Beta-conjugates of real algebraic numbers as Puiseux expansions

TL;DR

This paper introduces a new perspective on beta-conjugates of real algebraic numbers using Puiseux expansions, linking dynamical systems, algebraic geometry, and number theory to analyze their properties.

Contribution

It provides a novel definition of beta-conjugates via Puiseux expansions and decomposes associated germs into irreducible elements, connecting dynamical zeta functions with algebraic factorizations.

Findings

Beta-conjugates are characterized through Puiseux series expansions.

The germ decomposition reveals conjugacy classes of beta-conjugates.

A product formula for the dynamical zeta function is established, analogous to the Euler product.

Abstract

The beta-conjugates of a base of numeration $\beta > 1$, $\beta$ being a Parry number, were introduced by Boyd, in the context of the R\'enyi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are canonically associated with $\beta$. Let $\beta > 1$ be a real algebraic number. A more general definition of the beta-conjugates of $\beta$ is introduced in terms of the Parry Upper function $f_{\beta}(z)$ of the beta-transformation. We introduce the concept of a germ of curve at $(0,1/\beta) \in \mathbb{C}^{2}$ associated with $f_{\beta}(z)$ and the reciprocal of the minimal polynomial of $\beta$. This germ is decomposed into irreducible elements according to the theory of Puiseux, gathered into conjugacy classes. The beta-conjugates of $\beta$, in terms of the Puiseux expansions, are given a new equivalent definition in this new context. If $\beta$ is a Parry number the (Artin-Mazur) dynamical zeta function $\zeta_{\beta}(z)$ of the beta-transformation, simply related to $f_{\beta}(z)$, is expressed as a product formula, under some assumptions, a sort of analog to the Euler product of the Riemann zeta function, and the factorization of the Parry polynomial of $\beta$ is deduced from the germ.