Degree bound for toric envelope of a linear algebraic group

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 $G \subset \mathrm{GL}_n(C)$ 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 $\mathrm{GL}_n(C)$ 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.