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

## TL;DR

This paper establishes exponential lower bounds on the degrees of invariants needed to generate invariant rings and define null cones for specific cubic form and tensor actions, advancing understanding of invariant complexity.

## Contribution

The paper introduces a new method based on the Grosshans Principle to prove exponential lower bounds for invariant degrees in two key tensor actions.

## Key findings

- Proves exponential lower bounds for invariants of cubic forms.
- Establishes exponential lower bounds for tensor actions involving multiple SL groups.
- Provides a systematic approach to lower bound proofs in invariant theory.

## 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.