arXiv subjects

Jonathan Montaño

Publications and source records attributed to Jonathan Montaño.

At least 19 recordsLinked to original sources

arXiv2026

The symbolic and divisorial analytic spreads are finite

Let $R$ be a ring that is essentially of finite type over a field and $I\subseteq R$ be an ideal. In this article it is shown that the minimal number of generators of the symbolic powers $I^{(n)}$ is bounded above by a polynomial in $n$. Furthermore, if $R$ is assumed to be a domain and $\mathcal I=\{I_n\}_{n\ge 0}$ is an $\mathbb{R}$-divisorial filtration, then the minimal number of generators of the ideals $I_n$ is again bounded above by a polynomial in $n$.

math.AC↗

arXiv2026

The multiplicity sequence of monomial ideals

We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.

math.AC↗

arXiv2026

Matroid correspondence

Motivated by algebraic correspondences and linear operators associated with volume and Lorentzian polynomials, we introduce matroid correspondences and their polymatroid analogues. A matroid correspondence defines a functor between poset categories of matroids whose morphisms are matroid quotients, and various standard functors, including deletion, contraction, free extension, truncation, intersection, union, and pullback, arise in this way. We show that these correspondences preserve representability and algebraicity under natural hypotheses. In the polymatroid setting, we establish compatibility with multisymmetric lifts. Finally, we relate this construction to the supports of linear operators with Lorentzian symbols.

math.CO↗

arXiv2026

Homology of Vietoris-Rips complexes of hypercube graphs via group actions

The Vietoris-Rips complex of a metric space is the simplicial complex whose faces are the subsets of points with pairwise distance bounded above by a given scale $r$. In this paper, we study Vietoris-Rips complexes on the vertex set of the $n$-dimensional hypercube equipped with the Hamming distance. These complexes are stable under the action of the automorphism group of the hypercube graph, also known as the hyperoctahedral group, which therefore acts on their homology groups. Our results completely describe the decomposition of these homology groups into irreducible representations of the hyperoctahedral group at scales $r\leqslant 3$ and $r=n-1$.

math.CO↗

arXiv2026

Ladder determinantal varieties and their symbolic blowups

In this article we show that the symbolic Rees algebra of a mixed ladder determinantal ideal is strongly $F$-regular. Furthermore, we prove that the symbolic associated graded algebra of a mixed ladder determinantal ideal is $F$-pure. The latter implies that mixed ladder determinantal rings are $F$-pure. We also show that ideals of the poset of minors of a generic matrix give rise to $F$-pure algebras with straightening law.

math.AC↗

arXiv2025

Quasipolynomial behavior via constructibility in multigraded algebra

Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert-Samuel multiplicity, arithmetic degree, and homological degree.

math.AC↗

arXiv2025

Safe Explicable Policy Search

When users work with AI agents, they form conscious or subconscious expectations of them. Meeting user expectations is crucial for such agents to engage in successful interactions and teaming. However, users may form expectations of an agent that differ from the agent's planned behaviors. These differences lead to the consideration of two separate decision models in the planning process to generate explicable behaviors. However, little has been done to incorporate safety considerations, especially in a learning setting. We present Safe Explicable Policy Search (SEPS), which aims to provide a learning approach to explicable behavior generation while minimizing the safety risk, both during and after learning. We formulate SEPS as a constrained optimization problem where the agent aims to maximize an explicability score subject to constraints on safety and a suboptimality criterion based on the agent's model. SEPS innovatively combines the capabilities of Constrained Policy Optimization and Explicable Policy Search to introduce the capability of generating safe explicable behaviors to domains with continuous state and action spaces, which is critical for robotic applications. We evaluate SEPS in safety-gym environments and with a physical robot experiment to show its efficacy and relevance in human-AI teaming.

cs.AI↗

arXiv2025

K-polynomials of multiplicity-free varieties

We describe the twisted $K$-polynomial of multiplicity-free varieties in a multiprojective setting. More precisely, for multiplicity-free varieties, we show that the support of the twisted $K$-polynomial is a generalized polymatroid. As applications, we show that the support of the Möbius function of a linear polymatroid is a generalized polymatroid, and we settle a conjecture of Monical, Tokcan and Yong regarding Grothendieck polynomials for the case of zero-one Schubert polynomials.

math.AG↗

arXiv2025

Degree functions of graded families of ideals

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite.

math.AC↗

arXiv2025

Rational powers, invariant ideals, and the summation formula

We provide explicit descriptions for the rational powers and Rees valuations of several classes of ideals invariant under natural actions of tori and products of general linear groups, in terms of polyhedra and lattice points. This allows us to show that a version of Mustaţă-Takagi's summation formula for multiplier ideals also holds for the rational powers of these ideals. Moreover, for arbitrary ideals in normal domains that are finitely generated over algebraically closed fields, we prove a weaker version of this formula that holds for sufficiently large rational numbers.

math.AC↗

arXiv2025

Multiplicities and degree functions in local rings via intersection products

We prove a theorem on the intersection theory over a Noetherian local ring $R$, which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings $R$, using the theory of rational equivalence developed by Thorup, and the Snapper-Mumford-Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self-contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert-Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over $2$-dimensional excellent local rings in terms of our intersection product.

math.AC↗

arXiv2024

Exterior powers and Tor-persistence

A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of Avramov, et. al. asks whether any such $R$ is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with $\mathfrak{m}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.

math.AC↗

arXiv2024

Blowup algebras of determinantal ideals in prime characteristic

We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results we develop the notion of $F$-split filtrations and symbolic $F$-split ideals.

math.AC↗

arXiv2023

Computing multiplicity sequences

The MultiplicitySequence package for Macaulay2 computes the multiplicity sequence of a graded ideal in a standard graded ring over a field, as well as several invariants of monomial ideals related to integral dependence. We discuss two strategies implemented for computing multiplicity sequences: one via the bivariate Hilbert polynomial, and the other via the technique of general elements.

math.AC↗

arXiv2023

Double Schubert polynomials do have saturated Newton polytopes

We prove that double Schubert polynomials have the Saturated Newton Polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study non-standard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen-Macaulay prime ideal, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

math.AC↗

arXiv2023

Purity of monoids and characteristic-free splittings in semigroup rings

Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent.

math.AC↗

arXiv2023

The core of monomial ideals

The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$ coincides with the largest monomial ideal contained in a general reduction of $I$. We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an explicit formula for the core, and make progress towards this conjecture.

math.AC↗

arXiv2022

Frobenius methods in combinatorics

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for $F$-pure rings that were inspired by work done for Stanley-Reisner rings.

math.AC↗