Computation of residual polynomial operators of inductive valuations

Abstract

Let ( K , v ) be a valued field, and μ an inductive valuation on K [ x ] extending v . Let G μ be the graded algebra of μ over K [ x ] , and κ the maximal subfield of the subring of G μ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field κ and the residual polynomial operator R μ : K [ x ] → κ [ 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.