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

2019/02/27 by Derksen, Harm, Makam, Visu · 1 citation
#13A50 #14L24 #15A72 #17B22 #37J15 #Commutative Algebra (math.AC) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1902.10773

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 Sym³(V)^{⊕ 4}, the space of 4-tuples of cubic forms, and the second is the action of \rm SL(V) × \rm SL(W) × \rm 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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