The NumericalHilbert package 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.
Methods for computing and manipulating local dual spaces:
Auxiliary numerical linear algebra methods:
The algorithm used for computing truncated dual spaces is that of B. Mourrain ("Isolated points, duality and residues." J. Pure Appl. Algebra, 117/118:469–493, 1997). To compute the initial ideal and Hilbert regularity of positive dimensional ideals we use the algorithm of R. Krone ("Numerical algorithms for dual bases of positive-dimensional ideals." Journal of Algebra and Its Applications, 12(06):1350018, 2013.). This package depends on the package NAGtypes.
This documentation describes version 0.2 of NumericalHilbert.
The source code from which this documentation is derived is in the file NumericalHilbert.m2. The auxiliary files accompanying it are in the directory NumericalHilbert/.