Interpolation of Ideals

TL;DR

This paper investigates how ideals in polynomial rings over algebraically closed fields can be reconstructed from their cross sections, establishing conditions and bounds for unique determination based on the size and properties of the ideal.

Contribution

It introduces new conditions and bounds under which an ideal can be uniquely recovered from its cross sections, including cases for primary, principal, and degree-bounded ideals.

Findings

Ideals with primary components are uniquely determined by their cross sections over infinite sets.

A function B(d,n) bounds the size of sets needed for reconstruction of degree-d generated ideals.

Principal ideals can be reconstructed with at least 2d cross sections, and this bound is proven to be sharp.

Abstract

Let K denote an algebraically closed field. We study the relation between an ideal I in K[x1,...,xn] and its cross sections I_a=I+<x1-a>. In particular, we study under what conditions I can be recovered from the set I_S={(a,I_a):a in S} with S a subset of K. For instance, we show that an ideal I=cap_i Q_i, where Q_i is primary and Q_i cap K[x1]={0}, is uniquely determined by I_S when S is infinite. Moreover, there exists a function B(d,n) such that, if I is generated by polynomials of degree at most d, then I is uniquely determined by I_S when |S|>=B(d,n). If I is also known to be principal, the reconstruction can be done when |S|>=2d, and in this case, we prove that the bound is sharp.