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 fβ(z) of the beta-transformation. We introduce the concept of a germ of curve at (0, 1/β) ∈ C2 associated with fβ(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 ζβ(z) of the beta-transformation, simply related to fβ(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.