Fast computation of determinants of Bézout matrices and application to curve implicitization

Abstract

When using bivariate polynomial interpolation for computing the implicit equation of a rational plane algebraic curve given by its parametric equations, the generation of the interpolation data is the most costly of the two stages of the process. In this work a new way of generating those interpolation data with less computational cost is presented. The method is based on an efficient computation of the determinants of certain constant Bézout matrices.

Computing the Determinant of a Matrix with Polynomial Entries by Approximation

“…Notice that the coefficients of system (2) are Vandermonde matrices, the reference [24] by the Newton's interpolation method presented a progressive algorithm which is significantly more efficient than previous available methods in O(d 2 1 ) arithmetic operations in Algorithm 1.…”

“…In the scientific computing and engineering fields, such as computing multipolynomial resultants [1], computing the implicit equation of a rational plane algebraic curve given by its parametric equations [2], and computing Jacobian determinant in multidomain unified modeling [3], computing the determinant of a matrix with polynomial entries (also called symbolic determinant) is inevitable. Therefore, computing symbolic determinants is an active area of research [4][5][6][7][8][9][10][11][12].…”