Polynomial bound for partition rank in terms of analytic rank - Milićević, Luka (Geometric and functional analysis, 2019)
Let G 1 , … , G k be vector spaces over a finite field F = F q with a non-trivial additive character χ . The analytic rank of a multilinear form α : G 1 × ⋯ × G k → F is defined as arank ( α ) = - log q E x 1 ∈ G 1 , … , x k ∈ G k χ ( α ( x 1 , … , x k ) ) . The partition rank prank ( α ) of α is the smallest number of maps of partition rank 1 that add up to α , where a map is of partition rank 1 if it can be written as a product of two multilinear forms, depending on different coordinates. It is easy to see that arank ( α ) ≤ O ( prank ( α ) ) and it has been known that prank ( α ) can be bounded from above in terms of arank ( α ) . In this paper, we improve the latter bound to polynomial, i.e. we show that there are quantities C , D depending on k only such that prank ( α ) ≤ C ( arank ( α ) D + 1 ) . As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.