Taylor Polynomials of Rational Functions
Algebraic Geometry
2023-04-04 v1 Symbolic Computation
Commutative Algebra
Abstract
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.
Cite
@article{arxiv.2304.00712,
title = {Taylor Polynomials of Rational Functions},
author = {Aldo Conca and Simone Naldi and Giorgio Ottaviani and Bernd Sturmfels},
journal= {arXiv preprint arXiv:2304.00712},
year = {2023}
}
Comments
20 pages