An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

Abstract

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-ofthe-art implementations, that is, Lgp and Maple's Isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.

Efficient real root approximation

“…The problem of accurate root approximation is omnipresent in mathematical applications; certified methods are of particular importance in the context of computations with algebraic objects, e.g., when computing the topology of algebraic curves [10,6] or when solving systems of multivariate equations [3].…”

“…tal comparisons in the context of [3] have shown that the approximate version of QIR gives significantly better running times than its exact counterpart. These observations underline the practical relevance of our approximate version and suggest a practical comparison with state-of-the-art solvers mentioned above as further work.…”

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

“…Then, for each distinct x * an eigenproblem of size equal to the dimension of its kernel should be solved. Another approach is to eliminate both x and y separately and then for each pair of projected coordinates check whether it constitutes a solution of the given system or not [8,14,44]. For many applications, a nontrivial final task is to determine which of the solutions are real.…”

“…For a fair comparison, each algorithm Table 3.2 Properties of the initial moderate-degree problem set, containing a total of 16 systems. All of these systems with the exception of lebesgue were taken from [8].…”

Solving bivariate polynomial systems on a GPU

“…A GPU implementation of a bivariate solver over the real numbers is reported in [3]. Solving polynomial systems is a driving subject in symbolic computation with many successful results from theoretical to practical aspects.…”