Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

Understand

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions.

• Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of $\SL_n(\C)$-representations.
• We provide simple examples that disprove this conjecture (under standard complexity assumptions).
• We develop a general framework, denoted \emph{algebraic circuit search problems}, that captures many important problems in algebraic complexity and computational invariant theory.