Semigroup associated with a free polynomial

TL;DR

This paper generalizes the Abhyankar-Moh theory to irreducible polynomials over formal power series rings with exponents in a line free cone, associating characteristic exponents and semigroups of values, and relating these to approximate roots.

Contribution

It extends the theory of characteristic exponents and semigroups to a broader class of polynomials in power series rings with cone-based exponents, including their computation via approximate roots.

Findings

Semigroup of values can be derived from approximate roots.

Polynomials over power series rings fit into the generalized framework.

The theory applies to polynomials in ${ m K}[[x]]$ with specific cones.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $\mathbb{K}_{C}[[x_{1},...,x_{e}]]$ be the ring of formal power series in several variables with exponents in a line free cone $C$. We consider irreducible polynomials $f=y^n+a_1(\underline{x})y^{n-1}+\ldots+a_n(\underline{x})$ in $\mathbb{K}_{C}[[x_{1},...,x_{e}]][y]$ whose roots are in $\mathbb{K}_{C}[[x_{1}^{\frac{1}{n}},...,x_{e}^{\frac{1}{n}}]]$. We generalize to these polynomials the theory of Abhyankar-Moh. In particular we associate with any such polynomial its set of characteristic exponents and its semigroup of values. We also prove that the set of values can be obtained using the set of approximate roots. We finally prove that polynomials of ${\mathbb K}[[\underline{x}]][y]$ fit in the above set for a specific line free cone (see Section 4).