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 Fp. We prove that d × d matrices with entries in A determine almost |A|3+ 1 45 distinct determinants and almost |A|21 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
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