Can we define Taylor polynomials on algebraic curves?
Volume 118 / 2016
Annales Polonici Mathematici 118 (2016), 1-24
MSC: Primary 41A05, 41A63.
DOI: 10.4064/ap3996-9-2016
Published online: 10 October 2016
Abstract
We study the problem of finding the correct definition of a Taylor polynomial of degree $d$ at a point $\mathbf {a}$ for a function defined on an irreducible algebraic curve $V$ in $\mathbb {C}^2$. We show that a satisfactory definition can be given if and only if the point $\mathbf {a}$ is $d$-Taylorian, which holds for all but finitely many points of $V$. We provide an application to the study of the limit of certain Lagrangian interpolation operators when points coalesce.
