Beta-conjugates of real algebraic numbers as Puiseux expansions

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

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