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Lars Kastner

Publications and source records attributed to Lars Kastner.

At least 19 recordsLinked to original sources
arXiv2026

Fast Isotopy Computation for T-Curves

A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot \Delta_2$ together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves.

math.AG↗
arXiv2026

Limits of combinatorial patchworking

It is shown that there are real plane algebraic curves of degree eight that cannot be realized as T-curves, i.e., via combinatorial patchworking. In fact, this holds for several real schemes (i.e., ambient isotopy types) with the maximal number of real components, called $M$-curves. On the other hand, each nonempty real scheme of lower degree, maximal or not, arises as a T-curve. By constructing one patchwork of the dilated triangle $d\cdot\Delta_2$ for each nonempty real scheme of degree $d\leq 7$, we provide an explicit method for constructing polynomials realizing these real schemes. This resolves a question of Itenberg and Viro (1996).

math.AG↗
arXiv2025

Tropical elliptic curves in 3-space

We classify trivalent graphs with 16 vertices and 16 edges that arise from intersecting two quadratic surfaces in tropical 3-space. There are 4,009 such graphs, representing maximally degenerate stable models of elliptic curves realized as tropical complete intersections of two quadrics. Our classification is derived from 405,246,030 regular unimodular triangulations of the 4-dimensional Cayley polytope.

math.CO↗
arXiv2024

Regular Flips in mptopcom

A triangulation of a point configuration is regular if it can be given by a height function, that is every point gets lifted to a certain height and projecting the lower convex hull gives the triangulation. Checking regularity of a triangulation usually is done by solving a linear program. However when checking many flip-connected triangulations for regularity, one can instead ask which flips preserve regularity. When traversing the flip graph for enumerating all regular triangulations, this allows for vast reduction of the linear programs needing to be solved. At the same time the remaining linear programs will be much smaller.

math.CO↗
arXiv2024

Confirmable Workflows in OSCAR

We discuss what is special about the reproducibility of workflows in computer algebra. It is emphasized how the programming language Julia and the new computer algebra system OSCAR support such a reproducibility, and how users can benefit for their own work.

cs.MS↗
arXiv2024

Subdivisions of Hypersimplices: with a View Toward Finite Metric Spaces

The secondary fan $\Sigma(k,n)$ is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices $\Delta(k,n)$. We find new infinite families of rays of $\Sigma(k,n)$, and we compute the fans $\Sigma(2,7)$ and $\Sigma(3,6)$. In the special case $k=2$ the fan $\Sigma(2,n)$ is closely related to the metric fan $\mathop{MF}(n)$, which forms a natural parameter space for the metric spaces on $n$ points. So our results yield a classification of the finite metric spaces on seven points.

math.CO↗
arXiv2023

Toric Geometry in OSCAR

We report on the computer implementation for toric geometry in the computer algebra system $\texttt{OSCAR}$. The main architectural feature of $\texttt{OSCAR}$ is that its four fundamental tools $\texttt{Antic}$ (Hecke, Nemo), $\texttt{GAP}$, $\texttt{Polymake}$ and $\texttt{Singular}$ are $\mathit{integral~components}$, rather than external software. Toric geometry benefits greatly from this architecture. $\texttt{Julia}$ is a high-performance programming language designed for numerical and scientific computing. The growing ecosystem of $\texttt{Julia}$ packages ensures its continued viability for scientific computing and data analysis. Indeed, $\texttt{OSCAR}$ is written in $\texttt{Julia}$. This implies that the performance of $\texttt{OSCAR}$ should be comparable or even better than many other implementations.

math.AG↗
arXiv2022

Research-Data Management Planning in the German Mathematical Community

In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle.

math.HO↗
arXiv2020

Tropical compactification via Ganter's algorithm

We describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case, the procedure is analogous to the tropical compactifications of subvarieties of tori. We give an analysis of the combinatorial structure of the compactification and show that its Hasse diagram can be computed via Ganter's algorithm. Our algorithm is implemented in and shipped with polymake.

math.AG↗
arXiv2020

Hyperplane arrangements in polymake

Hyperplane arrangements form the latest addition to the zoo of combinatorial objects dealt with by polymake. We report on their implementation and on a algorithm to compute the associated cell decomposition. The implemented algorithm performs significantly better than brute force alternatives, as it requires less convex hulls computations.

math.CO↗
arXiv2019

The Newton polytope of the discriminant of a quaternary cubic form

We determine the $166\,104$ extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with $D$-equivalence classes of regular triangulations of the $3$-dilated tetrahedron. We describe how to compute these triangulations and their $D$-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer amount of triangulations effectively, as well as devising a suitably fast algorithm for computation of a $D$-equivalence class.

math.CO↗
arXiv2019

Random growth on a Ramanujan graph

The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which asymptotically maximize the spectral gap. Further, we consider Erd\H{o}s--R\'enyi random graphs and compare our theoretical results with computational experiments on flip graphs of point configurations. The latter is relevant for enumerating triangulations.

math.CO↗
arXiv2018

Immaculate line bundles on toric varieties

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.

math.AG↗
arXiv2017

Parallel Enumeration of Triangulations

We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before.

math.CO↗
arXiv2016

Cellular sheaf cohomology in Polymake

This chapter provides a guide to our polymake extension cellularSheaves. We first define cellular sheaves on polyhedral complexes in Euclidean space, as well as cosheaves, and their (co)homologies. As motivation, we summarise some results from toric and tropical geometry linking cellular sheaf cohomologies to cohomologies of algebraic varieties. We then give an overview of the structure of the extension cellularSheaves for polymake. Finally, we illustrate the usage of the extension with examples from toric and tropical geometry.

math.AG↗
arXiv2016

Ext and Tor on two-dimensional cyclic quotient singularities

Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain combinatorial objects using methods from toric geometry. In particular, it is enough to give a combinatorial description of the $\mathop{Ext}\nolimits^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher $\mathop{Ext}\nolimits^i$-groups are then reduced to the case of $\mathop{Ext}\nolimits^1$ via a quiver. We use this description to show that $\mathop{Ext}\nolimits^1_{X}(\mathcal{O}(D),\mathcal{O}(K-D')) = \mathop{Ext}\nolimits^1_{X}(\mathcal{O}(D'),\mathcal{O}(K-D))$, where $K$ denotes the canonical divisor on $X$. Furthermore, we show that $\mathop{Ext}\nolimits^{i+2}_{X}(\mathcal{O}(D),\mathcal{O}(D'))$ is the Matlis dual of $\mathop{Tor}\nolimits_{i}^{X}(\mathcal{O}(D),\mathcal{O}(D'))$.

math.AG↗