An exponential lower bound for the degrees of invariants of cubic forms and tensor actions

Abstract

Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's null cone. We consider two actions: The first is the action of $\mathrm{SL}(V)$ on ${\mathrm{Sym}}^3(V{)}^{\oplus 4}$, the space of $4$-tuples of cubic forms, and the second is the action of $\mathrm{SL}(V)\times \mathrm{SL}(W)\times \mathrm{SL}(Z)$ on the tensor space $(V\otimes W\otimes Z{)}^{\oplus 9}$. In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone.