A Fast Method for Sampling Real Algebraic Sets

Title: A Fast Method for Sampling Real Algebraic Sets

Abstract: This research article introduces a new method for finding points in real algebraic sets, specifically those of the form Z(p(Q(X)), Kn), where p and Q are polynomials. The method, which operates over an arbitrary real closed field K, significantly reduces the computational complexity compared to previous methods. It uses symbolic computation and real univariate representations to efficiently compute a set of sampling points that intersects nontrivially each connected component of the set. The method is particularly useful when the number of variables is large, as it is capable of performing these computations in polynomial time.

Research Question: How can we develop an efficient method for finding points in real algebraic sets, specifically those of the form Z(p(Q(X)), Kn), while maintaining a high level of accuracy and computational efficiency?

Methodology: The method presented in this article involves symbolic computation and the use of real univariate representations. It extends the real field K by infinitesimals and performs necessary geometric transformations. The algorithm finds a point in each connected component of the set Z(p(Q(X)), Kn), with a computational complexity of ( dn)O(k). This is a significant improvement over previous methods, which had exponential dependence on the number of variables.

Results: The research article presents a procedure that computes, in ( dn)O(k) arithmetic operations in D, a set S of sampling points that intersects nontrivially each connected component of Z(p(Q(X))). This set is constructed in such a way that it intersects each connected component of Z(p(Q(X))). The method is particularly useful when the number of variables is large, as it is capable of performing these computations in polynomial time.

Implications: The new method presented in this article has several important implications. Firstly, it significantly reduces the computational complexity of finding points in real algebraic sets, making it more accessible for large-scale applications. Secondly, it provides a more efficient way of performing these computations, which can lead to faster and more accurate results. Lastly, the method is general and can be applied to a wide range of real algebraic sets, making it a valuable tool in the field of real algebraic geometry.

Link to Article: https://arxiv.org/abs/0403008v1 Authors: arXiv ID: 0403008v1