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.
翻译:与线性代数组合作的代数组通常通过定义多数值方程式来代表它们。 人们总是可以选择定义一个代数组的方程, 该代数组在最大程度上应具有代数多样性。 但是, 线性代数组的数值可以任意地很大, 即使对于美元 = 1美元 。 赫鲁索夫斯基计算伽罗瓦方程式组计算线性差方程式的算法的关键成分之一, 是用一个“ 类似” 组“ 接近” 每个代数组的方程, 以“ 接近” 每个代数组的数值, 以使后者的数值一致以美元为单位。 使这一统一的组合计算能够使算法实用化非常关键。 在本文中, 我们为这种直线性差方方方程( 我们称之为“ 直线性方程” 的算法的关键成分之一, 是用“ 接近” 每个代数组的代数分组“ 接近” 。 我们改进了“ 基数” 的每个代数分组“ ” 的“ 近似” 的代数,,, 以美元为“, 等值, 等值 等值, 等值 的 平面法 平面法,, 等值 的 的 等值 等值法, 的 等值法 等值法将 的 的 的,,, 等值法,, 等值 。