arXiv subjects

Arne Kuhrs

Publications and source records attributed to Arne Kuhrs.

6 recordsLinked to original sources

arXiv2026

Layered mixed matrices and reaction networks

The purpose of this work is twofold. In the first part, we consider layered mixed matrices introduced by Murota, relate them to existing notions in combinatorial commutative algebra, and investigate the irreducibility of their determinants. Furthermore, for a layered mixed matrix in combinatorial canonical form, we determine the sparsity structure of its inverse. That is, we characterize which entries of the inverse are nonzero. In the second part, we establish for the first time a formal connection between these algebraic results and the theory of buffering structures for reaction networks developed by Mochizuki and Okada. We identify the lattice of buffering structures with the lattice of order ideals of the block poset of the combinatorial canonical form of the associated layered mixed matrix. This allows us to characterize the reducibility of the symbolic Jacobian determinant as a polynomial in the reaction-rate derivatives, as well as the nonzero sensitivity responses of species concentrations to reaction-rate perturbations.

math.CO↗

arXiv2026

The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces

For a real closed field $K$ with a non-Archimedean absolute value, we introduce the signed Goldman--Iwahori space, the space of homothety classes of signed seminorms on a finite-dimensional vector space over $K$. This space merges two geometric perspectives: it is the linear-algebraic analogue of the real analytification of projective space introduced by Jell, Scheiderer, and Yu, and a signed refinement of the Goldman--Iwahori space of seminorms studied in our previous work. Our first main result identifies it as the inverse limit of all real tropicalizations of projective space, the real analogue of a theorem of Payne. Our second main result gives a matroid-theoretic description in terms of the universal realizable oriented valuated matroid on the underlying vector space. In the constant coefficient case for $K = \mathbb{R}$, signed seminorms are exactly the diagonalizable ones which is a consequence of hyperplane separation that fails for all other real closed fields. In this case, this yields an explicit description of the signed Goldman--Iwahori space in terms of signed flags of subspaces and lets us show that the space coincides with the real Bergman fan of the universal oriented matroid.

math.AG↗

arXiv2026

On Bands and Limit Theorems in Tropical Geometry

We review the basic theory of bands and band schemes introduced by Baker-Jin-Lorscheid, which is an algebraic framework for tropicalization, analytification, and $\mathbb{F}_1$-geometry. For an affine scheme $X$ over a non-Archimedean valued field $k$, one can associate to every affine embedding $ι$ of $X$ a naturally defined affine band scheme $Y_ι$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,ι)$. We prove that $X$ is the limit of the $Y_ι$ in the category of band schemes, thereby obtaining a scheme-theoretic enhancement of Payne's limit theorem. By taking $\mathbb{T}$-rational points, this recovers Payne's theorem for affine tropicalizations from the perspective of band scheme theory and the same method provides an analogous result in the real tropical setting.

math.AG↗

arXiv2026

Scheme theory for commutative semirings

In this survey, we describe two different approaches to constructing affine schemes for commutative semirings: one based on prime ideals, and another based on prime kernels (also called subtractive ideals). We then explain how these two approaches are related through the theory of universal valuations.

math.AG↗

arXiv2025

Tropical reductive groups and principal bundles on metric graphs

We propose an elementary tropical analogue of a reductive group that combines the datum of a Weyl group and the tropicalization of a fixed maximal torus. For the classical groups, as well as $G_2$, these tropical reductive groups admit descriptions as tropical matrix groups that resemble their classical counterparts. Employing this perspective, we introduce tropical principal bundles on metric graphs and study their explicit presentations as pushforwards of line bundles along covers with symmetries and extra data. Our main result identifies the essential skeleton of the moduli space of semistable principal bundles on a Tate curve with its tropical analogue.

math.AG↗

arXiv2023

Buildings, valuated matroids, and tropical linear spaces

Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of $\mathrm{PGL}$ parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings $ι\colon\mathbb{P}^r\hookrightarrow\mathbb{P}^n$ and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.

math.AG↗