The number of roots of polynomials of large degree in a prime field

TL;DR

This paper provides asymptotic upper bounds on the number of roots modulo large primes for specific high-degree polynomials derived from truncated power series satisfying differential equations.

Contribution

It introduces new bounds for the roots of polynomials of degree comparable to the prime modulus, constructed from truncated power series with rational coefficients.

Findings

Established asymptotic upper bounds for roots modulo p.

Analyzed polynomials with degree p from truncated power series.

Applied to polynomials satisfying simple differential equations.

Abstract

We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by truncating certain power series with rational coefficients that satisfy simple differential equations.