We show an $\Omega \left(\frac{n^3}{(\ln n)^3}\right)$ lower bound on
the number of gates of any depth three ($\Sigma \Pi \Sigma$) arithmetic circuit computing an
explicit multilinear polynomial in $n$ variables over any field. This
improves upon the previously known quadratic lower bound by Shpilka
and Wigderson (1999, 2001).

In the previous version, the lower bound was for the number of edges which we now strengthen to a lower bound on the number of gates in the depth three circuit.

We show an $\Omega \left(\frac{n^3}{(\ln n)^2}\right)$ lower bound on the size of any depth three ($\sum\prod\sum$) arithmetic circuit computing an explicit multilinear polynomial in $n$ variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson.

We just fixed a small problem with a TeX command in the abstract.

We show an $\Omega \left(\frac{n^3}{(\ln n)^2}\right)$ lower bound on the size of any depth three ($\SPS$) arithmetic circuit computing an explicit multilinear polynomial in $n$ variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson.