Harvard University Cambridge, MA 02138, USA [email protected]
Abstract For a prime power q, we study the distribution of permanents of n × n matrices over a finite field q of q elements. We show that if A is a sufficient large subset of q then the set of permanents of n × n matrices with entries in A covers all (or almost) ∗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. We also study this problem in more general settings. Keywords: permanent, matrices over finite fields.
1
Motivation
Throughout this paper, let q = pr where p is an odd prime and r is a positive integer. Let q be a finite field of q elements. The prime base field p of q may then be naturally identified with p . The distribution of the determinant of matrices with entries in a finite field q has been studied by various researchers. Suppose that the ground field q is fixed and Md is a random 1571-0653/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2009.07.086
520
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
d × d matrices with entries chosen independently from q . If the entries are chosen uniformly from q , then it is well known that Pr(Md is nonsingular) → (1) (1 − q −i ) as d → ∞. i1
It is interesting that (1) is quite robust. Specifically, J. Kahn and J. Koml´os [6] proved a strong necessary and sufficient condition for (1). Theorem 1.1 ([6]) Let Md be a random d × d matrix with entries chosen according to some fixed non-degenerate probability distribution μ on q . Then (1) holds if and only if the support of μ is not contained in any proper affine subfield of q . An extension of the uniform limit is to random matrices with μ depending on n was considered by Kovalenko, Leviskaya and Savchuk [7]. They proved that the standard limit (1) under the condition that the entries mij of Md are independent and Pr(mij = α) > (log d + ω(1))/d for all α ∈ q . The behavior of the nullity of Md for 1 − μ(0) close to log d/d and μ(α) = (1 − μ(0)/(q − 1) for α = 0 was also studied by Bl¨omer, Karp and Welzl [2]. Another direction is to fix the dimension of matrices. For an integer number d and a subset E ⊆ dq , let Md (E) denote the set of d × d matrices with rows in E. For any t ∈ q , let Nd (E; t) be the number of d × d matrices in Md (E) having determinant t. Ahmadi and Shparlinski [1] studied some natural classes of matrices over finite fields p of p elements (p is a prime) with components in a given subinterval [−H, H] ⊆ [−(p − 1)/2, (p − 1)/2]. They showed that 2 (2H + 1)d d Nd ([−H, H] ; t) = (1 + o(1)) (2) p if t ∈ ∗q and H p3/4 . In the case n = 2, they improved the lower bound to H p1/2+ε for any constant ε > 0. Covert et al. [3] studied this problem in more general settings. When E ⊆ 3q is an arbitrary set, they obtained the following result. Theorem 1.2 ([3, Theorem 2.10]) Suppose that E ⊆ 3q of cardinality |E| Cq 2 for a sufficiently large constant C > 0. There exists c > 0 such that | vol(E)| cq. In [10], the author showed that, under the same condition, E determines all possible volumes. More precisely, we obtained the following general result. Theorem 1.3 ([10]) When E ⊆
d q
and |E| ≥ (d − 1)q d−1 , vol(E) =
q.
521
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
Note that the assumption |E| ≥ (d−1)q d−1 is sharp up to a factor of (d−1) as taking E to be a (d − 1)-hyperplane through the origin shows. A subset E ⊆ nq is called a product-like set if |Hd ∩ E| |E|d/n for any d-dimensional subspace Hd ⊂ nq . Covert et al. [3] showed that if E ⊂ 3q is a product-like set of cardinality |E| q 15/8 , then q ⊂ vol(E). When E is a product set, i.e. E = A×. . .×A, using the geometry incidence machinery developed in [3], and some properties of non-singular matrices, the author [9] obtained the following result for higher dimensional cases (d ≥ 4): 2
|A|d Nd (A ; t) = (1 + o(1)) , q d
d
if t ∈ ∗q and A ⊆ q of cardinality |A| q 2d−1 . On the other hand, little has been known about the permanent. The only known uniform limit similar to (1) for the permanent is due to Lyapkov and Sevast’yanov [8]. They proved that the permanent of a random n × m matrix Mnm with elements from p and independent rows has the limit distribution of the form lim Pr(Per(Mnm ) = k) = ρm δk0 + (1 − ρm )/p, k ∈
n→∞
p,
where δk0 is Kronecker’s symbol. The purpose of this paper is to study the distribution of the permanent when the dimension of matrices is fixed. For any t ∈ q and E ⊂ dq , 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)}. Throughout the abstract, the implied constants in the symbols O, o, , and may depend on integer parameter d. We recall that the notation U = O(V ) and U V are equivalent to the assertion that the inequality |U | ≤ cV holds for some constant c > 0. The notation U = o(V ) is equivalent to the assertion that U = O(V ) but V = O(U ).
2
Statement of results
The main result of this paper is that, if E is a sufficient large subset of nq then Pn (E) covers q . More precisely, our first result is the following theorem. Theorem 2.1 Suppose that q is an odd prime power and gcd(q, n) = 1. n+1 a) If E ∩ ( ∗q )n = ∅, and |E| > cq 2 , then ∗q ⊆ Pn (E). b) If E ⊂ nq of cardinality |E| > nq n−1 , then ∗q ⊆ Pn (E). Note that the bound in Part b) of Theorem 2.1 is tight up to a factor of
522
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
n. For example, |{x1 = 0}| = q n−1 and Pn ({x1 = 0}) = 0. When E is a product-set, i.e. E = A × . . . × A, we can improve the bound in Theorem 2.1. Theorem 2.2 Suppose that q is an odd prime power and gcd(q, n) = 1. 1 1 a) If A ⊂ q of cardinality |A| q 2 + 2n , then ∗q ⊆ Pn (An ). b) If A ⊆ q of cardinality |A| q 2/3 , then for each t ∈ ∗q P3 (A3 ; t) = (1 + o(1))
|E|3 . q
When E is a product-like set, we can get a positive proportion of the permanents under a weaker assumption. Theorem 2.3 Suppose that q is an odd prime power and gcd(q, n) = 1. If E⊂
n q
n2
is a product-like set of cardinality |E| q 2(n−1) , then |Pn (E)| (1 − o(1))q.
In the special case E = A × . . . × A, we have the following corollary. Corollary 2.4 Suppose that q is an odd prime power and gcd(q, n) = 1. If 1 1 A ⊆ q of cardinality |A| q 2 + 2(n−1) , then |Pn (An )| (1 − o(1))q. Furthermore, if we restrict our study to matrices over a finite field p of p elements (p is a prime) with components in a given interval, we obtain a stronger result. Theorem 2.5 For any prime p and an interval I := [a + 1, a + b] ⊆ b → ∞, p → ∞, 1/2 p log p we have 2 bn Pn (I n ; t) = (1 + o(1)) p for any t ∈ p .
p
with
As we will see in the proof of Theorem 2.5, the method applies for a large variety of multi-homogeneous forms. In particular, we have the following result for determinant. Theorem 2.6 For any prime p and an interval I := [a + 1, a + b] ⊆ b → ∞, p → ∞, 1/2 p log p
p
with
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
523
we have 2
Nd (I d ; t) = (1 + o(1)) for any t ∈
bd p
p.
References [1] O. Ahmadi and I. E. Shparlinski, Distribution of matrices with restricted entries over finite fields, Inda. Mathem. 18(3) (2007), 327–337. [2] J. Bl¨ omer, R. Karp, and E. Welzl, The rank of sparse random matrices over finite fields, Random Structures and Algorithms 10 (1997) 407–419. [3] D. Covert, D. Hart, A. Iosevich, D. Koh, and M. Rudnev, Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields, European Journal of Combinatorics, to appear. [4] D. Hart and A. Iosevich, Sums and products in finite fields: an integral geometric viewpoint, Contemporary Mathematics, Radon transforms, geometry, and wavelets, 464 (2008). [5] D. Hart, A. Iosevich, D. Koh, and M. Rudnev, Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erd¨ os-Falconer distance conjecture, preprint (2007). [6] J. Kahn and J. Koml´ os, Singularity probabilities for random matrices over finite fields, Combinatorics, Probability and Computing 10 (2001), 137–157. [7] I. N. Kovalenko, A. A. Leviskaya, and M. N. Savchuk, Selected Problems in Probabilistic Combinatorics (1986), Naukova Dumka, Kiev. In Russian. [8] L. A. Lyapkov and B. A. Sevast’yanov, The limiting probability distribution of a permanent of a random matrix in a GF (p) field, Diskr. Matem., 8(2) (1996), 3–13. [9] L. A. Vinh, On the distribution of determinants of matrices with restricted entries over finite fields, Journal of Combinatorics and Number Theory, to appear. [10] L. A. Vinh, On volume set of point sets in vector spaces over finite fields, submitted.
On the distribution of permanents of matrices over finite fields Le Anh Vinh Mathematics Department Harvard University Cambridge, MA 02138, USA [email protected]
Abstract For a prime power q, we study the distribution of permanents of n × n matrices over a finite field q of q elements. We show that if A is a sufficient large subset of q then the set of permanents of n × n matrices with entries in A covers all (or almost) ∗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. We also study this problem in more general settings. Keywords: permanent, matrices over finite fields.
1
Motivation
Throughout this paper, let q = pr where p is an odd prime and r is a positive integer. Let q be a finite field of q elements. The prime base field p of q may then be naturally identified with p . The distribution of the determinant of matrices with entries in a finite field q has been studied by various researchers. Suppose that the ground field q is fixed and Md is a random 1571-0653/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2009.07.086
520
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
d × d matrices with entries chosen independently from q . If the entries are chosen uniformly from q , then it is well known that Pr(Md is nonsingular) → (1) (1 − q −i ) as d → ∞. i1
It is interesting that (1) is quite robust. Specifically, J. Kahn and J. Koml´os [6] proved a strong necessary and sufficient condition for (1). Theorem 1.1 ([6]) Let Md be a random d × d matrix with entries chosen according to some fixed non-degenerate probability distribution μ on q . Then (1) holds if and only if the support of μ is not contained in any proper affine subfield of q . An extension of the uniform limit is to random matrices with μ depending on n was considered by Kovalenko, Leviskaya and Savchuk [7]. They proved that the standard limit (1) under the condition that the entries mij of Md are independent and Pr(mij = α) > (log d + ω(1))/d for all α ∈ q . The behavior of the nullity of Md for 1 − μ(0) close to log d/d and μ(α) = (1 − μ(0)/(q − 1) for α = 0 was also studied by Bl¨omer, Karp and Welzl [2]. Another direction is to fix the dimension of matrices. For an integer number d and a subset E ⊆ dq , let Md (E) denote the set of d × d matrices with rows in E. For any t ∈ q , let Nd (E; t) be the number of d × d matrices in Md (E) having determinant t. Ahmadi and Shparlinski [1] studied some natural classes of matrices over finite fields p of p elements (p is a prime) with components in a given subinterval [−H, H] ⊆ [−(p − 1)/2, (p − 1)/2]. They showed that 2 (2H + 1)d d Nd ([−H, H] ; t) = (1 + o(1)) (2) p if t ∈ ∗q and H p3/4 . In the case n = 2, they improved the lower bound to H p1/2+ε for any constant ε > 0. Covert et al. [3] studied this problem in more general settings. When E ⊆ 3q is an arbitrary set, they obtained the following result. Theorem 1.2 ([3, Theorem 2.10]) Suppose that E ⊆ 3q of cardinality |E| Cq 2 for a sufficiently large constant C > 0. There exists c > 0 such that | vol(E)| cq. In [10], the author showed that, under the same condition, E determines all possible volumes. More precisely, we obtained the following general result. Theorem 1.3 ([10]) When E ⊆
d q
and |E| ≥ (d − 1)q d−1 , vol(E) =
q.
521
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
Note that the assumption |E| ≥ (d−1)q d−1 is sharp up to a factor of (d−1) as taking E to be a (d − 1)-hyperplane through the origin shows. A subset E ⊆ nq is called a product-like set if |Hd ∩ E| |E|d/n for any d-dimensional subspace Hd ⊂ nq . Covert et al. [3] showed that if E ⊂ 3q is a product-like set of cardinality |E| q 15/8 , then q ⊂ vol(E). When E is a product set, i.e. E = A×. . .×A, using the geometry incidence machinery developed in [3], and some properties of non-singular matrices, the author [9] obtained the following result for higher dimensional cases (d ≥ 4): 2
|A|d Nd (A ; t) = (1 + o(1)) , q d
d
if t ∈ ∗q and A ⊆ q of cardinality |A| q 2d−1 . On the other hand, little has been known about the permanent. The only known uniform limit similar to (1) for the permanent is due to Lyapkov and Sevast’yanov [8]. They proved that the permanent of a random n × m matrix Mnm with elements from p and independent rows has the limit distribution of the form lim Pr(Per(Mnm ) = k) = ρm δk0 + (1 − ρm )/p, k ∈
n→∞
p,
where δk0 is Kronecker’s symbol. The purpose of this paper is to study the distribution of the permanent when the dimension of matrices is fixed. For any t ∈ q and E ⊂ dq , 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)}. Throughout the abstract, the implied constants in the symbols O, o, , and may depend on integer parameter d. We recall that the notation U = O(V ) and U V are equivalent to the assertion that the inequality |U | ≤ cV holds for some constant c > 0. The notation U = o(V ) is equivalent to the assertion that U = O(V ) but V = O(U ).
2
Statement of results
The main result of this paper is that, if E is a sufficient large subset of nq then Pn (E) covers q . More precisely, our first result is the following theorem. Theorem 2.1 Suppose that q is an odd prime power and gcd(q, n) = 1. n+1 a) If E ∩ ( ∗q )n = ∅, and |E| > cq 2 , then ∗q ⊆ Pn (E). b) If E ⊂ nq of cardinality |E| > nq n−1 , then ∗q ⊆ Pn (E). Note that the bound in Part b) of Theorem 2.1 is tight up to a factor of
522
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
n. For example, |{x1 = 0}| = q n−1 and Pn ({x1 = 0}) = 0. When E is a product-set, i.e. E = A × . . . × A, we can improve the bound in Theorem 2.1. Theorem 2.2 Suppose that q is an odd prime power and gcd(q, n) = 1. 1 1 a) If A ⊂ q of cardinality |A| q 2 + 2n , then ∗q ⊆ Pn (An ). b) If A ⊆ q of cardinality |A| q 2/3 , then for each t ∈ ∗q P3 (A3 ; t) = (1 + o(1))
|E|3 . q
When E is a product-like set, we can get a positive proportion of the permanents under a weaker assumption. Theorem 2.3 Suppose that q is an odd prime power and gcd(q, n) = 1. If E⊂
n q
n2
is a product-like set of cardinality |E| q 2(n−1) , then |Pn (E)| (1 − o(1))q.
In the special case E = A × . . . × A, we have the following corollary. Corollary 2.4 Suppose that q is an odd prime power and gcd(q, n) = 1. If 1 1 A ⊆ q of cardinality |A| q 2 + 2(n−1) , then |Pn (An )| (1 − o(1))q. Furthermore, if we restrict our study to matrices over a finite field p of p elements (p is a prime) with components in a given interval, we obtain a stronger result. Theorem 2.5 For any prime p and an interval I := [a + 1, a + b] ⊆ b → ∞, p → ∞, 1/2 p log p we have 2 bn Pn (I n ; t) = (1 + o(1)) p for any t ∈ p .
p
with
As we will see in the proof of Theorem 2.5, the method applies for a large variety of multi-homogeneous forms. In particular, we have the following result for determinant. Theorem 2.6 For any prime p and an interval I := [a + 1, a + b] ⊆ b → ∞, p → ∞, 1/2 p log p
p
with
L.A. Vinh / Electronic Notes in Discrete Mathematics 34 (2009) 519–523
523
we have 2
Nd (I d ; t) = (1 + o(1)) for any t ∈
bd p
p.
References [1] O. Ahmadi and I. E. Shparlinski, Distribution of matrices with restricted entries over finite fields, Inda. Mathem. 18(3) (2007), 327–337. [2] J. Bl¨ omer, R. Karp, and E. Welzl, The rank of sparse random matrices over finite fields, Random Structures and Algorithms 10 (1997) 407–419. [3] D. Covert, D. Hart, A. Iosevich, D. Koh, and M. Rudnev, Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields, European Journal of Combinatorics, to appear. [4] D. Hart and A. Iosevich, Sums and products in finite fields: an integral geometric viewpoint, Contemporary Mathematics, Radon transforms, geometry, and wavelets, 464 (2008). [5] D. Hart, A. Iosevich, D. Koh, and M. Rudnev, Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erd¨ os-Falconer distance conjecture, preprint (2007). [6] J. Kahn and J. Koml´ os, Singularity probabilities for random matrices over finite fields, Combinatorics, Probability and Computing 10 (2001), 137–157. [7] I. N. Kovalenko, A. A. Leviskaya, and M. N. Savchuk, Selected Problems in Probabilistic Combinatorics (1986), Naukova Dumka, Kiev. In Russian. [8] L. A. Lyapkov and B. A. Sevast’yanov, The limiting probability distribution of a permanent of a random matrix in a GF (p) field, Diskr. Matem., 8(2) (1996), 3–13. [9] L. A. Vinh, On the distribution of determinants of matrices with restricted entries over finite fields, Journal of Combinatorics and Number Theory, to appear. [10] L. A. Vinh, On volume set of point sets in vector spaces over finite fields, submitted.