# extrasForTheKernel -- constructions of prime fields and artinian reductions

## Description

Overview: A collection of a few functions that will be handy in many cases.

Perhaps the most interesting is the funcion randomKRationalPoint, which for a finite ground field K of moderate size, will find a K-rational points on a positive dimensional projective algebraic set X under the asumption that the degree of X is small compared to the char K.

Setup:

This package requires Macaulay2 version 1.4 or newer.

Install this Package by doing

installPackage("extrasForTheKernel")

## Version

This documentation describes version 0.1.1 of extrasForTheKernel.

## Source code

The source code from which this documentation is derived is in the file extrasForTheKernel.m2.

## Exports

• Functions and commands
• artinianReduction -- compute an artinian quotient of an Cohen-Macaulay ideal
• criticalKoszulDegrees -- compute all d with $\sum_{p+q=d} (-1)^q dim K_{p,q}(S/I)==0$ for an artinian ideal I
• expectedSyzygies -- compute the expected Betti numbers of an artinian module
• koszulMap -- compute the Koszul homology map whose kernel correspond the Koszulzycles for K_(p,q)(S/I)
• koszulMaps -- compute all Koszul maps of total degree p for an artinian ideal I