# Package invarFC0

This article is about a function from ApCoCoA-2. |

This page is about the InvFC0 package, which is about **inv**ariant theory for **f**inite groups in **c**haracteristic **0**. For a complete function list, see Category:Package invarFC0.

## Mathematical Definitions

Let be the polynomial ring over the rational numbers in *n* indeterminates and let be the column vector containing the indeterminates of *P*. Then a matrix group acts on via

where

for . David Hilbert has proven in 1890 that the set

is a finitely generated graded *Q*-subalgebra of *P* as long as *G* is a finite group.
This package provides a function for computing a minimal generating set of this subalgebra and a function for computing a SAGBI basis of it.

For details about the algorithms, see Harm Derksen, Gregor Kemper: *Computational invariant theory* or B. Andraschko, *Computational Invariant Theory For Finite Matrix Groups*.

## Package Description

Given a list `G`

of matrices over a polynomial ring `P`

that form a multiplicative group and a positive integer `d`

, one can call the functions

InvFC0.SAGBI(G); InvFC0.SAGBI(G,d); InvFC0.MinGens(G); InvFC0.MinGens(G,d);

In the first case, a SAGBI basis of is computed if a finite one exists. If no finite SAGBI basis exists, this function does not terminate, therefore one can use the second call to specify a degree `d`

such that a `d`

-truncated SAGBI basis is computed. Note that as it is the case for Gröbner bases, the term ordering is given by the ring `P`

.

The last two calls work in the same way, but produce a minimal homogeneous system of generators for . By Hilbert's finiteness theorem, both of them terminate, but it is also possible to specify a degree to obtain a truncated homogeneous generating system.