Polystability in positive characteristic and degree lower bounds for invariant rings

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our based on Kempf’s theory of optimal subgroups and we make some improvements simplify the from computational perspective. exhibit our many examples particular, give an algorithm to decide if symmetric polynomial $n$-variables has $\operatorname{SL}\_n$-orbit. As important application, prove exponential lower bounds maximal degree system generators invariant rings two actions are perspective Geometric Complexity Theory (GCT). The first action $\operatorname{SL}(V)$ $S^3(V)^{\oplus 3}$, space $3$-tuples cubic forms, second $\operatorname{SL}(V) \times \operatorname{SL}(W) \operatorname{SL}(Z)$ tensor $(V \otimes W Z)^{\oplus 5}$. In both these cases, bound invariants generate ring or define null cone.