Title: Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley,
Abstract: We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman]
it was shown that the method of shifted partial derivatives cannot be used to
separate the padded permanent from the determinant. Mulmuley asked if this
"no-go" result could be extended to a model without padding. We prove this is
indeed the case using the iterated matrix multiplication polynomial. We also
provide several examples of polynomials with maximal space of partial
derivatives, including the complete symmetric polynomials. We apply Koszul
flattenings to these polynomials to have the first explicit sequence of
polynomials with symmetric border rank lower bounds higher than the bounds
attainable via partial derivatives.