The evaluation of geometric queries: constraint databases and quantifier elimination

We model the algorithmic task of geometric elimination (e.g., quantifier elimination in the elementary field theories of real and complex numbers) by means of certain constraint database queries, called geometric queries. As a particular case of such a geometric elimination task, we consider sample point queries. We show exponential lower complexity bounds for evaluating geometric queries in the general and in the particular case of sample point queries. Although this paper is of theoretical nature, its aim is to explore the possibilities and (complexity-)limits of computer implemented query evaluation algorithms for Constraint Databases, based on the principles of the most advanced geometric elimination procedures and their implementations, like, e.g., the software package "Kronecker".