Explicit Polynomial Sequences with Maximal Spaces of Partial Derivatives and a Question of K. Mulmuley

TY - JOUR

T1 - Explicit Polynomial Sequences with Maximal Spaces of Partial Derivatives and a Question of K. Mulmuley

AU - Gesmundo, Fulvio

AU - Landsberg, Joseph M.

PY - 2019/9

Y1 - 2019/9

N2 - We answer a question of K. Mulmuley. Efremenko et al. (Math. Comp., 2018) have 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.

AB - We answer a question of K. Mulmuley. Efremenko et al. (Math. Comp., 2018) have 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.

M3 - Journal article

SN - 1557-2862

VL - 15

JO - Theory of Computing

JF - Theory of Computing

ER -