  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 depending on the analytic rank. Moreover, we show that a plausible
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, ideally over fixed finite
fields, is a central question in additive combinatorics, and was reiterated by
multiple authors. As a corollary we prove, allowing the field to depend on the
value of the norm, the Polynomial Gowers Inverse Conjecture in the d vs. d-1
case.
