Okutsu-Montes representations of prime ideals of one-dimensional integral closures

Nart, Enric

doi:10.5565/publmat_55211_01

Cited by 3 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For each i we have v p (Disc(G i )) = 2 ind p (G i ) + v p (Disc(L Gi /K p )). Therefore, (10) and (11) show that:…”

Section: 2mentioning

confidence: 96%

Local computation of differents and discriminants

Nart¹

2012

Preprint

View full text Add to dashboard Cite

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 neither Disc(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.

show abstract

“…For each i we have v p (Disc(G i )) = 2 ind p (G i ) + v p (Disc(L Gi /K p )). Therefore, (10) and (11) show that:…”

Section: 2mentioning

confidence: 96%

Local computation of differents and discriminants

Nart¹

2012

Preprint

View full text Add to dashboard Cite

show abstract

“…It was first published in [5], based on the theoretical background developed in [6]. A short review may be found in the survey [12] as well.…”

Section: The Om Factorization Algorithmmentioning

confidence: 99%

“…It is fully described in [GMN11], in terms of the theoretical background developed in [GMN12]. For a short review the reader may check the survey [Nar11].…”

Section: The Factorization Algorithm Of Ore Maclane and Montesmentioning

confidence: 99%

Complexity of OM factorizations of polynomials over local fields

Bauch¹,

Nart²,

Stainsby³

2013

LMS J. Comput. Math.

View full text Add to dashboard Cite

Let k be a locally compact complete field with respect to a discrete valuation v. Let O be the valuation ring, m the maximal ideal and F (x) ∈ O[x] a monic separable polynomial of degree n. Let δ = v(Disc(F )). The Montes algorithm computes an OM factorization of F . The singlefactor lifting algorithm derives from this data a factorization of F (mod m ν ), for a prescribed precision ν. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of O(n 2+ + n 1+ δ 2+ + n 2 ν 1+ ) word operations for the complexity of the computation of a factorization of F (mod m ν ), assuming that the residue field of k is small.

show abstract

“…Since the local discriminant ideal Disc(L P /K p ) is the norm of the local different ideal Diff(L P /K p ), the routine Different leads in an obvious way to a routine to compute v p (Disc(L/K)). This routine does not require the previous computation of the discriminant Disc(f ) of the polynomial f (x); actually v p (Disc(f )) may be deduced from the identity v p (Disc(f )) = v p (Disc(L/K)) + 2 ind p (f ), where (10) ind p (f ) :…”

Section: Local Computation Of Discriminants the Discriminant Of L/k mentioning

confidence: 99%

“…The Montes algorithm is fully described in [5], in terms of the theoretical background developed in [6]. For a short review, we recommend [10] or [1,Sec. 4].…”

mentioning

confidence: 99%

Local computation of differents and discriminants

Nart¹

2013

Math. Comp.

View full text Add to dashboard Cite

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.

show abstract

Okutsu-Montes representations of prime ideals of one-dimensional integral closures

Abstract: This is a survey on Okutsu-Montes representations of prime ideals of certain one-dimensional integral closures. These representations facilitate the computational resolution of several arithmetic tasks concerning prime ideals of global fields.

Cited by 3 publications

References 14 publications

Local computation of differents and discriminants

Local computation of differents and discriminants

Complexity of OM factorizations of polynomials over local fields

Local computation of differents and discriminants

Contact Info

Product

Resources

About