On the set of zero coefficients of a function satisfying a linear differential equation

TL;DR

This paper generalizes the Skolem-Mahler-Lech theorem by showing that solutions to certain polynomial recurrence relations have zero sets composed of finitely many arithmetic progressions and a finite set, extending classical results.

Contribution

It extends the Skolem-Mahler-Lech theorem to polynomial recurrence relations with polynomial coefficients, describing the structure of zero sets of solutions.

Findings

Zero set of solutions is a union of finitely many arithmetic progressions and a finite set.

Generalizes classical Skolem-Mahler-Lech theorem to polynomial coefficient recurrences.

Connects zero coefficients of differential equations to recurrence solutions.

Abstract

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that the set of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.