Finding Inverse Systems from Coordinates

TL;DR

This paper explores how to recover a polynomial $F$ from the coordinates of points in a Gorenstein scheme, extending Macaulay's classical correspondence between ideals and polynomials.

Contribution

It provides a partial method to describe the polynomial $F$ associated with an Artinian Gorenstein ideal of a reduced Gorenstein scheme.

Findings

$F$ can be obtained from the coordinates of the scheme's points

The work extends Macaulay's theorem to certain Gorenstein schemes

Provides a new perspective on inverse systems for zero-dimensional schemes

Abstract

Let $I$ be a homogeneous ideal in $R=\mathbb K[x_0,\ldots,x_n]$, such that $R/I$ is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance $I$ is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial $F$. A major question related to this result is to be able to describe $F$ in terms of the ideal $I$. In this note we give a partial answer to this question, by analyzing the case when $I$ is the Artinian reduction of the ideal of a reduced (arithmetically) Gorenstein zero-dimensional scheme $\Gamma\subset\mathbb P^n$. We obtain $F$ from the coordinates of the points of $\Gamma$.