arXiv subjects

Teresa Yu

Publications and source records attributed to Teresa Yu.

17 recordsLinked to original sources

arXiv2026

Partial fraction decompositions on hyperplane arrangements

We study partial fraction decompositions (PFDs) in several variables using tools from commutative algebra. We give criteria for when a rational function with poles on a hyperplane arrangement has a desirable PFD. Our criteria are obtained by examining the primary decomposition of ideals coming from hyperplane arrangements. We then present an algorithm for finding a PFD that satisfies properties desired for simplifying the calculation of scattering amplitudes. We demonstrate the effectiveness of this algorithm by computing practical examples coming from Feynman integrals.

math.AC↗

arXiv2025

Symmetric modules over the infinite polynomial ring I: nilpotent quotients

Cohen proved that the infinite variable polynomial ring $R=k[x_1,x_2,\ldots]$ is noetherian with respect to the action of the infinite symmetric group $\mathfrak{S}$. The first two authors began a program to understand the $\mathfrak{S}$-equivariant algebra of $R$ in detail. In previous work, they classified the $\mathfrak{S}$-prime ideals of $R$. An important example of an $\mathfrak{S}$-prime is the ideal $\mathfrak{h}_s$ generated by $(s+1)$st powers of the variables. In this paper, we study the category of $R/\mathfrak{h}_s$-modules. We obtain a number of results, and mention just three here: (a) we determine the Grothendieck group of the category; (b) we show that the Krull--Gabriel dimension is $s$; and (c) we obtain generators for the derived category. This paper will play a key role in subsequent work where we study general modules.

math.AC↗

arXiv2025

Decomposing conditional independence ideals with hidden variables

We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gr\"obner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. Moreover, we show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.

math.AC↗

arXiv2024

Moment varieties of the inverse Gaussian and gamma distributions are nondefective

We show that the parameters of a $k$-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first $3k-1$ moments, and rationally identifiable from the first $3k+2$ moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti--Mella.

math.AG↗

arXiv2024

Parabolic-equivariant modules over polynomial rings in infinitely many variables

We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{GL}(\mathbf{C}^\infty)$ consisting of elements fixing a flag in $\mathbf{C}^\infty$ with each graded piece infinite-dimensional. We decompose the category into simpler pieces that can be described combinatorially, and prove a number of finiteness results, such as finite generation of local cohomology and rationality of Hilbert series. Furthermore, we show that this category is equivalent to the category of representations of a particular combinatorial category generalizing $\mathbf{FI}$.

math.AC↗

arXiv2024

Jets and principal components of monomial ideals, and very well-covered graphs

Motivated by using combinatorics to study jets of monomial ideals, we extend a definition of jets from graphs to clutters. We offer some structural results on their vertex covers, and show an interesting connection between the cover ideal of the jets of a clutter and the symbolic powers of the cover ideal of the original clutter. We use this connection to prove that jets of very well-covered graphs are very well-covered. Next, we turn our attention to principal jets of monomial ideals, describing their primary decomposition and minimal generating sets. Finally, we give formulas to compute various algebraic invariants of principal jets of monomial ideals, including their Hilbert series, Betti numbers, multiplicity and regularity.

math.AC↗

arXiv2024

Polynomial functors on flags

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite rank groups, and we apply these descriptions to study polynomial representations and representation stability of parabolic subgroups of general linear groups.

math.RT↗

arXiv2023

Moment varieties from inverse Gaussian and gamma distributions

Motivated by previous work on moment varieties for Gaussian distributions and their mixtures, we study moment varieties for two other statistically important two-parameter distributions: the inverse Gaussian and gamma distributions. In particular, we realize the moment varieties as determinantal varieties and find their degrees and singularities. We also provide computational evidence for algebraic identifiability of mixtures, and study the identifiability degree and Euclidean distance degree.

math.AG↗

arXiv2023

Standard monomial theory modulo Frobenius in characteristic two

Over a field of characteristic two, we develop a theory of standard monomials for polynomial rings modulo a Frobenius power of the maximal ideal generated by all variables. As a result, we obtain a filtration by modular GL_n-representations whose characters are given by particular truncated Schur polynomials, thus proving a conjecture by Gao--Raicu--VandeBogert in the characteristic two case.

math.AC↗

arXiv2020

On Virtually Cohen-Macaulay Simplicial Complexes

We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces. In particular, we provide sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). We also show that all balanced simplicial complexes are virtually Cohen-Macaulay.

math.AC↗

arXiv2020

Completions of Uncountable Local Rings with Countable Spectra

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that uncountable local domains with countable spectra are more common than previously thought. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and completions of uncountable noncatenary local unique factorization domains with countable spectra.

math.AC↗

arXiv2020

Completions of Countable Excellent Domains and Countable Noncatenary Domains

We find necessary and sufficient conditions for a complete local ring containing the rationals to be the completion of a countable excellent local (Noetherian) domain. Furthermore, we find necessary and sufficient conditions for a complete local ring to be the completion of a countable noncatenary local domain, as well as necessary and sufficient conditions for it to be the completion of a countable noncatenary local unique factorization domain.

math.AC↗

arXiv2020

Gonality sequences of graphs

To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic curve. We study gonality sequences for graphs of low genus, proving that for genus up to $5$, the gonality sequence is determined by the genus and the first gonality. We then prove that any reasonable pair of first two gonalities is achieved by some graph. We also develop a modified version of Dhar's burning algorithm more suited for studying higher gonalities.

math.CO↗

arXiv2019

Multi-color forcing in graphs

Let $G=(V,E)$ be a finite connected graph along with a coloring of the vertices of $G$ using the colors in a given set $X$. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the multi-color forcing process terminates regardless of the number of colors used. We give an upper bound on the number of steps required to terminate a forcing procedure in terms of the number of vertices in the graph on which the procedure is being applied. We then focus on multi-color forcing with three colors and analyze the end states of certain families of graphs, including complete graphs, complete bipartite graphs, and paths, based on various initial colorings. We end with a few directions for future research.

math.CO↗

arXiv2019

Extended Nestohedra and their Face Numbers

Nestohedra are a family of convex polytopes that includes permutohedra, associahedra, and graph associahedra. In this paper, we study an extension of such polytopes, called extended nestohedra. We show that these objects are indeed the boundaries of simple polytopes, answering a question of Lam and Pylyavskyy. We also study the duals of (extended) nestohedra, giving a complete characterization of isomorphisms (as simplicial complexes) between the duals of extended nestohedra and a partial characterization of isomorphisms between the duals of nestohedra and extended nestohedra. In addition, we give formulas for their $f$-, $h$-, and $\gamma$-vectors. This includes showing that the $f$-vectors of the extended nestohedron corresponding to a forest $F$ and the nestohedron corresponding to the line graph of $F$ are the same, as well as showing that all flag extended nestohedra have nonnegative $\gamma$-vectors, thus proving Gal's conjecture for a large class of flag simple polytopes. We also relate the $f$- and $h$-vectors of the nestohedra and extended nestohedra, as well as give explicit formulas for the $h$- and $\gamma$-vectors in terms of descent statistics for a certain class of flag extended nestohedra. Finally, we define a partial ordering on partial permutations that is a join semilattice quotient of the weak Bruhar order on the symmetric group, and such that any linear extension of the partial order provides a shelling of the dual of the stellohedron.

math.CO↗

arXiv2018

Graphs of gonality three

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge- and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

math.CO↗

arXiv2018

Treewidth and gonality of glued grid graphs

We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs.

math.CO↗