  We give deterministic black-box polynomial identity testing algorithms for
multilinear read-once oblivious algebraic branching programs (ROABPs), in
n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the
variables. This is the first sub-exponential time algorithm for this model.
Furthermore, our result has no known analogue in the model of read-once
oblivious boolean branching programs with unknown order, as despite recent work
there is no known pseudorandom generator for this model with sub-polynomial
seed-length (for unbounded-width branching programs).
  This result extends and generalizes the result of Forbes and Shpilka that
obtained a n^(lg n)-time algorithm when given the order. We also extend and
strengthen the work of Agrawal, Saha and Saxena that gave a black-box algorithm
running in time exp((lg n)^d) for set-multilinear formulas of depth d. We note
that the model of multilinear ROABPs contains the model of set-multilinear
algebraic branching programs, which itself contains the model of
set-multilinear formulas of arbitrary depth. We obtain our results by
recasting, and improving upon, the ideas of Agrawal, Saha and Saxena. We phrase
the ideas in terms of rank condensers and Wronskians, and show that our results
improve upon the classical multivariate Wronskian, which may be of independent
interest.
  In addition, we give the first n^(lglg n) black-box polynomial identity
testing algorithm for the so called model of diagonal circuits. This model,
introduced by Saxena has recently found applications in the work of Mulmuley,
as well as in the work of Gupta, Kamath, Kayal, Saptharishi. Previously work
had given n^(lg n)-time algorithms for this class. More generally, our result
holds for any model computing polynomials whose partial derivatives (of all
orders) span a low dimensional linear space.
