On Polynomial Interpolation on Arbitrary Varieties

TL;DR

This paper introduces and studies polynomial interpolation on arbitrary algebraic varieties, exploring solvability conditions and linking the problem to boundary value problems, extending previous results in the field.

Contribution

It is the first work to systematically analyze polynomial interpolation on arbitrary varieties and connect it to boundary value problem solutions.

Findings

Established solvability conditions for polynomial interpolation on varieties.

Linked polynomial interpolation to boundary value problems.

Extended classical results to more general algebraic varieties.

Abstract

To the best of our knowledge this paper is the first attempt to introduce and study polynomial interpolation of the polynomial data given on arbitrary varieties. In the first part of the paper we present results on the solvability of such problems. In the second part of the paper we relate the interpolation problem to polynomial solution of some boundary values problems. In particular, we extend a result of W. K. Hayman and Z. G. Shanidze.