Heuristics to sift extraneous factors in Dixon resultants

Abstract

Dixon resultant is an efficient and practical method for simultaneously eliminating many variables from a parametric polynomial system P with coefficients in a field. It has been widely used in a variety of scientific fields including automated theorem proving, biological systems, computer vision, robot kinematics, and so on. However, Dixon resultant method is subjected to extraneous factors likewise other resultant methods. These extraneous factors are undesirable and they create the troublesome problems in certain applications. In other words, we compute the Dixon resultant possibly multiplied with some extraneous factors. Therefore, there is a need to develop or construct techniques that can eliminate, or at least reduce the number of extraneous factors from the Dixon resultant. Given a set of factors in Dixon resultant, we present a heuristic method to sift extraneous factors. It relies on the specialized system P by using the specialization of the parameters in the polynomial systems. Meanwhile, it needs to obtain the regular chain from the specialized system P. From this approach, the parallelization of the method arises naturally. By using the specialization formula this principle can be generalized to other resultant methods. We report on some benchmark examples of our algorithm sifting extraneous factors in Dixon resultants.