Numerical Computation of the Hilbert Function and Regularity of a Zero Dimensional Scheme

Abstract

Let $R = \mathbb{C}[x_{1},\ldots,x_{N}]$ and let $F =\{ f_{1},\ldots,f_{t}\} \subset R$ be a set of generators for an ideal I. Let $Y =\{ y_{1},\ldots,y_{\ell}\} \subset {\mathbb{C}}^{N}$ be a subset of the set of isolated solutions of the zero locus of F. Let $\mathfrak{m}_{y_{i}}$ denote the maximal ideal of y _i and let $\mathcal{P}_{y_{i}}$ denote the $\mathfrak{m}_{y_{i}}$-primary component of I. Let $J = \cap _{i=1}^{l}\mathcal{P}_{y_{i}}$ and let $\mathcal{Z}$ denote the corresponding zero dimensional subscheme supported on Y. This article presents a numerical algorithm for computing the Hilbert function and the regularity of $\mathcal{Z}$. In addition, the algorithm produces a monomial basis for R∕J. The input for the algorithm is the polynomial system F and a numerical approximation of each element in Y.