Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding algorithms that exploit our results.

翻译：切比雪夫簇是由切比雪夫多项式或其多变量推广参数化的代数簇。我们确定了这些簇的维数、度数、奇异轨迹以及定义方程。我们阐释了当单项式被切比雪夫多项式替代时，它们在稀疏多项式求根中如何扮演环面簇的角色。我们提出了利用这些结果的数值求根算法。