# Determinant structure for tau-function of holonomic deformation of linear differential equations

## TL;DR

This paper explores tau-functions linked to holonomic deformations of linear differential equations, providing a determinant formula for their ratios by leveraging Hermite's approximation problems.

## Contribution

It introduces a new determinant formula for tau-function ratios in the context of holonomic deformations, connecting Hermite's approximation problems with Schlesinger transformations.

## Key findings

- Derived a determinant formula for tau-quotients.
- Connected Hermite's approximation problems with tau-functions.
- Extended understanding of holonomic deformations in differential equations.

## Abstract

In our previous works, a relationship between Hermite's two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study tau-functions associated with holonomic deformations of linear differential equations by using Hermite's two approximation problems. As a result, we present a determinant formula for the ratio of tau-functions (tau-quotient).