131 research outputs found
The Koszul property for spaces of quadrics of codimension three
In this paper we prove that, if is an algebraically closed field
of characteristic different from 2, almost all quadratic standard graded
-algebras such that are Koszul. More
precisely, up to graded -algebra homomorphisms and trivial fiber
extensions, we find out that only two (or three, when the characteristic of
is 3) algebras of this kind are non-Koszul.
Moreover, we show that there exist nontrivial quadratic standard graded
-algebras with , that are Koszul but do not admit a Gr\"obner basis of quadrics even after a
change of coordinates, thus settling in the negative a question asked by Conca.Comment: 26 pages. Comments are very welcome
Determinantal sets, singularities and application to optimal control in medical imagery
Control theory has recently been involved in the field of nuclear magnetic
resonance imagery. The goal is to control the magnetic field optimally in order
to improve the contrast between two biological matters on the pictures.
Geometric optimal control leads us here to analyze mero-morphic vector fields
depending upon physical parameters , and having their singularities defined by
a deter-minantal variety. The involved matrix has polynomial entries with
respect to both the state variables and the parameters. Taking into account the
physical constraints of the problem, one needs to classify, with respect to the
parameters, the number of real singularities lying in some prescribed
semi-algebraic set. We develop a dedicated algorithm for real root
classification of the singularities of the rank defects of a polynomial matrix,
cut with a given semi-algebraic set. The algorithm works under some genericity
assumptions which are easy to check. These assumptions are not so restrictive
and are satisfied in the aforementioned application. As more general strategies
for real root classification do, our algorithm needs to compute the critical
loci of some maps, intersections with the boundary of the semi-algebraic
domain, etc. In order to compute these objects, the determinantal structure is
exploited through a stratifi-cation by the rank of the polynomial matrix. This
speeds up the computations by a factor 100. Furthermore, our implementation is
able to solve the application in medical imagery, which was out of reach of
more general algorithms for real root classification. For instance,
computational results show that the contrast problem where one of the matters
is water is partitioned into three distinct classes
Combinatorial resultants in the algebraic rigidity matroid
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs
Positive Geometries for Scattering Amplitudes in Momentum Space
Positive geometries provide a purely geometric point of departure for
studying scattering amplitudes in quantum field theory. A positive geometry is
a specific semi-algebraic set equipped with a unique rational top form - the
canonical form. There are known examples where the canonical form of some
positive geometry, defined in some kinematic space, encodes a scattering
amplitude in some theory. Remarkably, the boundaries of the positive geometry
are in bijection with the physical singularities of the scattering amplitude.
The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical
positive geometry. It lives in momentum twistor space and describes tree-level
(and the integrands of planar loop-level) scattering amplitudes in maximally
supersymmetric Yang-Mills theory.
In this dissertation, we study three positive geometries defined in on-shell
momentum space: the Arkani-Hamed-Bai-He-Yan (ABHY) associahedron, the Momentum
Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes
tree-level scattering amplitudes for different theories in different spacetime
dimensions. The three positive geometries share a series of interrelations in
terms of their boundary posets and canonical forms. We review these
relationships in detail, highlighting the author's contributions. We study
their boundary posets, classifying all boundaries and hence all physical
singularities at the tree level. We develop new combinatorial results to derive
rank-generating functions which enumerate boundaries according to their
dimension. These generating functions allow us to prove that the Euler
characteristics of the three positive geometries are one. In addition, we
discuss methods for manipulating canonical forms using ideas from computational
algebraic geometry.Comment: PhD Dissertatio
Gorenstein colength of local Artin k-algebras
[spa] En esta tesis abordamos el problema de la aproximación de anillos locales por anillos de Gorenstein en el caso cero-dimensional. Nos centramos en el estudio y el cálculo efectivo de la colongitud de Gorenstein, una noción propuesta por Ananthnarayan para medir qué tan cerca está una k-álgebra
artiniana de satisfacer la propiedad de Gorenstein.
Extendemos la caracterización de los anillos de Teter a k-álgebras de baja colongitud de Gorenstein en términos de sus sistemas inversos de Macaulay y ciertos ideales auto-duales generalizando resultados de Huneke-Vraciu, Ananthnarayan y Elias-Silva. Estudiamos ciertas propiedades de las coberturas Gorenstein minimales de un anillo, como su función de Hilbert y su dimensión de embedding.
La herramienta de los sistemas inversos resulta clave para la definición y cálculo efectivo de la variedad de coberturas Gorenstein minimales vía el método de integración introducido por Mourrain.
En codimensión 2, extendemos la parametrización de Conca-Valla para ideales del anillo de polinomios al anillo de series, obteniendo un método para el cálculo de coberturas Gorenstein basado en el estudio de matrices canónicas de Hilbert-Burch.
Todos los algoritmos propuestos se han implementado en una librería del software de álgebra communtativa Singular
Combinatorial Resultants in the Algebraic Rigidity Matroid
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs
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