Computing Limit Points of Quasi-components of Regular Chains and its Applications

Computing limit is a fundamental task in mathematics and different mathematical concepts are defined in terms of limit computations. Among these mathematical concepts, we are interested in three different types of limit computations: first, computing the limit points of solutions of polynomial systems represented by regular chains, second, computing tangent cones of space curves at their singular points which can be viewed as computing limit of secant lines, and third, computing the limit of real multivariate rational functions. For computing the limit of solutions of polynomial systems represented by regular chains, we present two different methods based on Puiseux series expansions and linear changes of coordinates. The first method, which is based on Puiseux series expansions, addresses the problem of computing real and complex limit points corresponding to regular chains of dimension one. The second method studies regular chains under changes of coordinates. It especially computes the limit points corresponding to regular chains of dimension higher than one for some cases. we consider strategies where these changes of coordinates can be either generic or guided by the input. For computing the Puiseux parametrizations corresponding to regular chains of dimension one, we rely on extended Hensel construction (EHC). The Extended Hensel Construction is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. We also outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a Standard basis. We also present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and V´elez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components

Gröbner's problem and the geometry of GT-varieties

[eng] Within the framework of algebraic geometry and commutative algebra, this thesis makes advances in (1) the Gröbner's longstanding problem of determining whether a monomial projection of the Veronese variety Vn,d is an aCM variety; (2) it contributes to the fundamental problem of describing the internal structure of the ring of invariants of a finite subgroup of GL(n+1,K). Our approach towards these subjects involves combinatorics with an application to the Lefschetz properties of artinian ideals. The heart of this dissertation is expounded in four chapters with an introductory Chapter 1 collecting all the basic notions and results needed onwards; and an Appendix A containing two algorithms and implementations with the software Wolfram Mathematica. In Chapter 2, we treat Gröbner's problem and we study the invariants of the cyclic extension bG of a finite diagonal abelian group G in GL(n+1,K)$ of order d. We prove that the set B1 of monomial invariants of G of degree d minimally generates the ring RbG of invariants. We establish that B1 parameterizes an aCM monomial projection Xd of Vn,d, which we call a bG-variety with group G. They form a family of aCM monomial projections of Vn,d blending commutative algebra, algebraic geometry, combinatorics and the Lefschetz properties. In Chapter 2, we study the geometry of bG-varieties Xd with group G. We investigate their Hilbert function and series from the perspectives of invariant theory and combinatorics. We prove that their homogeneous ideals I(Xd) are generated by binomials of degree at most 3 and we exhibit examples reaching this bound. We identify the canonical module ωXd of Xd with an ideal I(relint(HA)) of RbG and we prove that it is generated by monomial of degree d and 2d. We characterize the Castelnuovo-Mumford regularity of Xd in terms of ωXd. In Chapter 3, we investigate the invariants of finite supgroups of SL(3,K) and we relate them to the weak Lefschetz property. We consider the cyclic extension of a representation in SL(n+1,K) of the dihedral group D2d of order 2d. We prove that its ring of invariants is minimally generated by a set of monomials and binomials of degree 2d which generates a non monomial GT-system with group D2d and parameterizes an aCM projection SD2d of V2,d. We describe a minimal graded free resolution of SD2d and we compute a minimal set of generators of I(SD2d) of degree 2. In Chapter 4, we introduce RL-varieties Xd: a family of smooth rational non aCM monomial projections of Vn,d related to bG-varieties Xd with group G. They are parameterized by a set of monomials of degree d determined by ωXd which defines an embedding of Pn. These properties allow us to describe their normal bundles NXd and to contribute to the classical problem of computing the dimension of the cohomology of the normal bundle of projective varieties.[spa] La tesis doctoral contribuye principalmente a dos problemas remarcables en geometría algebraica y álgebra conmutativa. Por un lado tenemos el problema, propuesto por Gröbner en 1967, de determinar cuándo una proyección monomial de la variedad de Veronese es aritméticamente Cohen-Macaulay (aCM). Por otro lado, el problema clásico y fundamental de describir la estructura interna del anillo de invariantes de un grupo finito. El enfoque desarrollado en la disertación es combinatorio con una aplicación a las propiedades de Lefschetz de los ideales artinianos. La disertación se ha organizado en cuatro capítulos principales, un Capítulo 1 introductorio donde se recopilan las nociones y resultados básicos que se necesitan en el cuerpo del texto; y un Apéndice A donde se recogen dos algoritmos y sus implementaciones en el programa Wolfram Mathematica, los cuales se han utilizado en la ejemplicación de los resultados principales de la tesis. En el Capítulo 2, consideramos el problema de Gröbner y demostramos que dado un grupo G lineal, diagonal, abeliano y finito de orden d, el conjunto de invariantes monomiales de G de grado d es un sistema minimal de generadores del anillo de invariantes de su extensión cíclica GG. Demostramos que dicho conjunto de monomios parametriza una proyección monomial aCM de la variedad de Veronese.Llamamos a dichas projecciones GG-variedades con grupo G, forman una familia de variedades aCM que conectan la geometria algebraica, el álgebra conmutativa, la combinatoria y las propiedades de Lefschetz. En el Capítulo 3, estudiamos la geometría de las GG-variedades con grupo G en aras de determinar una resolución libre y minimal de su anillo de coordenades homogéneo. Analizamos su función y serie de Hilbert desde el punto de vista de la combinatoria y la teoria de invariantes. Demostramos que su ideal homogéneo está minimalmente generado por binomios de grado como máximo 3 y exhibimos ejemplos que alcanzan dicha cota. Identificamos su módulo canónico con un ideal del anillo de invariantes de GG y probamos que está generado por monomios de grado d y 2d. En el Capítulo 4, investigamos la relación entre los invariantes de grupos lineales no abelianos y la propiedad débil de Lefschetz. Demostramos que el anillo de invariantes de una representación del grupo diedral D de orden 2d está mínimamente generado por monomios y binomios de grado 2d. Esto nos permite probar que generan un GT-sistema y que parametriza una GT-variedad aCM. A continua -ción determinamos una resolución libre y minimal de su anillo de coordenadas homogéneo y determinamos un sistema minimal de generadores de su ideal homogéneo de grado 2. En el Capítulo 5, introducimos la família de RL-variedades. Se trata de proyecciones monomiales no aCM lisas y racionales de la variedad de Veronese relacionadas con las GG-variedades con grupo G. Describimos el fibrado vectorial normal de una RL-variedad y contribuimos al problema clásico de determinar la dimensión de la cohomología del fibrado normal de variedades proyectivas

The symmetric signature

This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of K\"ahler differentials and of the the top-dimensional syzygy module of the residue field respectively. We study and compute them for different classes of rings: two-dimensional ADE singularities, two-dimensional cyclic singularities, and cones over plane elliptic curves (for the differential symmetric signature). The values obtained coincide with the F-signature of such rings in positive characteristic. The thesis contains also a short survey on the Auslander correspondence for quotient singularities.Comment: The main results of the thesis appeared also in arXiv:1602.07184 and arXiv:1603.0642

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Noether normalization guided by monomial cone decompositions

This paper explains the relevance of partitioning the set of standard monomials into cones for constructing a Noether normalization for an ideal in a polynomial ring. Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials – also known as a Stanley decomposition – is constructed in the context of Janet bases, in order to come up with sparse coordinate changes which achieve Noether normal position for the given ideal

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

On tautological classes of fibre bundles and self-embedding calculus

In this thesis we study the ring of characteristic classes of smooth fibre bundles with a particular focus on tautological classes and the ring that they generate. In the first part we use tools from rational homotopy theory to compute analogues of tautological rings over fibrations, which provides upper bounds on the tautological rings of fibre bundles. In some cases we find an upper bound on the Krull dimension that is sharp. In the second part, we study the classifying space using the calculus of embeddings which provides a homotopy theoretic approximation. We construct cohomology classes on the self-embedding tower which extend certain characteristic classes that were introduced by Kontsevich. This construction is based on introducing configuration space integrals over the tower itself, which also has some consequences for tautological classes that we explore.Cambridge Trust and King's College Martin Thackeray Scholarshi

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2