Person: Luengo Velasco, Ignacio
Universidad Complutense de Madrid
Faculty / Institute
Álgebra, Geometría y Topología
Now showing 1 - 10 of 64
PublicationFPGA implementation of post-quantum DME cryptosystem(Institute of Electrical and Electronics, 2020-05) Imaña Pascual, José Luis; Luengo Velasco, IgnacioThe rapid development of quantum computing constitutes a significant threat to modern Public-Key Cryptography (PKC). The use of Shor's algorithm with potential powerful quantum computers could easily break the two most widely used public key cryptosystems, namely, RSA and Elliptic Curve Cryptography (ECC), based on integer factorization and discrete logarithm problems. For this reason, Post-Quantum Cryptography (PQC) based on alternative mathematical features has become a fundamental research topic due to its resistance against quantum computers. The National Institute of Standards and Technology (NIST) has even opened a call for proposals of quantum-resistant PKC algorithms in order to standardize one or more PQC algorithms. Cryptographic systems that appear to be extremely difficult to break with large quantum computers are hash -based cryptography, lattice -based cryptography, code -based cryptography, and multivariate -quadratic cryptography. Furthermore, efficient hardware implementations are highly required for these alternative quantum -resistant cryptosystems. PublicationClasificación de puntos dobles de superficies algebroides sumergidas(Universitat Autònoma de Barcelona, 1981-02) Luengo Velasco, IgnacioIt is a well-known result that one can associate to each embedded algrebroid surfaces (E.A.S.), such that mult(s)=2, and algebroid curve c(v) (not necesarilly reduced). On the other hand, is  we associate to each E.A.S. W a finite weighted tree Ar (W). In this work we prove that if S and S' are E.A.S. with m(S) = m(S')= 2, then Ar(S')= Ar(S) if and only if C(S) and C(s') are equisingular as non-reduced curves. PublicationBifurcations and topology of meromorphic germs(Kluwer Academic Publishers, 2001) Gusein-Zade, Sabir Medgidovich; Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Siersma, D.; Wall, Charles Terence Clegg; Zakalyukin, V.A meromorphic germ at the origin in the complex space Cn is a ratio of two holomorphic germs on (Cn,0). After presentation of the basic definitions in the general context of arbitrary meromorphic germs the authors study the monodromy by calculating its zeta function. Then they give some results on homology splitting and bouquet-type theorems for the global case of meromorphic functions on compact complex manifolds. Some applications to traditional cases of rational functions on CPn, including in particular polynomial functions on Cn, are considered. PublicationPower structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points(Michigan Mathematical Journal, 2006) Gusein-Zade, Sabir Medgidovich; Luengo Velasco, Ignacio; Melle Hernández, AlejandroThe power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety as an exponent of that for the complex affine space of the same dimension. Specializations of this relation give formulae for generating series of such invariants of the Hilbert schemes of points as the Euler characteristic and the Hodge-Deligne polynomial. PublicationOn ν-quasi-ordinary power series: factorization, Newton trees and resultants(American Mathematical Society, 2011) Artal Bartolo, Enrique; Cassou-Noguès, Pierrette; Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Cogolludo Agustín, José Ignacio; Hironaka, ErikoThe concept of ν-quasi-ordinary power series, which is a generalization of quasi-ordinary power series, was first introduced by H. Hironaka. In the paper under review, the authors study ν-quasi-ordinary power series and give a factorization theorem for ν-quasi-ordinary power series in the first part. The proof of the theorem uses Newton maps. In the second part of the paper, using the factorization theorem, they introduce the Newton tree to encode the Newton process for any hypersurface singularity defined by a power series germ as in Notation 1.1. Finally, the authors describe a condition for two ν-quasi-ordinary power series to have an "intersection multiplicity " by using Newton trees and they can also compute this generalized intersection multiplicity, resultants and discriminant. PublicationLinks and analytic invariants of superisolated singularities(American Mathematical Society, 2005) Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Némethi, A.Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the ``Seiberg-Witten invariant conjecture''(of Nicolaescu and the third author), the ``Universal abelian cover conjecture'' (of Neumann and Wahl) and the ``Geometric genus conjecture'' fail (at least at that generality in which they were formulated). Moreover, we also show that for Gorenstein singularities (even with integral homology sphere links) besides the geometric genus, the embedded dimension and the multiplicity (in particular, the Hilbert-Samuel function) also fail to be topological; and in general, the Artin cycle does not coincide with the maximal (ideal) cycle. PublicationHigher-order orbifold Euler characteristics for compact Lie group actions(Cambridge University Press, 2015-12) Gusein-Zade, Sabir Medgidovich; Luengo Velasco, Ignacio; Melle Hernández, AlejandroWe generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well PublicationOn rational cuspidal plane curves, open surfaces and local singularities(World Scientific Publishing Co., 2007) Fernández de Bobadilla de Olarzábal, Javier José; Luengo Velasco, Ignacio; Melle Hernández, Alejandro; Némethi, A.; Cheniot, D.; Dutertre, D.; Murolo, C.; Trotman, D.; Pichon, A.Let C be an irreducible projective plane curve in the complex projective space P(2). The classification of such curves, up to the action of the automorphism group PGL(3, C) on P(2), is a very difficult open problem with many interesting connections. The main goal is to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of singularities of given topological type. In this note we are mainly interested in the case when C is a rational curve. The aim of this article is to present some of the old conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors. PublicationMonodromy conjecture for some surface singularities(Société Mathématique de France, 2002-07) Melle Hernández, Alejandro; Artal Bartolo, Enrique; Cassou-Noguès, Pierrette; Luengo Velasco, IgnacioIn this work we give a formula for the local Denef–Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef–Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler–Poincaré characteristic is three.