Gröbner fan and universal characteristic sets of prime differential ideals

AbstractThe concepts of Gröbner cone, Gröbner fan, and universal Gröbner basis are generalized to the case of characteristic sets of prime differential ideals. It is shown that for each cone there exists a set of polynomials which is characteristic for every ranking from this cone; this set is called a strong characteristic set, and an algorithm for its construction is given. Next, it is shown that the set of all differential Gröbner cones is finite for any differential ideal. A subset of the ideal is called its universal characteristic set, if it contains a characteristic set of the ideal w.r.t. any ranking. It is shown that every prime differential ideal has a finite universal characteristic set, and an algorithm for its construction is given. The question of minimality of this set is addressed in an example. The example also suggests that construction of a universal characteristic set can help in solving a system of nonlinear PDE’s, as well as maybe providing a means for more efficient parallel computation of characteristic sets

Tropical Positivity and Semialgebraic Sets from Polytopes

This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes. Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part? In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part. Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4. In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane. Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope $P_G$ of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of $P_G$, and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction 1. Background 2. Tropical Positivity and Determinantal Varieties 3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes 4. Combinatorics of Correlated Equilibri

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

On some local cohomology spectral sequences

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The 1st type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain by applying a family of functors to a single module. For the 2nd type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their 2nd page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules of Stanley–Reisner rings given by Hochster.Peer ReviewedPostprint (author's final draft

On some local cohomology spectral sequences

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set.The 1st type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain by applying a family of functors to a single module. For the 2nd type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their 2nd page. As a consequence we obtain some decomposition theorems that greatly generalize the wellknown decomposition formula for local cohomology modules of Stanley-Reisner rings given by Hochster

Enhanced Koszulity in Galois cohomology

Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois cohomology is a Koszul algebra. In particular, we show that for all known cases where the maximal pro-$p$-quotient of the absolute Galois group is finitely generated, Galois cohomology is universally Koszul. Assuming the Elementary Type conjecture, this gives us infinitely many refinements of the Bloch-Kato Conjecture. We moreover obtain several unconditional results. Lastly, we show that all forms of enhanced Koszulity are preserved under certain natural operations, which generalizes results that were only known to hold in the commutative case

Cohomologies of coherent sheaves and massless spectra in F-theory

In this PhD thesis we investigate the significance of Chow groups for zero mode counting and anomaly cancellation in F-theory vacua. The major part of this thesis focuses on zero mode counting. We explain that elements of Chow group describe a subset of gauge backgrounds and give rise to a line bundle on each matter curve. The sheaf cohomologies of these line bundles are found to encode the chiral and anti-chiral localised zero modes in this compactification. Therefore, it is of prime interest to compute these sheaf cohomologies. Unfortunately, the line bundles in question are in general non-pullback line bundles. In particular, this is the case for the hypercharge flux employed in F-theory models of grand unified theories (GUTs). Consequently, existing methods, such as the cohomCalg-algorithm, cannot be applied. In collaboration with the mathematician Mohamed Barakat, we have therefore implemented algorithms which determine the sheaf cohomologies of all coherent sheaves on toric varieties. These algorithms are provided by the gap-package SheafCohomologiesOnToricVarieties which extends the homalg_project of Mohamed Barakat. We exemplify these algorithms in explicit (toy-)models of F-theory GUTs. As a spin-off of this analysis, we proved that in an entire class of F-theory vacua, the matter surface fluxes satisfy a number of relations in the Chow ring, which we related to anomaly cancellation. Based on this evidence we conjecture that the well-known anomaly cancellation conditions in F-theory -- typically phrased as intersections in the cohomology ring -- can be extended even to relations in the Chow ring

Finiteness of leading monomial ideals and critical cones of characteristic varieties

Diese Dissertation besteht aus zwei Teilen. Im ersten Teil geben wir einen in sich abgeschlossenen und einheitlichen Ansatz zu einigen Endlichkeitsergebnissen über Leitmonomideale von Idealen im Polynomring bezüglich verschiedener Typen von totalen Monomordnungen. Die Ergegnisse in diesem Teil sind weitgehend nicht neu und können in den Arbeiten anderer Autoren gefunden werden, entweder basierend auf verschiedenen Ansätzen oder angewandt auf verschiedene Kontexte. Wir verallgemeinern einen Teil dieser Resultate auf Vektorräume, die zum Polynomring isomorph sind, einen Teil auf die grosse Klasse der zulässigen Algebren, welche zumindest die Klasse der Algebren von auflösbarem Typ umfasst. In der Literatur werden Leitmonomideale meistens nur bezüglich Monoidordnungen von Nt0 mit t ∈ N studiert, weil diese Ordnungen eine ergebnisreiche Divisionstheorie induzieren. In diesem Rahmen stellt der Macaulay’sche Basissatz den Schlüssel zu den Endlichkeitsresultaten für Leitmonomideale dar. Wir betrachten Leitmonomideale bezüglich Totalordnungen, Gradordnungen, Halbgruppenordnungen, Monoidordnungen, und gradverträglicher Monoidordnungen. Es stellt sich heraus, dass ein Ideal im Polynomring höchstens endlich viele bezüglich der Inklusion minimale Leitmonomideale besitzt, die aus Totalordnungen stammen. Weiter besitzt ein Ideal höchstens endlich viele minimale Leitmonomideale bezüglich Gradordnungen. Durch Monoidordnungen induzierte Leitmonomideale sind wegen einer hier bewiesenen leicht verallgemeinerten Version des Macaulay’schen Basissatzes minimal, und es folgt so, dass es nur endlich viele Leitmonomideale bezüglich Monoidordnungen zu einem gegebenen Ideal gibt. Anfangs hatten wir geplant, die Existenz von universellen Gröbnerbasen in zulässigen Algebren mithilfe der erwähnten Endlichkeitsresultate durch Nachahmung des klassischen Beweises im Polynomring zu zeigen. Das war unsere ursprüngliche Motivation, diese Endlichkeitseigenschaften zu untersuchen. In der Tat folgt aber die Existenz universeller Gröbnerbasen schon aus der Tatsache, dass die Totalordnungen auf einer gegebenen Menge einen kompakten topologischen Raum bilden und die zulässigen Algebren noethersch sind. Mit diesem Thema beenden wir den ersten Teil der vorliegenden Arbeit. Der zweite und innovative Teil dieser Dissertation stellt den Inhalt unseres Artikels [13] dar, welcher im Dezember 2010 zur Publikation in den Transactions of the American Mathematical Society angenommen worden ist. Hier widmen wir uns den charakteristischen Varietäten von Moduln über Weylalgebren. Diese affinen Variet¨aten werden mit gewichteten Gradfiltrierungen eines endlich erzeugten Moduls über einer Weilalgebra konstruiert. Zun¨achst erinnern wir also einige Tatsachen über filtrierte Moduln und deren assoziierte graduierte Moduln. Für filtrierte Moduln über filtrierten kommutativen Ringen zeigen wir, dass der Annullator des assoziierten graduierten Moduls radikalgleich ist zum assoziierten graduierten Ideal des filtrierten Annullators. Ein klassischer Satz von Bernstein besagt, dass die einem gegebenen Modul zugehörigen charakteristischen Varietäten nach dem Grad und nach der Ordnung die gleiche Krulldimension haben. In der Tat haben alle charakteristischen Varietäten eines Moduls die gleiche Krulldimension. Dies wird üblicherweise durch homologische Methoden gezeigt. Wir betten den erwähnten Dimensionssatz in den grösseren Zusammenhang einer Deformationstheorie von gewichteten Gradfiltrierungen und Monomordnungen ein. Unser deformationstheoretischer Ansatz wendet universelle Gröbnerbasen an, und die erwähnte Dimensionsgleichheit folgt als Korollar aus einem tieferen Resultat. Charakteristische Varietäten zeigen nämlich ein bemerkenswertes Verhalten, wenn man ihre definierenden Filtrierungen durch gewisse Adjustierungen der Gewichtung deformiert. Genauer wird eine charakteristische Varietät durch solche Deformationen in ihren eigenen kritischen Kegel übergeführt. Dies erlaubt, eine nichtendliche Filtrierung so zu deformieren, dass die entstehende Filtrierung endlich wird und die zu ihr assoziierte charakteristische Variet¨at gerade der kritische Kegel der ursprünglichen Varietät ist. Daraus folgt die Dimensionsgleichheit. Ein Grund hierfür ist, dass eine affine Variet¨at die gleiche Krulldimension wie ihr kritischer Kegel hat. Ein weiterer Grund ist, dass die Krull- und die GK-Dimension eines endlich erzeugten Moduls über einer endlich erzeugten kommutativen K-Algebra übereinstimmen. Ein dritter Grund ist, dass die GK-Dimension eines endlich filtrierten Moduls beim Übergang zum assoziierten graduierten Modul erhalten bleibt. Unser Resultat stellt auch einen ersten Schritt zur Klassifikation der charakteristischen Varietäten dar. Wir waren aber nicht in der Lage, eine solche Klassifikation in voller Allgemeinheit durchzuführen. Wir haben uns deshalb auf charakteristische Varietäten von zyklischen Moduln über der ersten Weylalgebra beschränkt und eine approximierte Klassifikation durch ein Computerexperiment berechnet. Das Experiment zeigt, dass der Gewichtsraum N20 r{(0, 0)} der Gradfiltrierungen in halbkegelförmige Gebiete unterteilt werden kann, welche jeweils zur selben charakteristische Varietät führen. Auf Grund dieses Experiments können wir auch eine obere Schranke für die Anzahl dieser charakteristischen Varietäten in Termen von Totalgraden der Elemente einer universellen Gröbnerbasis vermuten. Im Hinblick auf eine Arbeit von Aschenbrenner und Leykin [2] kann diese obere Schranke auch in Termen von Totalgraden von Erzeugern des Ideals angegeben werden, das den gegebenen zyklichen Modul definiert. Wir beenden den zweiten Teil mit einem Resultat von Skoda über die Lokalisierung von filtrierten Moduln. Mithilfe eines leichten Lemmas können wir Skodas Ergebnis eine geometrische Interpretation in unserem Kontext geben. Im ersten Anhang geben wir einen direkteren Beweis der Existenz von universellen Gröbnerbasen in Weylalgebren basierend auf den Divisionseigenschaften dieser Algebren und auf der Kompaktheit des topologischen Raums der Monoidordnungen. Im zweiten Anhang listen wir das Computerprogramm auf, das wir für das erw¨ahnte Experiment geschrieben haben