Interpolation of ideals

Abstract

Let $\mathbb{K}$ denote an algebraically closed field. We study the relation between an ideal $I\subseteq \mathbb{K}[x_1,\dots ,x_n]$ and its cross sections $I_{\alpha }=I+⟨x_1-\alpha ⟩$. In particular, we study under what conditions $I$ can be recovered from the set $I_S=\{(\alpha ,I_{\alpha }):\alpha \in S\}$ with $S\subseteq \mathbb{K}$. For instance, we show that an ideal $I={\bigcap }_i{Q}_i$, where $Q_i$ is primary and $Q_i\cap \mathbb{K}[x_1]=\{0\}$, is uniquely determined by $I_S$ when $|S|=\infty$. Moreover, there exists a function $B(\delta ,n)$ such that, if $I$ is generated by polynomials of degree at most $\delta$, then $I$ is uniquely determined by $I_S$ when $|S|\ge B(\delta ,n)$. If $I$ is also known to be principal, the reconstruction can be made when $|S|\ge 2\delta$, and in this case, we prove that the bound is sharp.