A survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

Signature Gr\"obner bases in free algebras over rings

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page

The Ideal Membership Problem and Abelian Groups

Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications, for instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP($\Gamma$). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP($\Gamma$) where $\Gamma$ is a Boolean constraint language, while Bulatov and Rafiey [ArXiv'21] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over `affine' constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [ArXiv'21] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $\Gamma$ is an affine constraint language then IMP($\Gamma$) is solvable in polynomial time assuming the input polynomial has bounded degree

Axioms for a theory of signature bases

Twenty years after the discovery of the F5 algorithm, Gr\"obner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gr\"obner bases with signatures with the purpose of uncoupling the theory and the algorithms, and giving general results applicable in many different settings (e.g. Gr\"obner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.)

A survey on signature-based algorithms for computing Gröbner basis computations

International audienceThis paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research

Une approche par l’analyse algébrique effectivedes systèmes linéaires sur des algèbres de Ore

The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra - based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras - and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.Le but de ce papier est de présenter un état de l’art d’une approche par l’analyse algébrique effective de la théorie des systèmes linéaires avec des applications à la théorie du contrôle et à la physique mathématique.En particulier, nous montrons comment la combinaison des méthodes effectives de calcul formel - basées sur lestechniques de bases de Gröbner sur une classe d’algèbres polynomiales noncommutatives d’opérateurs fonctionnels appelée algèbres de Ore - et d’aspects constructifs de théorie des modules et d’algèbre homologique permet lacaractérisation de propriétés structurelles des systèmes linéaires fonctionnels. Des algorithmes sont donnés et uneimplémentation dédiée, appelée OREALGEBRAICANALYSIS, basée sur le package Mathematica HOLONOMIC-FUNCTIONS, est présenté

Bach-flat manifolds and conformally Einstein structures

Einstein manifolds, being critical for the Hilbert-Einstein functional, are central in Riemannian Geometry and Mathematical Physics. A strategy to construct Einstein metrics consists on deforming a given metric by a conformal factor so that the resulting metric is Einstein. In the present Thesis we follow this approach with special emphasis in dimension four. This is the first non-trivial case where the conformally Einstein condition is not tensorial and there are topological obstructions to the existence of Einstein metrics. The conformally Einstein condition is given by a overdetermined PDE-system. Hence the consideration of necessary conditions to be conformally Einstein are of special relevance: the Bach-flat condition is central. In this Thesis we classify four-dimensional homogeneous conformally Einstein manifolds and provide a large family of strictly Bach-flat gradient Ricci solitons. We show the existence of Bach-flat structures given as deformations of Riemannian extensions by means of the Cauchy-Kovalevskaya theorem

Tropical Geometry in Singular

Das Ziel dieser Dissertation ist die Entwicklung und Implementation eines Algorithmus zur Berechnung von tropischen Varietäten über allgemeine bewertete Körper. Die Berechnung von tropischen Varietäten über Körper mit trivialer Bewertung ist ein hinreichend gelöstes Problem. Hierfür kombinieren die Autoren Bogart, Jensen, Speyer, Sturmfels und Thomas eindrucksvoll klassische Techniken der Computeralgebra mit konstruktiven Methoden der konvexer Geometrie. Haben wir allerdings einen Grundkörper mit nicht-trivialer Bewertung, wie zum Beispiel den Körper der $p$-adischen Zahlen $\mathbb{Q}_p$, dann stößt die konventionelle Gröbnerbasentheorie scheinbar an ihre Grenzen. Die zugrundeliegenden Monomordnungen sind nicht geeignet um Problemstellungen zu untersuchen, die von einer nicht-trivialen Bewertung auf den Koeffizienten abhängig sind. Dies führte zu einer Reihe von Arbeiten, welche die gängige Gröbnerbasentheorie modifizieren um die Bewertung des Grundkörpers einzubeziehen.$$\phantom{newline}$$ In dieser Arbeit präsentieren wir einen alternativen Ansatz und zeigen, wie sich die Bewertung mittels einer speziell eingeführten Variable emulieren lässt, so dass eine Modifikation der klassischen Werkzeuge nicht notwendig ist. Im Rahmen dessen wird Theorie der Standardbasen auf Potenzreihen über einen Koeffizientenring verallgemeinert. Hierbei wird besonders Wert darauf gelegt, dass alle Algorithmen bei polynomialen Eingabedaten mit ihren klassischen Pendants übereinstimmen, sodass für praktische Zwecke auf bereits etablierte Softwaresysteme zurückgegriffen werden kann. Darüber hinaus wird die Konstruktion des Gröbnerfächers sowie die Technik des Gröbnerwalks für leicht inhomogene Ideale eingeführt. Dies ist notwendig, da bei der Einführung der neuen Variable die Homogenität des Ausgangsideal gebrochen wird.$$\phantom{newline}$$ Alle Algorithmen wurden in Singular implementiert und sind als Teil der offiziellen Distribution erhältlich. Es ist die erste Implementation, welches in der Lage ist tropische Varietäten mit $p$-adischer Bewertung auszurechnen. Im Rahmen der Arbeit entstand ebenfalls ein Singular Paket für konvexe Geometrie, sowie eine Schnittstelle zu Polymake