Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szántó. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds.

No 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.

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.