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Saiei-Jaeyeong Matsubara-Heo

Publications and source records attributed to Saiei-Jaeyeong Matsubara-Heo.

At least 19 recordsLinked to original sources
arXiv2026

Principal Matroid Determinants

We develop a theory of principal determinants and hypergeometric systems for realizable matroids. Our framework parallels the toric theory of Gel'fand, Kapranov, and Zelevinsky (GKZ), but with the combinatorics of matroids and their flats replacing the usual role of polytopes and their faces. In this analogy, the toric variety is replaced by a reciprocal linear space. The {principal $A$-determinant} is replaced by the {principal matroid determinant}, defined as a specialization of a resultant. The GKZ hypergeometric system is replaced by the {matroid hypergeometric system}, a holonomic $D$-module of combinatorial nature whose singular locus is conjectured to be the principal matroid~determinant.

math.AG↗
arXiv2026

Periods of Limiting Mixed Hodge Structures of Projective Hypersurfaces

For a generic one-parameter degeneration of projective hypersurfaces, we show that the periods of the limiting mixed Hodge structure are generated by certain special values of logarithm, Gamma and Dirichlet $L$-functions. Our proof is based on the analytic continuation of solutions to the GKZ system.

math.AG↗
arXiv2025

Hypergeometric Discriminants

Given a family of varieties, the Euler discriminant locus distinguishes points where Euler characteristic differs from its generic value. We introduce a hypergeometric system associated with a flat family of very affine locally complete intersection varieties. It is proven that the Euler discriminant locus is its singular locus and is purely one-codimensional unless it is empty. Of particular interest is a family of very affine hypersurfaces. We coin the term hypergeometric discriminant for the characteristic cycle of the hypergeometric system and establish a formula in terms of likelihood equations.

math.AG↗
arXiv2025

Differential Space of Feynman Integrals: Annihilators and $\mathcal{D}$-module

We present a novel algorithm for constructing differential operators with respect to external variables that annihilate Feynman-like integrals and give rise to the associated $\mathcal{D}$-modules, based on Griffiths-Dwork reduction. By leveraging the Macaulay matrix method, we derive corresponding relations among partial differential operators, including systems of Pfaffian equations and Picard-Fuchs operators. Our computational approach is applicable to twisted period integrals in projective coordinates, and we showcase its application to Feynman graphs and Witten diagrams. The method yields annihilators and their algebraic relations for generic regulator values, explicitly avoiding contributions from surface terms. In the cases examined, we observe that the holonomic rank of the $\mathcal{D}$-modules coincides with the dimension of the corresponding de Rham co-homology groups, indicating an equivalence relation between them, which we propose as a conjecture.

hep-th↗
arXiv2024

Euler Discriminant of Complements of Hyperplanes

The Euler discriminant of a family of very affine varieties is defined as the locus where the Euler characteristic drops. In this work, we study the Euler discriminant of families of complements of hyperplanes. We prove that the Euler discriminant is a hypersurface in the space of coefficients, and provide its defining equation in two cases: (1) when the coefficients are generic, and (2) when they are constrained to a proper subspace. In the generic case, we show that the multiplicities of the components can be recovered combinatorially. This analysis also recovers the singularities of an Euler integral. In the appendix, we discuss a relation to cosmological correlators.

math.AG↗
arXiv2024

Generalized hypergeometric functions with several variables

We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kempé de Fériet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the singularities, the rank and the condition for the reducibility of the system. We give complete local solutions of the system at many singular points of the system and solve the connection problem among these local solutions. Under some assumptions, the system is written as a KZ equation. We determine its spectral type in the direction of coordinates as well as simultaneous eigenspace decompositions of residue matrices. The system may or may not be rigid in the sense of N.~Katz viewed as an ordinary differential equation in some direction. We also show that the system is a special case of Gel'fand-Kapranov-Zelevinsky system. From this point of view, we discuss multivariate generalizations.

math.CA↗
arXiv2023

Four Lectures on Euler Integrals

These lecture notes provide a self-contained introduction to Euler integrals, which are frequently encountered in applications. In particle physics, they arise as Feynman integrals or string amplitudes. Our four selected topics demonstrate the diverse mathematical techniques involved in the study of Euler integrals, including polyhedral geometry, very affine varieties, differential equations, and computational algebra.

math-ph↗
arXiv2023

Twisted cohomology and likelihood ideals

A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood equations. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.

math.AG↗
arXiv2022

Localization formulas of cohomology intersection numbers

We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of the canonical form.

math.AG↗
arXiv2021

Global analysis of GG systems

This paper deals with some analytic aspects of GG system introduced by I.M.Gelfand and M.I.Graev: we compute the dimension of the solution space of GG system over the field of functions meromorphic and periodic with respect to a lattice. We describe the monodromy invariant subspace of the solution space. We give a connection formula between a pair of bases consisting of $Γ$-series solutions of GG system associated to a pair of regular triangulations adjacent to each other in the secondary fan.

math.AG↗
arXiv2020

Euler and Laplace integral representations of GKZ hypergeometric functions

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We establish a combinatorial method of constructing a basis of the rapid decay homology group associated to Euler-Laplace integral with a nice intersection property. This construction yields a remarkable expansion formula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application, we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions and their confluent analogue in terms of bipartite graphs.

math.CA↗
arXiv2020

Homology and cohomology intersection numbers of GKZ systems

We describe the homology intersection form associated to regular holonomic GKZ systems in terms of the combinatorics of regular triangulations. Combining this result with the twisted period relation, we obtain a formula of cohomology intersection numbers in terms of a Laurent series. We show that the cohomology intersection number depends rationally on the parameters. We also prove a conjecture of F. Beukers and C. Verschoor on the signature of the monodromy invariant hermitian form. This is a continuation of the previous work arXiv:1904.00565.

math.AG↗
arXiv2020

Computing cohomology intersection numbers of GKZ hypergeometric systems

In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in \cite{GM}, \cite{MH}, \cite{MT} and \cite{MT2}. We also discuss the relation between intersection theory and evaluation of an integral of a product of powers of absolute values of polynomials.

math.AG↗
arXiv2020

A tree expansion formula of a homology intersection numbers on the configuration space $\mathcal{M}_{0,n}$

In \cite{M}, Sebastian Mizera discovered a tree expansion formula of a homology intersection number on the configuration space $\mathcal{M}_{0,n}$. The formula originates in a study of Kawai-Lewellen-Tye relation in string theory. In this paper, we give an elementary proof of the formula. The basic ingredients are the combinatorics of the real moduli space $\overline{\mathcal{M}}_{0,n}(\R)$ and a combinatorial identity related to the face number of the associahedron.

math.CO↗
arXiv2018

Laplace, Residue, and Euler integral representations of GKZ hypergeometric functions

We consider four types of representations of solutions of GKZ system: series representations, Laplace integral representations, Euler integral representations, and Residue integral representations which will be introduced in this paper. In the former half of this paper, we provide a method for constructing integration cycles for Laplace, Residue, and Euler integral representations and relate them to series representations. In the latter half, we reformulate our integral representations in terms of direct images of $D$-modules and show their equivalence.

math.CA↗
arXiv2018

Residue current approach to Ehrenpreis-Malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags

We introduce the ring of partial differential operators with constant coefficients and commensurate time lags (we use the terminology D$Δ$ operators from now) initially defined by H. Glüsing-Lürßen for ordinary $DΔ$ operators and investigate its cohomological properties. Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $DΔ$ equations.

math.AP↗