Computation of residual polynomial operators of inductive valuations

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. As an application of the methods of the paper, we describe the structure of $G_\mu$ as a $G_v$-algebra, where $G_v$ is the graduate algebra of $v$ over $K$.