Complexity of Triangular Representations of Algebraic Sets

TL;DR

This paper provides explicit bounds on the degrees and number of components in the triangular decomposition of algebraic sets, enhancing understanding of the complexity of Szanto's algorithm.

Contribution

It offers the first complete degree and component bounds for the triangular decomposition algorithm, with explicit formulas.

Findings

Derived explicit bounds for polynomial degrees

Established bounds on the number of components

Enhanced understanding of algorithm complexity

Abstract

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. Szanto. 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.