Each graph gives rise to a graphic matroid, which in turn gives rise to a Bergman fan. We refer to stars of the latter as a tropical galaxy, and propose them an objects of study in hopes of obtaining better lower bounds for its realization number. The sharp lower bound is conjectured to be 2^(n-3), where n is the number of vertices. The best lower bound we currently have is 2. This project was part of an undergraduate summer research project of the first three authors.

In the same spirit as [HMRT19], this paper investigates the tropical geometry of space sextics and their tritangents with a particular emphasis on their (total) reality. We show that tropical tritangents naturally come in groups of eight, which all share the same reality. In particular, we use our tropical methods to construct space sextics with 64 and 120 totally real tritangents, the former being only the second example known for this count.

Vertically parametrized polynomial systems are a particularly nice class of parametrized polynomial equations. They are also featured in [HR25a] and [HHR24], and they encompass steady state equations of chemical reaction networks. In this paper, we study the task of embedding a polynomial system into a vertically parametrized family, which touches upon difficult alignment problems in complexity theory. We propose a heuristic algorithm, present a new OSCAR interface to ODEbase, and show that our heuristic algorithm performs exceptionally on it. This project was part of an undergraduate summer project of the third author.

In [HR18] it is shown that computing tropicalizations of polynomial ideals boils down to two tasks: the Groebner walk and zero-dimensional tropicalizations. This paper contains an improved algorithm and OSCAR implementation for the latter. This project was part of an undergraduate summer research project of the first author.

In [HR25a] we described how generic root counts can be expressed as tropical intersection numbers. This paper expands on that technique and explains how the tropical intersection points give rise to optimal homotopies. If said intersection involves tropical hypersurfaces, we recover the seminal result by Huber and Sturmfels on polyhedral homotopies.

Take your favourite graph. Think of its edges as bars and its vertices as joints. We say the graph is rigid or flexible, if the resulting structure is. For example, a rectangle in 2D is flexible, but a rectangle plus a single diagonal in 2D is rigid. The latter is also known as a minimally rigid graph. These minimally rigid graphs are the simplest graphs that only allow for finitely many embeddings with fixed edge length, and that number of embeddings is their realisation number. This paper uses the theory of [HR25a] and lays the foundation for studying realisation numbers using tropical geometry. We uncover uncovering interesting relations between the realisation number and the Tutte polynomial.

One of the biggest challenges in solving a polynomial system over the complex numbers fast is determining how many solution it has in the first place. More generally, one may ask how many solutions a parametrized polynomial system generically has. And while this seems to be a strictly harder question, if the parameters are distributed in the right way, it can make the problem much simpler using tropical geometry. This paper lays some necessary foundations in expression generic root counts as tropical intersection numbers, and discusses examples where this approach is particularly fruitful.

Coupled oscillators are more than meets the eye. They provide the foundational framework for understanding how complex systems interact, synchronise, and exchange energy. This paper contains a first introduction to tropical geometry in OSCAR and puts the theory of [HR25a] into practice for tropically transverse systems, a class of parametrized polynomial systems which contains many types of coupled oscillators.

This project is dedicated to the computation of tropical varieties using massive parallelization. It also contains a first algorithm for testing whether a maximal cell of the tropicalization belongs to its positive part. We combine both to verify a conjecture by Speyer and Williams on the positive part of the tropical Grassmannian TGr(3,7).

When comparing intersections and stable intersections of balanced polyhedral complexes, it only seems natural that every connected component of the former contains a point of the latter. This paper proves that fact in the case where the balanced polyhedral complexes are tropicalizations of polynomial ideals relying on some sophisticated techniques in algebraic geometry by Josephine Yu. The question remains open for general balanced polyhedral complexes.

This short magazine article explains how to create 3d printable tropical models. If you are interested in seeing the models in person, you can find them in my office.

In [HR18] it is shown that computing tropicalizations of polynomial ideals down to the Groebner walk and computing zero-dimensional tropicalizations. In this paper, we prove that the former dominates the latter in terms of complexity. This is done by developing a new algorithm for zerodimensional tropicalizations, which is not as performant as the one in [HR18], but which is more amenable to complexity analysis.

A maxout neural network is a generalisation of ReLU networks where the individual neurons are a maximum of affine linear functions, often used in voice recognition. In this paper, we derive sharp upper bounds on their complexity by generalising Zavlasky's work on hyperplane arrangements to tropical hypersurfaces and combining them with Weibel's approach of relating large arrangements to their smaller subarrangements.

Tropical bases play an important role in tropical geometry, both in its theory but also in its applications as seen in the seminal works by Hampton and Moeckel on the finiteness of central arrangements in the four body problem. This paper details the construction of certificates for the failure of polynomial systems to be tropical bases. We demonstrate our techniques to answer some open problems on Del Pezzo surfaces of degree 3 and valuated gaussoids on 4 elements.

Constructing well-behaved quotients is not straightforward in algebraic geometry, and one of the best methods are so-called GIT quotients. Their constructing is not unique however and their variation is described by so-called GIT fans. In this paper, we discuss the effect of residual symmetries on the GIT fan and showcase it on the GIT fan of the moduli space of stable rational curves with six marked points.

Based on the work in [RMT19], we propose a numerical value measuring cooperativity in the form of a minimal energy. We propose methods for computing it using polynomial optimisation and compare it to the maximal slope of the Hill plot.

In this paper, we explain how to compute tropical varieties over general fields with valuations, generalising a seminal work by Bogart, Jensen, Speyer, Sturmfels, and Thomas. The main idea is that tropical Groebner bases over fields with valuation can be computed via standard bases over the valuation ring. This work relies on the mathematical foundations laid out in [MRO17] and [MR17].

Following [KRSS18], which revolved around the computation of tritangents to space sextics, this paper provides an algorithm for reconstructing the space sextics from its tritangents, which is known to be possible by works of Caporaso and Sernesi. All algorithms are implemented in Magma.

In many surprising ways tropical geometry is a facsimile of algebraic geometry, and this paper is one such ways. We show that in tropical geometry abstract tropical curves of genus 3 can be embedded inside a tropical plane if and only if said curve is not realizably hyperelliptic.

This is a short technical report on the Singular library gfan.lib.

Ligands and ligand binding are an important topic in biochemistry with applications in cell signaling, enzyme regulation, immune recognition, sensory perception, and drug discovery. In this paper, we use tools from algebraic geometry to study a common ligand binding model arising from the grand canonical ensemble of statistical mechanics. Of particular interest is the elusive emerging property of cooperativity, which is of utmost importance in practise.

Tropicalizations of polynomial ideals are computed via a traversal on the Groebner complex. Hence it requires the computation of a single tropical point to start the traversal and repeated computations of tropical links to continue the traversal, the latter being a significant bottleneck of existing implementations at the time of writing. This paper proposes a fast algorithm for the former and also explains how it can be used for the latter, speeding up the computation by orders of magnitude.

This paper uses tools from computational and numerical algebraic geometry to construct a space sextics with 120 tritangents, answering a question by Emch in 1928.

This is a brief technical report on the Singular library polymake.lib.

This paper is one of two papers that lays the necessary mathematical foundations for computing tropical varieties over valued fields.

This paper is a whitepaper on challenges in the development of open source computer algebra systems.

This paper recalls Mora's computational treatment of localizations of coordinate rings at polynomial ideals and discusses its implementation in the Singular library graal.lib.

We translate the foundational work by the first author on Weyl groupoids to the language of toric geometry, obtaining a classification of smooth projective toric varieties which arise from arrangements of hyperplanes. We show that they can be written as products of varieties arising from triangulations of n-gons, reflection arrangements of type A, B, C, and D, or one of 74 sporadic varieties.

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