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

Let $A$ be a set in a prime field $F_{p}$ . We prove that $d \times d$ 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.

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Keywords

finite fields, sum-product estimates, determinant, permanent, matrix

Mathematical Subject Classification 2010

Primary: 05C50

Milestones

Received: 1 January 2018

Revised: 25 September 2018

Accepted: 25 September 2018

Published: 30 July 2019

Authors

Doowon Koh
Department of Mathematics Chungbuk National University Cheongju City South Korea

Thang Pham
Department of Mathematics University of California, San Diego La Jolla, CA United States

Chun-Yen Shen
Department of Mathematics National Taiwan University Taipei Taiwan

Le Anh Vinh
University of Education Vietnam Institute of Educational Sciences Cau Giay Hanoi 100000 Vietnam

Vol. 300, No. 2, 2019

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

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors