This paper considers the computation of the greatest common divisor (GCD) $d_{t_1,t_2}(x,y)$ of three bivariate Bernstein polynomials that are defined in a rectangular domain, where $t_1(t_2)$ is the degree of $d_{t_1,t_2}(x,y)$ 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 $t_1$ and $t_2$, 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.

本文考虑最大公约数（GCD）的计算 $d_{{Ť}_{\mathrm{1个}}，{Ť}_{\mathrm{2个}}}（X，ÿ）$ 在矩形域中定义的三个二元伯恩斯坦多项式的集合，其中 ${Ť}_{\mathrm{1个}}（{Ť}_{\mathrm{2个}}）$ 是的程度 $d_{{Ť}_{\mathrm{1个}}，{Ť}_{\mathrm{2个}}}（X，ÿ）$ 当它写成多项式时 $X（ÿ）$ 其系数为的多项式 $ÿ（X）$。Sylvester结果矩阵及其子结果矩阵用于计算GCD的度数和系数。结果表明，这些矩阵有四种形式，并且它们的计算属性不同。计算中最困难的部分是确定${Ť}_{\mathrm{1个}}$ 和 ${Ť}_{\mathrm{2个}}$，并介绍了两种计算方法。一种方法简单但效率低下，另一种方法将问题减少到两个单变量多项式的GCD阶数的计算上，效率更高。多项式的基本函数包括二项式项，即使对于中等程度的多项式，它们也跨越多个数量级。结果表明，在对多项式进行计算之前，对多项式进行三项运算后，会获得较大幅度范围的不利影响以及GCD程度对噪声的敏感性显着降低。本文中包含了证明该理论的示例。