Hilbert Functions, Shallow Circuits, and Other Approaches to GCT
summary
Goal is separating permanent from determinant by finding GL-modules of polynomials that vanish on the determinant orbit closure but not on the (padded) permanent. Standard tool shifted partial derivatives gives rank tests (determinant partials scale like ~binomial(n,k)^2, monomials ~binomial(n,k)) but provably fails to separate when the padding m < n/2. New geometric embeds (a Kazul flattening / exterior-type map) boost Waring/border-rank bounds roughly 2x improvement, and staircase-type partitions push the method further, giving the sharp product polynomial rank ~2^{n-1}, but determinant and permanent still sit at similar orders so the gap remains unresolved.
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