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 degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group G ⊂ GL n ⁡ ( C ) G \subset \operatorname {GL}_n(C) can be arbitrarily large even for n = 1 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 GL n ⁡ ( C ) \operatorname {GL}_n(C) by a “similar” group so that the degree of the latter is bounded uniformly in n 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 a toric envelope), which is qualitatively optimal. As an application, we improve the quintuply exponential bound due to Feng for the first step of Hrushovski’s algorithm to a single-exponential bound. For the cases n = 2 , 3 n = 2, 3 often arising in practice, we further refine our general bound.

Note on the construction of Picard-Vessiot rings for linear differential equations

“…which implies that stab(m) ⊂ stab(J) ⊂ H. By Lemma 5.1 of [1], stab(J) is also a toric envelope of stab(m). So J is a toric maximal δ-ideal.…”

“…a method to compute a maximal δ-ideal m. Denote by Gal(R/k) the Galois group of (1) over k that is defined to be the set of kautomorphisms of R which commute with δ. Let F be a fundamental matrix of (1) with entries in R. Then F induces an injective group homomorphism from Gal(R/k) to GL n (C). The image of this homomorphism can be described via the stabilizer of the maximal δ-ideal with F as a zero.…”

“…It is well-known that stab(I) is an algebraic subgroup of GL n (C). The following notion is introduced by Amzallag etc in [1].…”

Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz

“…The problem of calculating explicitly the differential Galois group of ∂ y = A y is old and still difficult. Among the several references, we cite [CS99], [Hru02], [vdH07], [Fen15] and [AMP18] that do not make any assumption on the order n of the system. Implemented (or implementable) algorithms exist only for small dimensions n. See for example [Kov86], [SU93], [vHRUW99], [Hes01], [vH02], [Per02], [NvdP10], [CS18].…”