$\mathbb{N}$-algebraicity of zeta functions of sofic-Dyck shifts

TL;DR

This paper proves that the zeta functions of sofic-Dyck shifts are $ $-algebraic, linking dynamical systems with visibly pushdown languages and their generating functions.

Contribution

It establishes that the multivariate zeta function of a sofic-Dyck shift corresponds to a visibly pushdown language's generating function, showing $ $-algebraicity.

Findings

Zeta function of a sofic-Dyck shift is the generating function of a visibly pushdown language.

The zeta function is an $ $-algebraic series.

Multivariate zeta functions relate to visibly pushdown languages.

Abstract

We prove that the multivariate zeta function of a sofic-Dyck shift is the commutative series of some visibly pushdown language. As a consequence the zeta function of a sofic-Dyck shift is the generating function of a visibly pushdown language and is thus an $\mathbb{N}$-algebraic series.