La théorie du polynôme de Bernstein-Sato pour les algèbres de Tate et de Dwork-Monsky-Washnitzer

Mebkhout, Zoghman; Macarro, Luis Narváez

doi:10.24033/asens.1627

Cited by 39 publications

(38 citation statements)

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“…In the same way as in Lemma (2.6) and Proposition (2.7) we prove that for any automorphism of C-algebras ϕ : C [25]). Now we are ready to state and prove the announced extension of Theorem (4.1) to the case of arbitrary logarithmic connections.…”

Section: More Generally For Any Q(s) ∈ C[s] We Consider the Q(s)-bermentioning

confidence: 80%

A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

Macarro

2015

Advances in Mathematics

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In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the D[s]-module D[s]h s admits a Spencer logarithmic resolution satisfies the symmetry property b(−s − 2) = ±b(s). This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection E and of its dual E * with respect to a free divisor of linear Jacobian type are related by the equality b E (s) = ±b E * (−s − 2). Our results are based on the behaviour of the modules D[s]h s and D[s]E[s]h s under duality.

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Section: More Generally For Any Q(s) ∈ C[s] We Consider the Q(s)-bermentioning

confidence: 80%

A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

Macarro

2015

Advances in Mathematics

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“…The crucial fact is that if K ⊂ k is a subfield containing all the coefficients of f , then the coefficients of b f (s) computed over k belong to K. This is because upon examining every step of Oaku's algorithm one sees that all calculations are done in K. Let K be a finite extension of Q containing the coefficients of f . Since one can embed K into C, the Bernstein-Sato polynomial of f over K is the same as over C. Now we are done by Kashiwara's result in conjunction with Mebkhout and Narváez-Macarro (1991).…”

Section: Rationality Of the Rootsmentioning

confidence: 98%

Constructibility of the Set of Polynomials with a Fixed Bernstein–Sato Polynomial: an Algorithmic Approach

Leykin

2001

Journal of Symbolic Computation

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Let n and d be positive integers, let k be a field and let P(n, d; k) be the space of the non-zero polynomials in n variables of degree at most d with coefficients in k. Let B(n, d) be the set of the Bernstein-Sato polynomials of all polynomials in P(n, d; k) as k varies over all fields of characteristic 0. G. Lyubeznik proved that B(n, d) is a finite set and asked if, for a fixed k, the set of the polynomials corresponding to each element of B(n, d) is a constructible subset of P(n, d; k).In this paper we give an affirmative answer to Lyubeznik's question by showing that the set in question is indeed constructible and defined over Q, i.e. its defining equations are the same for all fields k. Moreover, we construct an algorithm that for each pair (n, d) produces a complete list of the elements of B(n, d) and, for each element of this list, an explicit description of the constructible set of polynomials having this particular Bernstein-Sato polynomial.

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“…From this fact, as the global b-function is the least common multiple of the b-functions localized at any point (Mebkhout and Narváez-Macarro, 1991), then the global b-function has the same property.…”

Section: An Application: Comparing Modulesmentioning

confidence: 99%

Explicit Comparison Theorems for D -modules

Castro-Jiménez

Ucha-Enrı́quez

2001

Journal of Symbolic Computation

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La théorie du polynôme de Bernstein-Sato pour les algèbres de Tate et de Dwork-Monsky-Washnitzer

Cited by 39 publications

References 0 publications

A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

Constructibility of the Set of Polynomials with a Fixed Bernstein–Sato Polynomial: an Algorithmic Approach

Explicit Comparison Theorems for D -modules

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