An Improved Method for Computing Dixon Polynomial

Computational methods for manipulating sets of polynomial equations are becoming of greater importance due to the use of polynomial in various applications. Dixon resultant algorithm provides one of the most efficient methods for solving the system of polynomial equations or eliminating variables. When computing Dixon resultant, we first construct the Dixon polynomial for input polynomial system. However, as the entries in the determinant are symbolic, large intermediate symbolic expression is generated and leads to computer algebra systems run for a long time or crash. To avoid the intermediate expression swell, we propose using Zippel's multivariate probabilistic interpolation to compute Dixon polynomial. At first, the method of truncated formal power series, which converts the operation of division to multiplication, is used to preprocess the expression of Dixon polynomial. Secondly, Dixon polynomial is interpolated heuristically by Zippel's method. In order to solve the linear equation effectively in Zippel's method, Kaltofen, E.'s efficient algorithm[4] is introduced to solve the equations of transposed Vandermonde system in interpolation. Besides these, we combine Zippel's multivariate method with Lagrange interpolation so as to take advantage of the sparsity of polynomial system sufficiently. Intermediate symbolic expressions can be reduced because the algorithm converts the symbolic computation to the numerical.