A New Bound on Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation

TL;DR

This paper improves the theoretical complexity bound for Hrushovski's algorithm in computing Galois groups of linear differential equations, reducing it from sextuply exponential to triple exponential in the order of the equation.

Contribution

It introduces a new complexity bound using Szanto's triangular representation algorithm, advancing the understanding of the algorithm's efficiency.

Findings

New triple exponential bound on algorithm complexity

Application of Szanto's algorithm to differential Galois groups

Comparison with previous sextuply exponential bound

Abstract

The complexity of computing the Galois group of a linear differential equation is of general interest. In a recent work, Feng gave the first degree bound on Hrushovski's algorithm for computing the Galois group of a linear differential equation. This bound is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group and is sextuply exponential in the order of the differential equation. In this paper, we use Szanto's algorithm of triangular representation for algebraic sets to analyze the complexity of computing the Galois group of a linear differential equation and we give a new bound which is triple exponential in the order of the given differential equation.