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Distribution of the number of zeros of polynomials over a finite field

Ritik Jain, Han-Bom Moon and Peter Wu

Vol. 18 (2025), No. 4, 707–718
Abstract

We study the probability distribution of the number of zeros of multivariable polynomials with bounded degree over a finite field. We find the probability generating function for each set of bounded degree polynomials. In the single variable case, we show that, as the degree of the polynomials and the order of the field simultaneously approach infinity, the distribution converges to a Poisson distribution.

Keywords
finite field, probability distribution, number of roots
Mathematical Subject Classification
Primary: 05A15, 11T06, 60C05
Milestones
Received: 21 September 2023
Revised: 12 April 2024
Accepted: 19 April 2024
Published: 30 July 2025

Communicated by Anant Godbole
Authors
Ritik Jain
Department of Mathematics
Fordham University
New York, NY
United States
Han-Bom Moon
Department of Mathematics
Fordham University
New York, NY
United States
Peter Wu
Department of Mathematics
Fordham University
New York, NY
United States