A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Pad\'e approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fr\"oberg's Conjecture in commutative algebra.

翻译：泰勒簇由所有固定阶数的有理函数的泰勒多项式构成，其中变量数目以及分子与分母的次数固定。在单变量情形下，泰勒簇由汉克尔矩阵的秩约束刻画。其自然参数化的逆运算即为帕德逼近。我们研究泰勒簇的维数与定义理想。泰勒超曲面因其海森矩阵倾向于退化而在射影几何中具有重要价值。当变量数目为三个或更多时，存在维数小于参数数目的反常泰勒簇，本文利用交换代数中的弗勒贝格猜想对此加以解释。