# Computation of residual polynomial operators of inductive valuations

## TL;DR

This paper presents an algorithm to compute residual polynomial operators and the associated field in inductive valuations, enabling efficient factorization of certain polynomials over henselian fields without complex algebraic computations.

## Contribution

It introduces a novel algorithm for computing the residual polynomial operator and the field ppa in inductive valuations, simplifying factorization processes.

## Key findings

- Algorithm efficiently computes the field ppa and residual polynomial operator.
- Enables OM algorithm for factorization of separable defectless polynomials.
- Avoids complex computations in the graded algebra.

## 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.