A new bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation

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 (see Definition 2.7) and is sextuply exponential in the order of the differential equation. In this paper, we use Szántó'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.

On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

“…Hrushovski's work has been subsequently pursued by Feng [14] and Sun [37], leading to a triple exponential algorithm for computing the Galois group of a linear differential equation. A different approach was recently pursued by Amzallag et al, who introduced the notion of toric envelope to obtain a single exponential bound for the first step in Feng's formulation of Hrushovski's algorithm, which is qualitatively optimal [1,2].…”

Degree bound for toric envelope of a linear algebraic group

“…The algorithm was used, for example, to design algorithms for computing the Galois group of a linear differential equation with parameters in several cases [22,23]. For the last decade, it has been a challenge to understand the complexity of Hrushovski's algorithm and make it practical, see [10,32,28] for recent progress in this direction. One of the key ingredients of the algorithm is the following fact, which is of independent interest to the effective and computational theory of algebraic groups.…”

“…He also conjectured that the overall complexity of the algorithm is at most double-exponential [15,Remark 4.4]. Feng [10,Proposition B.14] found the first explicit formula for such a d(n) by presenting a function of quintuply exponential growth in n that could be used as d(n) in (H2) (see also [32] for a related bound for H).…”

On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

“…Hrushovski's algorithm was later simplified by Feng, who analyzed its complexity and obtained a sextuply exponential bound in the order of the differential equation for the degree of the proto-Galois group [Fen15]. A much improved, triple exponential bound, was given by Sun [Sun19] following the same approach but using a triangular representation for algebraic sets. A different approach was recently pursued by Amzallag, Minchenko and Pogudin, who introduced the notion of toric envelope to obtain a single exponential bound for the first step of the Hrushovski's algorithm due to Feng which is qualitatively optimal [AMP19, Amz18].…”