arXiv subjects

Felipe Rincón

Publications and source records attributed to Felipe Rincón.

At least 19 recordsLinked to original sources

arXiv2026

The rank-nullity ring of a matroid

We introduce the rank-nullity ring of a matroid $M$, which is a subring of the Chow ring of the permutahedral toric variety. This subring contains the tautological Chern classes of $M$, a fact we deduce from a highly symmetric formula for these classes. When the matroid $M$ is a uniform matroid, the rank-nullity ring coincides with the subring of $S_n$-invariants of the Chow ring of the permutahedral toric variety. In this case, we compute its Hilbert function explicitly and provide a Gröbner basis for the ideal of relations among its generators.

math.CO↗

arXiv2025

The Ehrhart polynomial of a matroid specializes to the beta invariant

We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $β$-invariant of the matroid. This yields a lattice-point counting formula for the $β$-invariant and establishes a new and unexpected positivity property of Ehrhart polynomials of matroid polytopes.

math.CO↗

arXiv2025

Accelerating wave simulations with neural dispersion correctors

We present a Fourier neural operator network, designed to correct dispersion errors in numerical wave simulations. The neural dispersion corrector enables the replacement of a computationally expensive high-accuracy simulation by a less expensive low-accuracy simulation. In contrast to neural network surrogates that fully replace a wave equation, the neural dispersion corrector has only a weak dependence on the distribution of model parameters, such as wave speeds. Consequently, the network can be trained with a significantly smaller dataset, while still generalising to unseen input parameters. Following a description of the network architecture and training, we provide examples for the 3-D elastic wave equation. After training with merely 1$\,$000 examples on one GPU, the neural corrector achieves a speed-up of 16$\times$ compared to a reference spectral-element simulation and a generalisation to a broad range of strongly heterogeneous wave speed distributions.

physics.geo-ph↗

arXiv2025

Moments, Sums of Squares, and Tropicalization

We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$, while "pseudo-moments" are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudo-moment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudo-moment cones.

math.AG↗

arXiv2025

Tropicalizing Principal Minors of Positive Definite Matrices

We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of $M$-concave functions on the discrete $n$-dimensional cube. We show it coincides with the intersection of the affine tropical flag variety with the submodular cone. In particular, any cell in the regular subdivision of the cube induced by a point in this tropicalization can be subdivided into base polytopes of realizable matroids. We use this tropicalization as a guide to discover new algebraic inequalities among the principal minors of positive semidefinite matrices of a fixed size. We also extend our results to positive semidefinite matrices via taking closures in the tropical semifield $\mathbb{R}\cup\{-\infty\}$.

math.CO↗

arXiv2024

Log-concavity for independent sets of valuated matroids

Recently, several proofs of the Mason--Welsh conjecture for matroids have been found, which asserts the log-concavity of the sequence that counts independent sets of a given size. In this article we use the theory of Lorentzian polynomials, developed by Brändén and Huh, to prove a generalization of the Mason-Welsh conjecture to the context of valuated matroids. In fact, we provide a log-concavity result in the more general setting of valuated discrete polymatroids, or equivalently, M-convex functions. Our approach is via the construction of a generic extension of a valuated matroid or M-convex function, so that the bases of the extension are related to the independent sets of the original matroid. We also provide a similar log-concavity result for valuated bimatroids, which, we believe, might be of independent interest.

math.CO↗

arXiv2023

Chern Classes of Tropical Manifolds

We extend the definitions of Chern-Schwartz-MacPherson (CSM) cycles of matroids to tropical manifolds. To do this, we provide an alternate description of CSM cycles of matroids which is invariant under integer affine transformations. Utilising results of Esterov and Katz-Stapledon, we prove correspondence theorems for the CSM classes of tropicalisations of subvarieties of toric varieties. We also provide an adjunction formula relating the CSM cycles of a tropical manifold and a codimension-one tropical submanifold. Lastly, we establish Noether's Formula for compact tropical surfaces with a Delzant face structure. This extends the class of surfaces for which the formula had been previously proved by the third author.

math.CO↗

arXiv2021

Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization

We study the moduli space of $d$-dimensional linear subspaces contained in a fixed $(d+1)$-dimensional linear variety $X$, and its tropicalization. We prove that these moduli spaces are linear subspaces themselves, and thus their tropicalization is completely determined by their associated (valuated) matroids. We show that these matroids can be interpreted as the matroid of lines of the hyperplane arrangement corresponding to $X$, and generically are equal to a Dilworth truncation of the free matroid. In this way, we can describe combinatorially tropicalized Fano schemes and tropicalizations of moduli spaces of stable maps of degree $1$ to a plane.

math.AG↗

arXiv2021

Tropical ideals do not realise all Bergman fans

Every tropical ideal in the sense of Maclagan-Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements, and in which all maximal cones have weight one.

math.CO↗

arXiv2021

Positively Hyperbolic Varieties, Tropicalization, and Positroids

A variety of codimension $c$ in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension $c$. Positively hyperbolic hypersurfaces are defined by stable polynomials. We give a new characterization of positively hyperbolic varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. We prove that positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type $A$ hyperplane arrangement defined by $x_i = x_j$, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of some results of Choe--Oxley--Sokal--Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we also give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.

math.CO↗

arXiv2021

Paving Tropical Ideals

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We show that paving tropical ideals of degree d+1 are in bijection with $\mathbb Z^n$-invariant d-partitions of $\mathbb Z^n$. This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with $\mathbb Z^n$-invariant 2-partitions of quotient groups of the form $\mathbb Z^n/L$. We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals.

math.CO↗

arXiv2020

Varieties of Tropical Ideals are Balanced

Tropical ideals, introduced in arXiv:1609.03838, define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of varieties) works for tropical ideals as in the classical case. The matroid condition that defines tropical ideals is crucial for these results.

math.AG↗

arXiv2017

Tropical schemes, tropical cycles, and valuated matroids

We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure.

math.AG↗

arXiv2017

Tropical Ideals

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing [Giansiracusa-Giansiracusa]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

math.AG↗

arXiv2016

Compressed sensing of data with a known distribution

Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted $\ell_1$-norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.

math.OC↗

arXiv2015

Stiefel tropical linear spaces

The tropical Stiefel map associates to a tropical matrix A its tropical Pluecker vector of maximal minors, and thus a tropical linear space L(A). We call the L(A)s obtained in this way Stiefel tropical linear spaces. We prove that they are dual to certain matroid subdivisions of polytopes of transversal matroids, and we relate their combinatorics to a canonically associated tropical hyperplane arrangement. We also explore a broad connection with the secondary fan of the Newton polytope of the product of all maximal minors of a matrix. In addition, we investigate the natural parametrization of L(A) arising from the tropical linear map defined by A.

math.CO↗

arXiv2013

Positively oriented matroids are realizable

We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.

math.CO↗

arXiv2013

Positroids and non-crossing partitions

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on [n] is connected equals 1/e^2 asymptotically.

math.CO↗