Revista Matemática Iberoamericana

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Volume 31, Issue 1, 2015, pp. 291–302
DOI: 10.4171/RMI/834

Published online: 2015-03-05

Interpolation of ideals

Martín Avendaño[1] and Jorge Ortigas-Galindo[2]

(1) Academia General Militar, Zaragoza, Spain
(2) Academia General Militar, Zaragoza, Spain

Let $\mathbb K$ denote an algebraically closed field. We study the relation between an ideal $I\subseteq\mathbb K[x_1,\ldots,x_n]$ and its cross sections $I_\alpha=I+\langle x_1-\alpha\rangle$. 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 $I=\bigcap_iQ_i$, 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 $|S|\geq B(\delta,n)$. If $I$ is also known to be principal, the reconstruction can be made when $|S|\geq 2\delta$, and in this case, we prove that the bound is sharp.

Keywords: Polynomial interpolation, ideals, complexity

Avendaño Martín, Ortigas-Galindo Jorge: Interpolation of ideals. Rev. Mat. Iberoam. 31 (2015), 291-302. doi: 10.4171/RMI/834