Local computation of differents and discriminants

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 Disc(f), and the resultant Res(f, g), for polynomials f (x), g(x) ∈ 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 either Disc(f) or 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.

Genetics of polynomials over local fields

“…All Okutsu invariants of F may be deduced from an OM representation of F by closed formulas. For instance, let us exhibit some more Okutsu invariants, taken from [14,Sec. 1], besides e(F ), f (F ) and δ 0 (F ) already mentioned in (2.1).…”

“…The concept of a type and the Montes algorithm were introduced in [13] for v a discrete valuation on a global field K. These results were reviewed in [5,6] and their computational implications were developed in a series of papers [2,7,8,9,14]. The derivation of these tools from the modern presentation of MacLane's valuations in the spirit of Vaquié, leads to a more elegant treatment of the subject and to its generalization to arbitrary discrete valued fields (K, v).…”

On the equivalence of types

“…He introduced types as computational objects which are able to construct MacLane's valuations and the higher residual polynomial operators foreseen by Ore. These ideas made the whole theory constructive and well-suited to computational applications, and led to the design of several fast algorithms to perform arithmetic tasks in global fields [3,5,6,8,12].…”

Residual ideals of MacLane valuations

“…These operators make the whole theory constructive and well-suited to computational applications. These ideas led to the design of several fast algorithms to perform arithmetic tasks in global fields [2,4,6,7,9,13].…”

“…This result reveals that MacLane's original approach is best-suited for computational applications, because the elements in the set M may be described in terms of discrete parameters which are easily manipulated by a computer. As a consequence, all algorithmic developments based on the Montes algorithm [4,6,7,9,13] admit a more elegant description and a natural extension to arbitrary discrete valued fields. However, we postpone the discussion of these computational aspects to a forthcoming paper [8].…”