arXiv subjects

Lucas Gierczak

Publications and source records attributed to Lucas Gierczak.

5 recordsLinked to original sources

arXiv2026

Tropical Weierstrass points and Weierstrass weights

In this paper, we study tropical Weierstrass points. These are the analogues for tropical curves of ramification points of line bundles on algebraic curves. For a divisor on a tropical curve, we associate intrinsic weights to the connected components of the locus of tropical Weierstrass points. These are obtained by analyzing the slopes of rational functions in the complete linear series of the divisor. We prove that for a divisor $D$ of degree $d$ and rank $r$ on a genus $g$ tropical curve, the sum of weights is equal to $d - r + rg$. We establish analogous statements for tropical linear series. In the case $D$ comes from the tropicalization of a divisor, these weights control the number of Weierstrass points that are tropicalized to each component. Our results provide answers to open questions originating from the work of Baker on specialization of divisors from curves to graphs. We conclude with multiple examples that illustrate interesting features appearing in the study of tropical Weierstrass points, and raise several open questions.

math.AG↗

arXiv2026

Equality of tropical rank and dimension for semimodules of tropical rational functions, and computational aspects

The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has remained elusive due to the challenges of computing it in practice. We establish that the tropical rank is, in fact, precisely equal to the topological dimension of the semimodule, one more than the dimension of the associated linear system of divisors. This implies that the equality of divisorial and tropical ranks in the definition of tropical linear series is equivalent to the pure dimensionality of the corresponding linear system. We then address the question of computing the tropical rank. In particular, we show that checking whether a given family of tropical rational functions is tropically independent is equivalent to solving a turn-based stochastic mean-payoff game, whereas calculating the tropical rank of a finitely generated semimodule of tropical rational functions is NP-hard. We conclude with several complementary results and questions regarding combinatorial and topological properties of the tropical rank.

math.AG↗

arXiv2024

Limit linear series: combinatorial theory

We develop a purely combinatorial theory of limit linear series on metric graphs. This will be based on the formalisms of hypercube rank functions and slope structures. We provide a full classification of combinatorial limit linear series of rank one, and discuss connections to other concepts in tropical algebra and combinatorial algebraic geometry.

math.AG↗

arXiv2024

Combinatorial flag arrangements

We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of arrangements of initial flags in a vector space. We give cryptomorphic axiomatic systems in terms of rank function, flats, circuits, and independent sets, and formulate a duality concept. We also provide precise links between matricubes, permutation arrays and matroids, and raise several open questions.

math.CO↗

arXiv2020

Unsteady Wave Drag on a Disturbance Moving Along an Arbitrary Trajectory

We derive analytical formulas for the wake and wave drag of a disturbance moving arbitrarily at the air-water interface. We show that, provided a constant velocity is reached in finite time, the unsteady surface displacement converges to its well-known steady counterpart as given by Havelock's famous formula. Finally we assess, in a specific situation, to which extent one can rightfully use Havelock's steady wave drag formula for non-uniform motion (quasi-static). Such an approach can be used to legitimize or discredit a number of studies which used steady wave drag formulas in unsteady situations.

physics.flu-dyn↗