Vindas Meléndez Co-Authors Paper on Polynomial Properties in Polyhedral Geometry

Andrés R. Vindas Meléndez, assistant professor of mathematics at Harvey Mudd College, co-authored a research paper that explores the behavior of local h-polynomials, a mathematical tool used in the study of lattice-point enumeration of high-dimensional geometric objects known as lattice polytopes. The paper, “Local h-polynomials for one-row Hermite normal form simplices,” was published in Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry and is the result of a collaboration that followed a 2022 workshop hosted by the American Institute of Mathematics.

Lattice polytopes are geometric shapes that sit within a structured grid, making them useful in combinatorial and discrete geometry. Vindas Meléndez’s study focuses on a special class of these polytopes that can be represented through matrices in Hermite normal form, a method for simplifying integer matrices. Specifically, Vindas Meléndez and his collaborators examined cases where the matrix contains only one non-zero row, analyzing how the coefficients of the associated polynomials behave as the size of the polytope increases.

“This kind of research is useful in the mathematical areas of combinatorics and discrete geometry, particularly in understanding how shapes relate to counting problems,” says Vindas Meléndez. “This research direction will give students who work with me an opportunity to pursue computational and experimental avenues to theoretical mathematics research.”

The results of this study contribute to a growing area of research examining how polynomial invariants—mathematical properties that remain unchanged under transformations—are distributed within lattice polytopes. Vindas Meléndez recently co-organized a workshop at the Institute for Pure and Applied Mathematics at UCLA, where researchers discussed broader questions related to computational interactions between algebra, combinatorics and discrete geometry.

This summer, Vindas Meléndez anticipates working with about six Harvey Mudd students on related projects to continue exploring combinatorially defined polytopes.

Through this research, Vindas Meléndez, his collaborators and student researchers are expanding the understanding of fundamental mathematical properties, reinforcing Harvey Mudd’s commitment to innovative and evolving education.