Apéry base and polar invariants for real valuations

Abstract

The main objective of this paper is to study real valuations μ defined on the polynomial ring K[T] with coefficients in a field K such that the restriction of μ to K is a proper valuation on K. For this, we consider the Apery base and the iterated sequence of valuations associated with μ and we show the main properties and the relation between both invariants. We also prove a factorization theorem for elements \({f\in K[T]}\) of the suitable degree such that μ(f) is in the Apery base of μ, obtaining strong properties of their irreducible factors. In particular, some results of M. Merle, R. Ephraim and A. Granja for irreducible algebroid plane curve singularity are a special case of this theorem.