The computation of the greatest common divisor of three bivariate Bernstein polynomials defined in a rectangular domain

This paper considers the computation of the greatest common divisor (GCD) dt1,t2 (x, y) of three bivariate Bernstein polynomials that are defined in a rectangular domain, where t1(t2) is the degree of dt1,t2(x, y) when it is written as a polynomial in x(y) whose coefficients are polynomials in y(x). The Sylvester resultant matrix and its subresultant matrices are used for the computation of the degrees and coefficients of the GCD. It is shown that there are four forms of these matrices and that they differ in their computational properties. The most difficult part of the computation is the determination of t1 and t2, and two methods for this computation are described. One method is simple but inefficient, and the other method reduces the problem to the computation of the degree of the GCD of two univariate polynomials, which is more efficient. The basis functions of the polynomials include binomial terms, which span many orders of magnitude, even for polynomials of moderate degrees. It is shown that the adverse effects of this wide range of magnitudes and a significant reduction in the sensitivity of the degree of the GCD to noise are obtained when the polynomials are processed by three operations before computations are performed on them. Examples that demonstrate the theory are included in the paper.