We exhibit an unambiguous $k$-DNF formula that requires CNF width $\tilde{\Omega}(k^{1.5})$. Our construction is inspired by the board game Hex and it is vastly simpler than previous ones, which achieved at best an exponent of $1.22$. Our result is known to imply several other improved separations in query and communication ... more >>>

The orbit of an $n$-variate polynomial $f(\mathbf{x})$ over a field $\mathbb{F}$ is the set $\mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}$. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>>

In this paper we study polynomials in VP$_e$ (polynomial-sized formulas) and in $\Sigma\Pi\Sigma$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group GL$^{aff}_n({\mathbb F})$, are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms.

As ... more >>>