Degree bound for toric envelope of a linear algebraic group

Abstract

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group G ⊂ GLn(C) can be arbitrarily large even for n = 1. One of the key ingredients of Hrushovski's algorithm for computing the Galois group of a linear differential equation was an idea to "approximate" every algebraic subgroup of GLn(C) by a "similar" group so that the degree of the latter is bounded uniformly in n. Making this uniform bound computationally feasible is crucial for making the algorithm practical.In this paper, we derive a single-exponential degree bound for such an approximation (we call it toric envelope), which is qualitatively optimal. As an application, we improve the quintuply exponential bound for the first step of the Hrushovski's algorithm due to Feng to a single-exponential bound. For the cases n = 2, 3 often arising in practice, we further refine our general bound.

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

“…The Zariski closure of SL 𝑛 (Z) is SL 𝑛 (Q). 2 We observe that any non-zero polynomial 𝑓 that vanishes on SL 𝑛 (Q) has (total) degree at least 𝑛. Indeed, evaluating 𝑓 on the diagonal matrix diag(𝑥, .…”

“…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].…”

Computing the Lie algebra of the differential Galois group: the reducible case

“…Using model theory, Hrushovski gave in [Hru02] the first general decision procedure computing the Galois group. It was recently clarified and improved by Feng in [Fen15], see also [AMP18,Sun18]. A symbolic-numeric algorithm was proposed by van der Hoeven in [vdH07], based on the Schlesinger-Ramis density theorems.…”