Diophantine approximation of polynomials over $\mathbb{F}_q[t]$ satisfying a divisibility condition

TL;DR

This paper investigates Diophantine approximation for polynomials over finite fields, establishing estimates for fractional parts under divisibility conditions and extending results to linear combinations, advancing understanding in finite field polynomial approximation.

Contribution

It introduces new estimates for fractional parts of polynomials over finite fields satisfying divisibility conditions and extends these results to linear combinations, filling a gap in finite field Diophantine approximation.

Findings

Established bounds for fractional parts of polynomials over finite fields

Extended approximation results to linear combinations of such polynomials

Analogous divisibility conditions to intersective polynomials in integers

Abstract

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.