Paper detail

Computation of residual polynomial operators of inductive valuations

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 $R_μ: K[x]\toκ[y]$, 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.