An Optimal Inverse Theorem

We prove that the partition rank and the analytic rank of tensors are equivalent up to a constant, over any large enough finite field (independently of the number of variables). The proof constructs rational maps computing a partition rank decomposition for successive derivatives of the tensor, 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.