On the complexity of computing Gr\"obner bases for weighted homogeneous systems

Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights $W=(w\_{1},\dots,w\_{n})$, $W$-homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree $\deg\_{W}(X\_{1}^{\alpha\_{1}},\dots,X\_{n}^{\alpha\_{n}}) = \sum w\_{i}\alpha\_{i}$. Gr\"obner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted homogeneous case. We show that in this case, the complexity estimate for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be divided by a factor $\left(\prod w\_{i} \right)^{\omega}$. For zero-dimensional systems, the complexity of Algorithm~\FGLM $nD^{\omega}$ (where $D$ is the number of solutions of the system) can be divided by the same factor $\left(\prod w\_{i} \right)^{\omega}$. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of $W$-degree $(d\_{1},\dots,d\_{n})$, these complexity estimates are polynomial in the weighted B\'ezout bound $\prod\_{i=1}^{n}d\_{i} / \prod\_{i=1}^{n}w\_{i}$. Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by the weighted Macaulay bound $\sum (d\_{i}-w\_{i}) + w\_{n}$, and this bound is sharp if we can order the weights so that $w\_{n}=1$. For overdetermined semi-regular systems, estimates from the homogeneous case can be adapted to the weighted case. We provide some experimental results based on systems arising from a cryptography problem and from polynomial inversion problems. They show that taking advantage of the weighted homogeneous structure yields substantial speed-ups, and allows us to solve systems which were otherwise out of reach

Optimized Lie-Trotter-Suzuki decompositions for two and three non-commuting terms

Lie-Trotter-Suzuki decompositions are an efficient way to approximate operator exponentials $\exp(t H)$ when $H$ is a sum of $n$ (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. We provide optimized decompositions up to order $t^6$. The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For $n=2$ terms, several of the optima we find are close to those in McLachlan, SlAM J. Sci. Comput. 16, 151 (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.Comment: 30 pages, 8 figures, 8 tables; added results, figures, and references, extended discussio

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}

An Atlas for the Pinhole Camera

We introduce an atlas of algebro-geometric objects associated with image formation in pinhole cameras. The nodes of the atlas are algebraic varieties or their vanishing ideals related to each other by projection or elimination and restriction or specialization respectively. This atlas offers a unifying framework for the study of problems in 3D computer vision. We initiate the study of the atlas by completely characterizing a part of the atlas stemming from the triangulation problem. We conclude with several open problems and generalizations of the atlas.Comment: 47 pages with references and appendices, final versio

Matroids, Feynman categories, and Koszul duality

We show that various combinatorial invariants of matroids such as Chow rings and Orlik--Solomon algebras may be assembled into "operad-like" structures. Specifically, one obtains several operads over a certain Feynman category which we introduce and study in detail. In addition, we establish a Koszul-type duality between Chow rings and Orlik--Solomon algebras, vastly generalizing a celebrated result of Getzler. This provides a new interpretation of combinatorial Leray models of Orlik--Solomon algebras.Comment: Should be an almost final versio

Moment ideals of local Dirac mixtures

In this paper we study ideals arising from moments of local Dirac measures and their mixtures. We provide generators for the case of first order local Diracs and explain how to obtain the moment ideal of the Pareto distribution from them. We then use elimination theory and Prony's method for parameter estimation of finite mixtures. Our results are showcased with applications in signal processing and statistics. We highlight the natural connections to algebraic statistics, combinatorics and applications in analysis throughout the paper.Comment: 26 pages, 3 figure

A Poly-algorithmic Approach to Quantifier Elimination

Cylindrical Algebraic Decomposition (CAD) was the first practical means for doing real quantifier elimination (QE), and is still a major method, with many improvements since Collins' original method. Nevertheless, its complexity is inherently doubly exponential in the number of variables. Where applicable, virtual term substitution (VTS) is more effective, turning a QE problem in $n$ variables to one in $n-1$ variables in one application, and so on. Hence there is scope for hybrid methods: doing VTS where possible then using CAD. This paper describes such a poly-algorithmic implementation, based on the second author's Ph.D. thesis. The version of CAD used is based on a new implementation of Lazard's recently-justified method, with some improvements to handle equational constraints

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