Algebraic aspects of hypergeometric differential equations

Abstract

We review some classical and modern aspects of hypergeometric differential equations, including A-hypergeometric systems of Gel$$'$$ ′ fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler–Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

Nonlinear algebra and applications

“…This ideal was further studied in [112,184,100]. While hypergeometric functions of one complex variable are well-studied in terms of GKZ systems [200], there is no intrinsic description of hypergeometric functions of a matrix argument in terms of D-modules yet. For [78] modeling the discrete statistical experiment of flipping a biased coin twice.…”

Nonlinear Algebra and Applications

“…This ideal was further studied in [95,107,176]. While hypergeometric functions of one complex variable are well-studied in terms of GKZ systems [191], there is no intrinsic description of hypergeometric functions of a matrix argument in terms of D -modules yet. For broadening the range of applications of D -modules in the sciences, one is in need of seeking out further functions that can be described by means of holonomic functions.…”

Local cohomology -- an invitation

“…We refer to [SST00] for extensive background on hypergeometric functions, their associated differential equations, and how they relate to hypergeometric modules M A pβq via a dehomogenization technique investigated in [BMW19]. The article [RSSW21] is a gentle introduction to hypergeometric D-modules, combined with a survey on recent applications to Hodge theory.…”

“…The Hodge and weight filtrations of hypergeometric systems have been studied in [RS20,Rei14,RW], showing connections to intersection homology of toric varieties. See [RSSW21] for a survey.…”