On fully split lacunary polynomials in finite fields

TL;DR

This paper estimates the variety of degree patterns for lacunary polynomials that split completely over finite fields, using bounds on zeros and graph theory techniques.

Contribution

It introduces a novel approach combining zero bounds and graph theory to analyze lacunary polynomials over finite fields.

Findings

Bound on the number of degree patterns for lacunary polynomials

Application of graph theory to polynomial splitting

Enhanced understanding of polynomial zero distribution

Abstract

We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.