An exponential lower bound for the degrees of invariants of cubic forms and tensor actions
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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 SL(V) on Sym^3(V)^⊕ 4, the space of 4-tuples of cubic forms, and the second is the action of SL(V) × SL(W) × SL(Z) on the tensor space (V ⊗ W ⊗ Z)^⊕ 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.
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