arXiv subjects
Publications and source records attributed to Stefano Mereta.
We formulate several open questions regarding the tropicalization of linear ODEs, aiming primarily to develop methods for calculating the radius of convergence of their classical solutions. To this aim it is of foremost importance to characterize the classes of equations that admit a finite differential tropical basis, as introduced in (Fink and Toghani, 2022). Our initial exploration examines the second- and third-order cases.
We study the semi-algebraicity of the neuromanifold $\mathcal{M}_\mathbf{d}$ of a feedforward ReLU neural network and its symmetries. We prove that $\mathcal{M}_\mathbf{d}$ is not a semi-algebraic quotient of the space of weights of the network. We introduce and study the notion of \emph{honest} open subset of the space of weights, where the network does not show any hidden symmetries. Finally, we conjecture that the maximal honest open is always semi-algebraic and prove that in the shallow case it is even Zariski.
This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical polynomials, ideals, congruences, and how the connection with tropical geometry is made via congruences of bend relations. Tropical geometry and matroid theory are telling us that we should focus attention on a narrow slice of the world of tropical algebra, and this leads to the theory of tropical ideals (as developed by Maclagan and Rinc\'{o}n) and an abundance of interesting open questions. In the second lecture we examine the relationship between Berkovich analytification and tropicalization from the perspective of bend relations, giving a refinement of Payne's influential limit theorem. In the third lecture we set aside geometry and focus on tropicalization via bend relations as a construction in commutative and non-commutative algebra. Constructions such as symmetric algebras, exterior algebras, matrix algebras, and Clifford algebras can be tropicalized. In the case of exterior algebras, the resulting tropical notion beautifully completes the picture of the Pl\"{u}cker embedding and gives a new perspective on the tropical Pl\"{u}cker relations. For matrix algebras and Clifford algebras, Morita theory becomes an interesting topic.
In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and discuss ideas beyond the algebraic domain. This work lays the foundations of a research direction bridging algebraic geometry and deep learning, that we refer to as neuroalgebraic geometry.
The fundamental theorem of tropical differential algebra has been established for formal power series solutions of systems of algebraic differential equations. It has been shown that the direct extension to formal Puiseux series solutions fails. In this paper, we overcome this issue by transforming the given differential system and such a generalization is presented. Moreover, we explain why such transformations do not work for generalizing the fundamental theorem to transseries solutions, but show that one inclusion still holds for this case even without using any transformation.
We introduce higher-dimensional cubical sliding puzzles that are inspired by the classical 15 Puzzle from the 1880s. In our puzzles, on a $d$-dimensional cube, a labeled token can be slid from one vertex to another if it is topologically free to move on lower-dimensional faces. We analyze the solvability of these puzzles by studying how the puzzle graph changes with the number of labeled tokens vs empty vertices. We give characterizations of the different regimes ranging from being completely stuck (and thus all puzzles unsolvable) to having only one giant component where almost all puzzles can be solved. For the Cube, the Tesseract, and the Penteract ($5$-dimensional cube) we have implemented an algorithm to completely analyze their solvability and we provide specific puzzles for which we know the minimum number of moves needed to solve them.
We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev's framework for tropical differential equations. We then prove a differential analogue of Payne's inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.
We prove a fundamental theorem for tropical partial differential equations, analogous to the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani for the case of trivial valuation as introduced by Grigoriev to differential equations with power series coefficients over any valued field. Crucial ingredients are the framework for tropical partial differential equations introduced by Giansiracusa and Mereta and a result on infinite intersections of projections of fibers of tropicalizations, which we prove using Hrushovski and Loeser's model-theoretic interpretation of Berkovich analytification. As a corollary of the fundamental theorem, we show that the radius of convergence of solutions of an ordinary differential equation over a nontrivially valued field can be computed tropically.
The notion of a pre-truss, that is, a set that is both a heap and a semigroup is introduced. Pre-trusses themselves as well as pre-trusses in which one-sided or two-sided distributive laws hold are studied. These are termed near-trusses and skew trusses respectively. Congruences in pre-trusses are shown to correspond to paragons defined here as sub-heaps satisfying particular closure property. Near-trusses corresponding to skew braces and near-rings are identified through their paragon and ideal structures. Regular elements in a pre-truss are defined leading to the notion of a (pre-truss) domain. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses as domains that satisfy the Ore condition are introduced and the pre-trusses of fractions are defined. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces.