arXiv subjects

Barbara Betti

Publications and source records attributed to Barbara Betti.

9 recordsLinked to original sources

arXiv2026

Incidence toric ideals and three-point functions

We study the ideal of the algebraic relations among 3-point functions from a combinatorial and topological perspective. We place this problem in the broader setting of incidence toric ideals associated with incidence matrices of t-subsets contained in k-subsets of n elements. Generators of these ideals admit combinatorial interpretations as null t-designs and topological interpretations as balanced orientable normal d-pseudomanifolds without boundary. Generators arising from octahedra play a fundamental role in the structure of these ideals.

math.AC↗

arXiv2026

Join-meet binomial algebras of distributive lattices

We investigate the defining ideal of the algebra over a field generated by the join-meet binomials coming from a finite distributive lattice. In the frame of algebras with straightening laws, the problem when the defining ideal is generated by quadrics is studied.

math.AC↗

arXiv2026

Diagonal F-thresholds for determinants and Pfaffians

We compute the diagonal F-thresholds of determinantal hypersurfaces arising from a generic matrix and from a generic symmetric matrix, as well as of the Pfaffian hypersurface arising from a generic skew-symmetric matrix of even size. The main ingredient is a cohomology vanishing theorem for certain line bundles on flag varieties in characteristic $p$. In the cases of the generic matrix and the generic skew-symmetric matrix, we show that the diagonal F-threshold attains its minimal possible value, namely the negative of the a-invariant. The symmetric case is more subtle and relies in addition on a polynomiality result for representations afforded by cohomology, building on work of the second author with VandeBogert.

math.AC↗

arXiv2025

Graphical Scattering Equations

The CHY scattering equations on the moduli space $M_{0,n}$ play a prominent role at the interface of particle physics and algebraic statistics. We study the scattering correspondence when the Mandelstam invariants are restricted to a fixed graph on $n$ vertices.

math.CO↗

arXiv2025

The SagbiHomotopy.jl package for solving polynomial systems

We present the Julia package SagbiHomotopy.jl for solving systems of polynomial equations using numerical homotopy continuation. The package introduces an optimal choice of a start system based on SAGBI homotopies. For square horizontally parameterized systems, where each equation is a linear combination of a given set of polynomials, SAGBI homotopies significantly reduce the number of solution paths to track compared to polyhedral homotopies currently used by default in most software for numerical homotopy continuation. We illustrate our framework with a variety of examples, including problems arising in chemistry and physics.

math.AG↗

arXiv2024

Proudfoot-Speyer degenerations of scattering equations

We study scattering equations of hyperplane arrangements from the perspective of combinatorial commutative algebra and numerical algebraic geometry. We formulate the problem as linear equations on a reciprocal linear space and develop a degeneration-based homotopy algorithm for solving them. We investigate the Hilbert regularity of the corresponding homogeneous ideal and apply our methods to CHY scattering equations.

math.AG↗

arXiv2024

Mukai lifting of self-dual points in $\mathbb{P}^6$

A set of $2n$ points in $\mathbb{P}^{n-1}$ is self-dual if it is invariant under the Gale transform. Motivated by Mukai's work on canonical curves, Petrakiev showed that a general self-dual set of $14$ points in $\mathbb{P}^6$ arises as the intersection of the Grassmannian ${\rm Gr}(2,6)$ in its Pl\"ucker embedding in $\mathbb{P}^{14}$ with a linear space of dimension $6$. In this paper we focus on the inverse problem of recovering such a linear space associated to a general self-dual set of points. We use numerical homotopy continuation to approach the problem and implement an algorithm in Julia to solve it. Along the way we also implement the forward problem of slicing Grassmannians and use it to experimentally study the real solutions to this problem.

math.AG↗

arXiv2023

F-threshold of determinantal rings

In this paper, by using a combinatorial approach, we establish a new upper bound for the F-threshold $c^\mm(\mm)$ of determinantal rings generated by maximal minors. We prove that $c^\mm(\mm)$ coincides with the $a$-invariant in the case of $3\times n$ and $4\times n$ matrices and we conjecture such equality holds for all matrices.

math.AC↗

arXiv2023

Solving Equations Using Khovanskii Bases

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Pl\"ucker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.

math.AG↗