Algebra

URI permanente para esta comunidadhttps://hdl.handle.net/11441/10803

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A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors
(Elsevier, 2015-08-20) Narváez Macarro, Luis; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Sevilla. FQM218: Geometria Algebraica, Sistemas Diferenciales y Singularidades
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 bE(s) = ±bE∗ (−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.
Arithmetic motivic Poincaré series of toric varieties
(Mathematical Sciences Publishers, 2013) Cobo Pablos, Helena; González Pérez, Pedro Daniel; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; Universidad de Sevilla. FQM218: Geometría Algebraica, Sistemas Diferenciales y Singularidades
The arithmetic motivic Poincaré series of a variety V defined over a field of characteristic zero is an invariant of singularities that was introduced by Denef and Loeser by analogy with the Serre–Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form that specializes to the Serre-Oesterlé series when V is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper, we study this motivic series when V is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we explicitly deduce a finite set of candidate poles for this invariant.
Decomposing Jacobians of curves over finite fields in the absence of algebraic structure
(Elsevier, 2015-11) Ahmadi, Omran; McGuire, Gary; Rojas León, Antonio; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)
We consider the issue of when the L-polynomial of one curve over Fq divides the L-polynomial of another curve. We prove a theorem which shows that divisibility follows from a hypothesis that two curves have the same number of points over infinitely many extensions of a certain type, and one other assumption. We also present an application to a family of curves arising from a conjecture about exponential sums. We make our own conjecture about L-polynomials, and prove that this is equivalent to the exponential sums conjecture.
Pierre Deligne
(Real Sociedad Matemática Española, 2013) Rojas León, Antonio; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; Junta de Andalucía; Universidad de Sevilla. FQM218: Geometría Algebraica, Sistemas Diferenciales y Singularidades
The Newton Procedure for several variables
(Elsevier, 2011-07-15) Soto Prieto, Manuel Jesús; Vicente Córdoba, José Luis; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Universidad de Sevilla. FQM218: Geometria Algebraica, Sistemas Diferenciales y Singularidades
Let us consider an equation of the form P(x, z) = zm + w1(x)zm−1 + · · · + wm−1(x)z + wm(x) = 0, where m>1, n>1, x=(x1⋯xn) is a vector of variables, k is an algebraically closed field of characteristic zero, View the MathML source and wm(x)≠0. We consider representations of its roots as generalized Puiseux power series, obtained by iterating the classical Newton procedure for one variable. The key result of this paper is the following: Theorem 1.The iteration of the classical Newton procedure for one variable gives rise to representations of all the roots of the equation above by generalized Puiseux power series in x1/d, d∈Z>0, whose supports are contained in an n-dimensional, lex-positive strictly convex polyhedral cone (see Section 5). We must point out that the crucial result is not the existence of these representations, which is a well-known fact; but the fact that their supports are contained in such a special cone. We achieve the proof of this theorem by taking a suitable affine chart of a toric modification of the affine space.
The stability of the Kronecker products of Schur functions
(Elsevier, 2011-04-01) Briand, Emmanuel; Orellana, Rosa C.; Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de álgebra; Ministerio de Ciencia e Innovación (MICIN). España; Junta de Andalucía
In the late 1930’s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.

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Examinando Algebra por Agencia financiadora "Ministerio de Ciencia e Innovación (MICIN). España"