On solutions of codimension-one $A$-hypergeometric systems

TL;DR

This paper explicitly constructs logarithmic series solutions for codimension-one $A$-hypergeometric systems at the origin, analyzing conditions for maximal unipotent monodromy and providing a comprehensive solution set when parameters are nonresonant.

Contribution

It provides explicit constructions of solutions and criteria for monodromy properties in codimension-one $A$-hypergeometric systems, advancing understanding of their solution structure.

Findings

Constructed explicit logarithmic series solutions at the origin.

Determined conditions for maximal unipotent monodromy.

Established solution completeness for nonresonant parameters.

Abstract

By a codimension-one system we mean a system whose lattice of relations has rank one. We consider codimension-one $A$-hypergeometric systems and explicitly construct some of the logarithmic series solutions at the origin. When the parameter vector $\beta$ is nonresonant we obtain a full set of logarithmic series solutions at the origin by this procedure. We also determine when a codimension-one system with nonresonant parameter can have maximal unipotent monodromy at the origin.