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The regular singular inverse problem in differential Galois theory

Thomas Serafini, Michael Wibmer

TL;DR

The paper resolves the regular singular inverse problem by showing that any linear algebraic group $G$ over an algebraically closed field $k$ of characteristic zero can be realized as the differential Galois group of a regular singular equation $ rac{d}{dz}y=Ay$ with $A\in k(z)^{n\times n}$, provided $G$ is generated by $d$ elements and the singularities can be placed at $d+1$ prescribed points $S\subset \mathbb{P}^1(k)$. The authors reduce the existence to the complex case $k=\mathbb{C}$ via a Feng–Wibmer specialization theorem for differential torsors, leveraging the Riemann–Hilbert correspondence and Schlesinger density to realize $G$ there, then transfer the realization back to general $k$ through carefully constructed finitely generated bases and specialization. Key contributions include a precise regular-singularity realization theorem with explicit singularity control, and a proalgebraic interpretation linking the result to the universal free proalgebraic group on $d$ generators. This work provides a tangible affirmative answer to the regular singular inverse problem and connects differential Galois theory with proalgebraic group theory and the geometry of parameter spaces.

Abstract

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.

The regular singular inverse problem in differential Galois theory

TL;DR

The paper resolves the regular singular inverse problem by showing that any linear algebraic group over an algebraically closed field of characteristic zero can be realized as the differential Galois group of a regular singular equation with , provided is generated by elements and the singularities can be placed at prescribed points . The authors reduce the existence to the complex case via a Feng–Wibmer specialization theorem for differential torsors, leveraging the Riemann–Hilbert correspondence and Schlesinger density to realize there, then transfer the realization back to general through carefully constructed finitely generated bases and specialization. Key contributions include a precise regular-singularity realization theorem with explicit singularity control, and a proalgebraic interpretation linking the result to the universal free proalgebraic group on generators. This work provides a tangible affirmative answer to the regular singular inverse problem and connects differential Galois theory with proalgebraic group theory and the geometry of parameter spaces.

Abstract

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.
Paper Structure (8 sections, 17 theorems, 4 equations)

This paper contains 8 sections, 17 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. The regular singular inverse problem
  3. Differential Galois theory
  4. Regular singular differential equations
  5. Generating algebraic groups
  6. Specializations
  7. Proof of the main result
  8. An interpretation in terms of proalgebraic groups

Key Result

Theorem 1

Let $G\leq\textup{GL}_{n,k}$ be a linear algebraic group over $k$ that can be generated by $d$ elements (as an algebraic group) and let $S$ be a subset of $\mathbb P^1(k)$ with $d+1$ elements. Then there exists a matrix $A\in k(z)^{n\times n}$ such that the differential equation $\partial(y)=Ay$ is

Theorems & Definitions (32)

  • Theorem : Theorem \ref{['thm:inv']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 22 more