Local computation of differents and discriminants

TL;DR

This paper develops local algorithms for computing discriminants and resultants of polynomials over Dedekind domains, avoiding direct computation of these quantities and improving efficiency for high-degree or large-coefficient polynomials.

Contribution

It introduces routines for p-adic valuation computation of discriminants and resultants without explicit calculation, enhancing efficiency in complex polynomial cases.

Findings

Efficient routines for p-adic valuation of discriminants and resultants.

Applications to polynomials over Dedekind domains with large degrees or coefficients.

No need to compute full discriminants or resultants explicitly.

Abstract

We obtain several results on the computation of different and discriminant ideals of finite extensions of local fields. As an application, we deduce routines to compute the $\p$-adic valuation of the discriminant $\dsc(f)$, and the resultant $\res(f,g)$, for polynomials $f(x),g(x)\in A[x]$, where $A$ is a Dedekind domain and $\p$ is a non-zero prime ideal of $A$ with finite residue field. These routines do not require the computation of neither $\dsc(f)$ nor $\res(f,g)$; hence, they are useful in cases where this latter computation is inefficient because the polynomials have a large degree or very large coefficients.