Semigroup associated with a free polynomial

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).