The complexity of cylindrical algebraic decomposition with respect to polynomial degree

TL;DR

This paper investigates how the complexity of cylindrical algebraic decomposition (CAD) can be reduced with equational constraints and Groebner Bases, aiming to lower the exponential degree dependence in the algorithm's complexity.

Contribution

The authors extend previous work by analyzing how Groebner Bases can refine CAD complexity bounds related to polynomial degree, moving closer to the expected theoretical improvements.

Findings

Demonstrated potential for reducing degree-related complexity in CAD using Groebner Bases.

Extended previous bounds by incorporating polynomial degree considerations.

Provided insights into further optimization of CAD algorithms with algebraic techniques.

Abstract

Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be improved by adapting to take advantage of any equational constraints (ECs): equations logically implied by the input. Intuitively, we expect the double exponent in the complexity to decrease by one for each EC. In ISSAC 2015 the present authors proved this for the factor in the complexity bound dependent on the number of polynomials in the input. However, the other term, that dependent on the degree of the input polynomials, remained unchanged. In the present paper the authors investigate how CAD in the presence of ECs could be further refined using the technology of Groebner Bases to move towards the intuitive bound for polynomial degree.