Shpilka and Wigderson (CCC 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth three arithmetic circuits with bounded bottom fanin over a field $\mathbb{F}$ of characteristic zero. We resolve this problem by proving a $N^{\Omega(\frac{d}{\tau})}$ lower bound for (nonhomogeneous) depth three arithmetic circuits with bottom fanin ... more >>>

We show an $\widetilde{\Omega}(n^{2.5})$ lower bound for general depth four arithmetic circuits computing an explicit $n$-variate degree $\Theta(n)$ multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey by Shpilka and Yehudayoff (FnT-TCS, 2010), no super-quadratic lower bound was known for depth four ... more >>>