Taylorian points of an algebraic curve and bivariate Hermite interpolation

Abstract

We introduce and study the notion of Taylorian points of algebraic curves in C 2 , which enables us to define intrinsic Taylor interpolation polynomials on curves. These polynomials in turn lead to the construction of a wellbehaved Hermitian scheme on curves, of which we give several examples. We show that such Hermitian schemes can be collected to obtain Hermitian bivariate polynomial interpolation schemes.

On D-invariant points and local Taylor interpolation on algebraic hypersurfaces in RN

“…The polynomial p is called the L-Taylor interpolation polynomial of f at a to the order d and denoted by T d L ( f ). The following theorem is an extension of [7,Theorem 3.2]. A similar result was obtained in [13] for the complex case.…”

“…In [6,7], Bos and Calvi studied polynomial interpolation on algebraic hypersurfaces in C N . They obtained many beautiful results.…”

“…In [7], the authors continued studying the least space and the local Taylor interpolation in C 2 . They introduced the notation of Taylorian points on algebraic curves.…”

“…, m d (V) − 1}. It is showed in [7], for d ≥ 1, every point on irreducible quadratic curve in C 2 is d-Taylorian and all but finitely many points on an irreducible algebraic curve in C 2 are d-Taylorian. The local Taylor interpolation at a Taylorian point has many interesting properties.…”

“…The local Taylor interpolation at a Taylorian point has many interesting properties. Theorem 3.2 in [7] points out that a regular point a ∈ V is Taylorian if and only if ker T d L is an ideal in the space of analytic functions at a on V ⊂ C 2 . It is an important characterization of a d-Taylorian point.…”