Guy Moshkovitz (CUNY): An Optimal Inverse Theorem

The partition rank and the analytic rank of a tensor measure algebraic structure and bias, respectively. We prove that they are equivalent up to a constant, over any large enough finite field (independently of the number of variables) of any characteristic. The proof constructs rational maps computing a rank decomposition for successive derivatives, on a carefully chosen subset of the kernel variety associated with the tensor. Proving the equivalence between these two quantities is the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, and was reiterated by multiple authors.Joint work with Alex Cohen.