An efficient algorithm for the sign condition problem in the semi-algebraic context

• Authors: Rafael Grimson
• Venue: GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
• Year: 2010

We study algebraic complexity of the sign condition problem for any given family of polynomials. Essentially, the problem consists in determining the sign condition satisfied by a fixed family of polynomials at a query point, performing as little arithmetic operations as possible. After defining precisely the sign condition and the point location problems, we introduce a method called the dialytic method to solve the first problem efficiently. This method involves a linearization of the original polynomials and provides the best known algorithm to solve the sign condition problem. Moreover, we prove a lower bound showing that the dialytic method is almost optimal. Finally, we extend our method to the point location problem. The dialytic method solves (non-uniformly) the sign condition problem for a family of s polynomials in R[X1,...,Xn] given by an arithmetic circuit $\Gamma_{\mathcal F}$ of non-scalar complexity L performing ${\mathcal O}((L+n)^5\log(s))$ arithmetic operations. If the polynomials are given in dense representation and d is a bound for their degrees, the complexity of our method is ${\mathcal O}(d^{5n} log(s))$. Comparable bounds are obtained for the point location problem.