#### Vol. 300, No. 2, 2019

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

### Doowon Koh, Thang Pham, Chun-Yen Shen and Le Anh Vinh

Vol. 300 (2019), No. 2, 405–417
##### Abstract

Let $A$ be a set in a prime field ${\mathbb{F}}_{p}$. We prove that $d×d$ matrices with entries in $A$ determine almost $|A{|}^{3+\frac{1}{45}}$ distinct determinants and almost $|A{|}^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough. Our proofs rely on recent advances in additive combinatorics and incidence machinery.

##### Keywords
finite fields, sum-product estimates, determinant, permanent, matrix
Primary: 05C50