# Interpolation of ideals

### Martín Avendaño

Academia General Militar, Zaragoza, Spain### Jorge Ortigas-Galindo

Academia General Militar, Zaragoza, Spain

## Abstract

Let $K$ denote an algebraically closed field. We study the relation between an ideal $I⊆K[x_{1},…,x_{n}]$ and its cross sections $I_{α}=I+⟨x_{1}−α⟩$. In particular, we study under what conditions $I$ can be recovered from the set $I_{S}={(α,I_{α}):α∈S}$ with $S⊆K$. For instance, we show that an ideal $I=⋂_{i}Q_{i}$, where $Q_{i}$ is primary and $Q_{i}∩K[x_{1}]={0}$, is uniquely determined by $I_{S}$ when $∣S∣=∞$. Moreover, there exists a function $B(δ,n)$ such that, if $I$ is generated by polynomials of degree at most $δ$, then $I$ is uniquely determined by $I_{S}$ when $∣S∣≥B(δ,n)$. If $I$ is also known to be principal, the reconstruction can be made when $∣S∣≥2δ$, and in this case, we prove that the bound is sharp.

## Cite this article

Martín Avendaño, Jorge Ortigas-Galindo, Interpolation of ideals. Rev. Mat. Iberoam. 31 (2015), no. 1, pp. 291–302

DOI 10.4171/RMI/834