Differential Galois Groups, Specializations, and Matzat’s Conjecture
Abstract
We study families of linear differential equations parametrized by an algebraic variety X \mathcal {X} and show that the set of all points x ∈ X x\in \mathcal {X} , such that the differential Galois group at the generic fibre specializes to the differential Galois group at the fibre over x x , is Zariski dense in X \mathcal {X} . As an application, we prove Matzat’s conjecture in full generality: The absolute differential Galois group of a one-variable function field over an algebraically closed field of characteristic zero is a free proalgebraic group.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Polynomial and algebraic computation
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- Memoirs of the American Mathematical Society
- Volume
- 319
- Issue
- 1622
- Publication date
- 28 April 2026
- Language
- EN