Ideals generated by the inner 2-minors of collections of cells

In 2012 Ayesha Asloob Qureshi connected collections of cells to Commutative Algebra assigning to every collection $\mathcal{P}$ of cells the ideal of inner 2-minors, denoted by $I_{\mathcal{P}}$, in the polynomial ring $S_{\mathcal{P}}=K[x_v:v\text{ is a vertex of }\mathcal{P}]$. Investigating the main algebraic properties of $K[\mathcal{P}]=S_{\mathcal{P}}/I_{\mathcal{P}}$ depending on the shape of $\mathcal{P}$ is the purpose of this research. Many problems are still open and they seem to be fascinating and exciting challenges.\\ In this thesis we prove several results about the primality of $I_{\mathcal{P}}$ and the algebraic properties of $K[\mathcal{P}]$ like Cohen-Macaulyness, normality and Gorensteiness, for some classes of non-simple polyominoes. The study of the Hilbert-Poincar\'e series and the related invariants as Krull dimension and Castelnuovo-Mumford regularity are given. Finally we provide the code of the package \texttt{PolyominoIdeals} developed for \texttt{Macaulay2}

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

Algorithms in Intersection Theory in the Plane

This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal $I\subseteq \mathbb K[x,y]$, \textit{i.e.} finding the points (maximal ideals of $\mathbb K[x,y]/I$) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation. Let $I\subseteq \mathbb K[x,y]$ be an ideal generated by $\mathcal F$, a subset of $\mathbb A[x,y]$ with $\mathbb A\hookrightarrow\mathbb K$ and $\mathbb K$ a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of $I$ or its primary component at the origin. We introduce an $\mathfrak m$-adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of $I$ if $\mathfrak m\subseteq \mathbb A$ is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal $\mathfrak m$ suitable for our filtration. When the field $\mathbb K$ is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete $\mathfrak m$-adic algorithm to recover $\mathcal G$, the Gr\"obner basis of $I$. We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of $n$ generators. We use this result to obtain a bound on the height of the coefficients of $\mathcal G$ and to control the probability of choosing a \textit{good} maximal ideal $\mathfrak m\subseteq\mathbb A$ to build the $\mathfrak m$-adic expansion of $\mathcal G$. Inspired by Pardue (1994), we also give a constructive proof to characterise a Zariski open set of $\mathrm{GL}_2(\mathbb K)$ (with action on $\mathbb K[x,y]$) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional $I$ becomes Borel-fixed when $|\mathbb K|$ is sufficiently large. This sharpens our analysis to obtain, when $\mathbb A=\mathbb Z$ or $\mathbb A=k[t]$, a complexity less than cubic in terms of the dimension of $\mathbb Q[x,y]/\langle \mathcal G\rangle$ and softly linear in the height of the coefficients of $\mathcal G$. We adapt the resulting method and present the analysis to find the $\langle x,y\rangle$-primary component of $I$. We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras

A Combinatorial Commutative Algebra Approach to Complete Decoding

Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí

A quasisymmetric function for matroids

AbstractA new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: •defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients;•is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid;•is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight;•behaves simply under matroid duality;•has a simple expansion in terms of P-partition enumerators;•is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis

Exploiting Torus Actions: Immaculate Line Bundles on Toric Varieties and Parametrizations of Gröbner Cells

This dissertation contains two chapters on the use of torus actions in algebraic geometry. In chapter 2 we study ”immaculate line bundles” on projective toric varieties. The cohomology groups of those line bundles vanish in all degrees, including the 0-th degree. Immaculate line bundles can be seen as building blocks of full exceptional sequences of line bundles of the variety. All the immaculate line bundles of a toric variety X = TV(Σ) can be identified in two steps. First identify those subsets of the rays Σ(1) whose geometric realization is not k-acyclic, they will be called tempting. Those subsets of the rays give ”maculate sets/regions” in the class group of the variety. A line bundle is immaculate, if it is not in any of those maculate sets. So the first step in finding immaculate line bundles is to find all tempting subsets. When X is projective, the main result for this is that primitive collections – subsets of the rays that do not span a cone, but each proper subset spans a cone – are always tempting. And a subset of rays can only be tempting if it is the union of primitive collections. The same has to hold for the complement, too. We give descriptions of the immaculate line bundles for different examples. In particular, we describe the immaculate locus for projective toric varieties of Picard rank 3. Most of the results have been published in [ABKW20]. In chapter 3 we study the Hilbert scheme of n points in affine plane. It describes all ideals in the polynomial ring of two variables whose quotient is an n-dimensional vector space. The Hilbert scheme can be decomposed into so called Gröbner cells. They consist of those ideals that have a prescribed leading term ideal with respect to a given term order. The Gröbner cells for the lexicographic and the degree-lexicographic order are parametrized in [CV08] and [Con11], respectively, by canonical Hilbert-Burch matrices. A Hilbert-Burch matrix of an ideal is a matrix generating the syzygies of the ideal. Its maximal minors also generate the ideal. These results are generalized in two directions. Firstly, we consider the ring of formal power series. Here we give a parametrization of the cells that respects the Hilbert function stratification of the punctual Hilbert scheme. In particular, this cellular decomposition restricts to a cellular decomposition of the subscheme consisting of ideals with a prescribed Hilbert function. We use the parametrization to describe subsets of the Gröbner cells associated to lex-segment ideals with a given minimal number of generators. These subsets are quasi-affine varieties inside the cell. Most of these results have been published in [HW21] and [HW23]. The second way of changing the setting is to consider a general term order on the polynomial ring. We give a surjection to the Gröbner cell with respect to this ordering and parametrizations of subsets of the cell, as well as a conjecture how the parametrization of the whole cell should look like. We also study intersections of Gröbner cells with respect to different term orders.Die vorliegende Dissertation besteht aus zwei Kapiteln zu zwei unterschiedlichen Anwendungen von Toruswirkungen in der algebraischen Geometrie. Die wichtigsten Objekte des Kapitels 2 sind unbefleckte Geradenbündel auf projektiven torischen Varietäten X = TV(Σ), Geradenbündel, deren Kohomologiegruppen alle verschwinden. Unbefleckte Geradenbündel können als Bausteine für exzeptionelle Sequenzen aus Geradenbündeln dienen und somit die derivierte Kategorie der Varietät beschreiben. Die Bestimmung von unbefleckten Geradenbündeln lässt sich in zwei Schritte aufteilen. Es lassen sich Teilmengen der Strahlen Σ(1) des die torische Varietät beschreibenden Fächers Σ identifizieren, deren geometrische Realisierungen nicht k-azyklisch sind. Diese verlockenden Teilmengen der Strahlen definieren befleckte Teilmengen der Klassengruppe Cl(X). Ein Geradenbündel ist genau dann unbefleckt, wenn es in keiner befleckten Teilmenge von Cl(X) liegt. Die Bestimmung aller unbefleckten Geradenbündel lässt sich also in zwei Schritte aufteilen. Das Bestimmen der verlockenden Teilmengen der Strahlen und das Bestimmen der zugehörigen befleckten Regionen. Primitive Kollektionen – Teilmengen der Strahlen, die selbst keinen Kegel des Fächers aufspannen, aber jede ihrer Teilmenge spannt einen Kegel des Fächers auf – sind verlockend und außerdem ist eine Teilmenge nur dann verlockend, wenn sie eine Vereinigung von primitiven Kollektionen ist. Dies muss auch für das Komplement gelten. Wir geben die Beschreibung für die unbefleckten Geradenbündel für verschiedene Beispielklassen von projektiven torischen Varietäten. Insbesondere beschreiben wir die unbefleckten Geradenbündel für projektive torische Varietäten von Picardrang 3. Die meisten dieser Ergebnisse sind in [ABKW20] erschienen. In Kapitel 3 geht es um das Hilbertschema von n Punkten in der affinen Ebene. Seine Punkte sind Ideale im Polynomenring k[x, y], deren Quotient ein n-dimensionaler k-Vektorraum ist. Das Hilbertschema kann in sogenannte Gröbnerzellen unterteilt werden. Sie umfassen Ideale, die bezüglich einer Termordnung τ ein festgelegtes Leitideal haben. In [CV08] und [Con11] werden für die lexikographische und gradlexikographische Termordnung Parametrisierung der Gröbnerzellen durch kanonische Hilbert-Burch Matrizen angegeben. Hilbert-Burch Matrizen beschreiben die Syzygien des Ideals und ihre maximalen Minoren erzeugen das Ideal. Die Ergebnisse werden in zwei Richtungen verallgemeinert. Zunächst betrachten wir Ideale im Ring der formalen Potenzreihen. Wir geben eine Parametrisierung der Zellen, bei der die lokale Struktur der Ideale berücksichtigt wird. Insbesondere lässt sich diese zelluläre Unterteilung des lokalen Hilbertschemas auf eine zelluläre Unterteilung des Unterschemas einschränken, das nur Ideale mit einer gegebenen Hilbertfunktion beinhaltet. Durch diese Parametrisierung lassen sich für Ideale in diesen Zellen kanonische Hilbert-Burch Matrizen definieren. Diese benutzen wir um Teilmengen der Gröbnerzellen mit einer vorgegebenen minimalen Anzahl von Erzeugern zu beschreiben. Diese Teilmengen sind quasi-affine Varietäten in der Gröbnerzelle. Die meisten der Resultate sind in [HW21] und [HW23] erschienen. Die zweite Möglichkeit das Setting zu ändern, ist beliebige Termordnungen auf dem Polynomenring zu betrachten. Im zweiten Teil von Kapitel 3 geben wir eine Surjektion auf diese Gröbnerzellen, sowie Parametrisierungen von Teilmengen und geben eine Vermutung, wie eine Parametrisierung der ganzen Zelle aussieht. Außerdem untersuchen wir Schnitte von Gröbnerzellen bezüglich verschiedener Termordnungen

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

Convex Algebraic Geometry Approaches to Graph Coloring and Stable Set Problems

The objective of a combinatorial optimization problem is to find an element that maximizes a given function defined over a large and possibly high-dimensional finite set. It is often the case that the set is so large that solving the problem by inspecting all the elements is intractable. One approach to circumvent this issue is by exploiting the combinatorial structure of the set (and possibly the function) and reformulate the problem into a familiar set-up where known techniques can be used to attack the problem. Some common solution methods for combinatorial optimization problems involve formulations that make use of Systems of Linear Equations, Linear Programs (LPs), Semidefinite Programs (SDPs), and more generally, Conic and Semi-algebraic Programs. Although, generality often implies flexibility and power in the formulations, in practice, an increase in sophistication usually implies a higher running time of the algorithms used to solve the problem. Despite this, for some combinatorial problems, it is hard to rule out the applicability of one formulation over the other. One example of this is the Stable Set Problem. A celebrated result of Lovász's states that it is possible to solve (to arbitrary accuracy) in polynomial time the Stable Set Problem for perfect graphs. This is achieved by showing that the Stable Set Polytope of a perfect graph is the projection of a slice of a Positive Semidefinite Cone of not too large dimension. Thus, the Stable Set Problem can be solved with the use of a reasonably sized SDP. However, it is unknown whether one can solve the same problem using a reasonably sized LP. In fact, even for simple classes of perfect graphs, such as Bipartite Graphs, we do not know the right order of magnitude of the minimum size LP formulation of the problem. Another example is Graph Coloring. In 2008 Jesús De Loera, Jon Lee, Susan Margulies and Peter Malkin proposed a technique to solve several combinatorial problems, including Graph Coloring Problems, using Systems of Linear Equations. These systems are obtained by reformulating the decision version of the combinatorial problem with a system of polynomial equations. By a theorem of Hilbert, known as Hilbert's Nullstellensatz, the infeasibility of this polynomial system can be determined by solving a (usually large) system of linear equations. The size of this system is an exponential function of a parameter $d$ that we call the degree of the Nullstellensatz Certificate. Computational experiments of De Loera et al. showed that the Nullstellensatz method had potential applications for detecting non-$3$-colorability of graphs. Even for known hard instances of graph coloring with up to two thousand vertices and tens of thousands of edges the method was useful. Moreover, all of these graphs had very small Nullstellensatz Certificates. Although, the existence of hard non-$3$-colorable graph examples for the Nullstellensatz approach are known, determining what combinatorial properties makes the Nullstellensatz approach effective (or ineffective) is wide open. The objective of this thesis is to amplify our understanding on the power and limitations of these methods, all of these falling into the umbrella of Convex Algebraic Geometry approaches, for combinatorial problems. We do this by studying the behavior of these approaches for Graph Coloring and Stable Set Problems. First, we study the Nullstellensatz approach for graphs having large girth and chromatic number. We show that that every non-$k$-colorable graph with girth $g$ needs a Nullstellensatz Certificate of degree $\Omega(g)$ to detect its non-$k$-colorability. It is our general belief that the power of the Nullstellensatz method is tied with the interplay between local and global features of the encoding polynomial system. If a graph is locally $k$-colorable, but globally non-$k$-colorable, we suspect that it will be hard for the Nullstellensatz to detect the non-$k$-colorability of the graph. Our results point towards that direction. Finally, we study the Stable Set Problem for $d$-regular Bipartite Graphs having no $C_4$, i.e., having no cycle of length four. In 2017 Manuel Aprile \textit{et al.} showed that the Stable Set Polytope of the incidence graph $G_{d-1}$ of a Finite Projective Plane of order $d-1$ (hence, $d$-regular) does not admit an LP formulation with fewer than $\frac{\ln(d)}{d}|E(G_{d-1})|$ facets. Although, we did not manage to improve this lower bound for general $d$-regular graphs, we show that any $4$-regular bipartite graph $G$ having no $C_4$ does not admit an LP formulation with fewer than $|E(G)|$ facets. In addition, we obtain computational results showing the $|E(G)|$ lower bound also holds for the Finite Projective Plane $G_4$, a $5$-regular graph. It is our belief that Aprile et al. bounds can be improved considerably

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