Numerical Hilbert functions for Macaulay2

TL;DR

The paper introduces NumericalHilbert, a Macaulay2 package that uses numerically stable algorithms with floating point arithmetic to compute local scheme data, offering an alternative to symbolic methods.

Contribution

It presents new numerical algorithms for computing local dual spaces, initial ideals, and local Hilbert functions within Macaulay2, enabling stable computations over complex numbers.

Findings

Algorithms are numerically stable and efficient.

Able to compute local Hilbert functions and regularity.

Provides an alternative to symbolic methods.

Abstract

The NumericalHilbert package for Macaulay2 includes algorithms for computing local dual spaces of polynomial ideals, and related local combinatorial data about its scheme structure. These techniques are numerically stable, and can be used with floating point arithmetic over the complex numbers. They provide a viable alternative in this setting to purely symbolic methods such as standard bases. In particular, these methods can be used to compute initial ideals, local Hilbert functions and Hilbert regularity.