The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (�$ge$ 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions.
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机译：可以通过找到以张量Chebyshev为基础表示的多项式插值的公共零来计算两个双变量函数的公共零。在这里，我们基于隐藏变量结果方法和具有多项式项的Béezout矩阵开发了一种双变量根查找算法。使用包括域细分，Béezoutian正则化和局部细化在内的技术，我们能够可靠且准确地计算两个平滑函数的简单公零，且多项式插值非常高（$ ge $ 1000）。我们通过注意Béezout矩阵是矩阵多项式来分析所得方法及其条件。我们强大的算法在Chebfun2的roots命令中实现，Chebfun2是用面向对象的MATLAB编写的软件包，用于使用双变量函数进行计算。
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