Finding Inverse Systems from Coordinates

Abstract

Let $I$ be a homogeneous ideal in $R=\mathbb{K}[x_0,\dots ,x_n]$, 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 $\Gamma$.

8 pages. From the previous version the following changes took place: changed the title, corrected some typos, removed the section about the comparison between inverse systems and XL-algorithm, since this comparison does not improve neither of the two methods of solving systems of polynomial equations

References in corpus (1)

• Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property