On the determinants and permanents of matrices with restricted entries over prime fields

Abstract

Let A be a set in a prime field F p . In this paper, we prove that d × d matrices with entries in A determine almost |A| 3+ 1 45 distinct determinants and almost |A| 2− 1 6 distinct permanents when |A| is small enough.

Expanding phenomena over matrix rings

“…We note that similar results in the setting of Heisenberg group over prime fields for small sets were obtained recently by Hegyvári and Hennecart in [11]. Some generalizations can be found in [12,13]. We refer the interested reader to [5,19] and references therein for related results in the setting of R or Z.…”

Expanding phenomena over higher dimensional matrix rings

“…Similar results in the setting of Heisenberg group over prime fields for small sets were obtained recently by Hegyvári and Hennecart in [8]. Some generalizations can be found in [9,10,17,19].…”

On growth of the number of determinants with restricted entries

“…Remark 3. In papers [3], [10], a problem similar to our was considered, but for permanents. In particular, there it was proved that the number of distinct permanents of matrices with entries in a set A is at least |A| 2− 1 6 +o(1) , where o(1) tends to zero with the growth of matrices size.…”

“…One considers a function of several variables and explores how big is the image of the function while the arguments run along a finite set A, see [1], [2]. on matrices and distributions (particularly, on distributions of their determinants) Some related problems are considered in papers [3], [4], [5], particularly a problem on the distribution of determinants. A continuous counterpart of the examining problem is presented in [6].…”