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

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.