Computation of residual polynomial operators of inductive valuations

Abstract

Let $(K,v)$ be a valued field, and $\mu$ an inductive valuation on $K[x]$ extending $v$. Let $G_{\mu }$ be the graded algebra of $\mu$ over $K[x]$, and $\kappa$ the maximal subfield of the subring of $G_{\mu }$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $\kappa$ and the residual polynomial operator $R_{\mu }:K[x]\to \kappa [y]$, where $y$ is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.