Polynomial interpolation and residue currents

TL;DR

This paper establishes a connection between polynomial interpolation on complex subspaces and residue currents, providing a criterion involving moment conditions for the existence of interpolants.

Contribution

It introduces a residue current-based criterion for polynomial interpolation on complex subspaces, linking algebraic geometry with interpolation theory.

Findings

Interpolation characterized by moment conditions involving residue currents

Residue currents encode geometric and algebraic information for interpolation

Applicable to finite point sets in complex space for polynomial interpolation

Abstract

We show that a global holomorphic section of $\mathscr{O}(d)$ restricted to a closed complex subspace $X \subset \mathbb{P}^n$ has an interpolant if and only if it satisfies a set of moment conditions that involves a residue current associated with a locally free resolution of $\mathscr{O}_X$. When $X$ is a finite set of points in $\mathbb{C}^n \subset \mathbb{P}^n$ this can be interpreted as a set of linear conditions that a function on $X$ has to satisfy in order to have a polynomial interpolant of degree at most $d$.