TITLE: What is the ring of invariants of GL acting on quaternary cubic forms?
QUESTION [6 upvotes]: Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute the ring of invariants? The case of personal interest is when $K$ is a finite field but the the answer over $\mathbb{C}$ would of course be more than useful.

REPLY [4 votes]: Don't be misled into thinking that the answer over $\Bbb C$ tells you very much about the answer over your finite field $K$.  The space of cubic forms in 4 variables is 20 dimensional.  The group $GL(4,K)$ is a finite group and so the ring of invariants you seek has Krull dimension 20.  In particular, it has at least 20 generators.  Probably, however, it has a few thousand generators if not many more.  
There are computer algebra packages (e.g. Magma) that include routines that can compute generators for your ring of invariants in principle, but in practice you will find that they run out of memory long before getting very far on a problem of this size.  They will be able to compute vector space bases for the invariants in low degrees.  On the other hand, it is doubtful that having a list of thousands of generators (or even just their degrees) for the ring of invariants will help you very much.  On the other hand if you want to know something about the properties of the ring of invariants as a ring, then it's possible that may be known.