JournalsrmiVol. 31, No. 1pp. 291–302

Interpolation of ideals

  • Martín Avendaño

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

    Academia General Militar, Zaragoza, Spain
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Abstract

Let K\mathbb K denote an algebraically closed field. We study the relation between an ideal IK[x1,,xn]I\subseteq\mathbb K[x_1,\ldots,x_n] and its cross sections Iα=I+x1αI_\alpha=I+\langle x_1-\alpha\rangle. In particular, we study under what conditions II can be recovered from the set IS={(α,Iα):αS}I_S=\{(\alpha,I_\alpha)\,:\,\alpha\in S\} with SKS\subseteq\mathbb K. For instance, we show that an ideal I=iQiI=\bigcap_iQ_i, where QiQ_i is primary and QiK[x1]={0}Q_i\cap\mathbb K[x_1]=\{0\}, is uniquely determined by ISI_S when S=|S|=\infty. Moreover, there exists a function B(δ,n)B(\delta,n) such that, if II is generated by polynomials of degree at most~δ\delta, then II is uniquely determined by ISI_S when SB(δ,n)|S|\geq B(\delta,n). If II is also known to be principal, the reconstruction can be made when S2δ|S|\geq 2\delta, 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