An Optimal Inverse Theorem

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over any large enough finite field. The proof constructs rational maps computing a partition rank decomposition for successive derivatives of the tensor, on an open subset of the kernel variety associated with the tensor. This largely settles the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, which was reiterated by Kazhdan and Ziegler, Lovett, and others.