Agrawal and Vinay (FOCS 2008) have recently shown that an exponential lower bound for depth four homogeneous circuits with bottom layer of $\times$ gates having sublinear fanin translates to an exponential lower bound for a general arithmetic circuit computing the permanent. Motivated by this, we examine the complexity of computing ... more >>>

We consider arithmetic formulas consisting of alternating layers of addition $(+)$ and multiplication $(\times)$ gates such that the fanin of all the gates in any fixed layer is the same. Such a formula $\Phi$ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree ... more >>>

Guo and Wang gave a deterministic polynomial-time black-box identity test for $\Sigma^{[3]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of arbitrary characteristic, for constant $\delta$, assuming that one product gate is squarefree. This note communicates an observation suggested by a large language model: the squarefreeness assumption can be removed by combining the normalization argument ... more >>>