Partition Rank and Analytic Rank are Uniformly Equivalent

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality, independently of the number of variables. Moreover, a quantitative improvement of our field cardinality requirement would imply that the ranks are equal up to 1+o(1) in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary we prove, over any mildly large field, the Polynomial Gowers Inverse Conjecture in the d vs. d-1 case.