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An exponential lower bound for the degrees of invariants of cubic forms and tensor actions (1902.10773v1)
Published 27 Feb 2019 in math.RT, cs.CC, math.AC, and math.RA
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 ${\rm SL}(V)$ on ${\rm Sym}3(V){\oplus 4}$, the space of $4$-tuples of cubic forms, and the second is the action of ${\rm SL}(V) \times {\rm SL}(W) \times {\rm SL}(Z)$ 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.