We show that a global holomorphic section of $\mathcal{O}\left(d\right)$ 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 ${\mathcal{O}}_X$. 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.

中文翻译：

---

多项式插值和剩余电流

我们证明了一个全局全纯截面$\mathcal{○}\left(d\right)$受限于一个封闭的复子空间$X\subset {\mathbb{磷}}^n$有一个插值当且仅当它满足一组矩条件，这些条件涉及与局部自由分辨率相关的残余电流${\mathcal{○}}_X$. 当X是${\mathbb{C}}^n\subset {\mathbb{磷}}^n$这可以解释为X上的函数必须满足的一组线性条件，以便具有最多d次的多项式插值。