Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible ... more >>>

We show that there is a sequence of explicit multilinear polynomials $P_n(x_1,\ldots,x_n)\in \mathbb{R}[x_1,\ldots,x_n]$ with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for $P_n$ must have size $\exp(\Omega(n)).$ This builds on (and strengthens) a result of Yehudayoff (2018) who showed a lower bound of $\exp(\tilde{\Omega}(\sqrt{n})).$

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Following the seminal work of [Williams, J. ACM 2014], in a recent breakthrough, [Murray and Williams, STOC 2018] proved that NQP (non-deterministic quasi-polynomial time) does not have polynomial-size ACC^0 circuits.

We strengthen the above lower bound to an average case one, by proving that for all constants c, ... more >>>