Complexity of OM factorizations of polynomials over local fields

Bauch, Jens-Dietrich; Nart, Enric; Stainsby, Hayden D.

doi:10.48550/arxiv.1204.4671

Cited by 3 publications

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“…If δ = 0, then B p = A p [θ]; thus, we assume δ > 0 in our analysis. By [1,Thm. 3.14], for the computation of a p-integral basis of B/A we may work modulo p δ+1 .…”

Section: 1mentioning

confidence: 99%

“…Let us briefly recall how the Montes algorithm proceeds to obtain further dissections of the subsets P ϕ . We use the version of the algorithm described in [1,Sec. 4], guaranteeing that the OM representations t P have order r P + 1, where r P is the Okutsu depth of F P .…”

Section: 1mentioning

confidence: 99%

“…The Montes algorithm has a cost of O n 2+ǫ + n 1+ǫ δ log q + n 1+ǫ δ 2+ǫ p-small operations [1,Thm. 5.15].…”

Section: 1mentioning

confidence: 99%

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Higher Newton polygons and integral bases

Guàrdia

Montes

Nart

2015

Journal of Number Theory

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Abstract. Let A be a Dedekind domain whose field of fractions K is a global field. Let p be a non-zero prime ideal of A, and Kp the completion of K at p. The Montes algorithm factorizes a monic irreducible separable polynomial f (x) ∈ A[x] over Kp, and it provides essential arithmetic information about the finite extensions of Kp determined by the different irreducible factors. In particular, it can be used to compute a p-integral basis of the extension of K determined by f (x). In this paper we present a new and faster method to compute p-integral bases, based on the use of the quotients of certain divisions with remainder of f (x) that occur along the flow of the Montes algorithm.

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“…If δ = 0, then B p = A p [θ]; thus, we assume δ > 0 in our analysis. By [1,Thm. 3.14], for the computation of a p-integral basis of B/A we may work modulo p δ+1 .…”

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Higher Newton polygons and integral bases

Guàrdia

Montes

Nart

2015

Journal of Number Theory

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“…The original version of the Montes algorithm produces a more comprehensive output, but we require only ind p(t) . Admitting fast multiplication, it is shown in [1,Theorem 5.14]…”

Section: Algebraic Function Fieldsmentioning

confidence: 99%

“…According to our tests, the running time of the genus computation is in most of the cases dominated by the computation and factorization of the discriminant of a defining polynomial of F . The complexity estimation for the Montes algorithm in [1] affords us concrete bounds for the number of operations in the finite constant field k, which are needed to compute the genus of F (Theorem 3.6). Unfortunately, these theoretical bounds do not fit well with the practical performance of the method.…”

Section: Introductionmentioning

confidence: 99%

Genus computation of global function fields

Bauch

2015

Journal of Symbolic Computation

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In this paper we present an algorithm that computes the genus of a global function field. Let F/k be function field over a field k, and let k 0 be the full constant field of F/k. By using lattices over subrings of F , we can express the genus g of F in terms of [k 0 : k] and the indices of certain orders of the finite and infinite maximal orders of F . If k is a finite field, the Montes algorithm computes the latter indices as a by-product. This leads us to a fast computation of the genus of global function fields. Our algorithm does not require the computation of any basis, neither of finite nor infinite maximal order.

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Local computation of differents and discriminants

Nart¹

2012

Preprint

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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.

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Complexity of OM factorizations of polynomials over local fields

Cited by 3 publications

References 0 publications

Higher Newton polygons and integral bases

Higher Newton polygons and integral bases

Genus computation of global function fields

Local computation of differents and discriminants

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