Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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A. Conca - S. Naldi - G. Ottaviani - B. Sturmfels

Taylor Polynomials of Rational Functions

created by naldi on 20 Jul 2023


Accepted Paper

Inserted: 20 jul 2023
Last Updated: 20 jul 2023

Journal: Acta Mathematica Vietnamica
Year: 2023

ArXiv: 2304.00712 PDF


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.

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