Deterministic Approximate Counting of Depth-2 Circuits

We describe deterministic algorithms which for a given depth-2 circuit $F$ approximate the probability that on a random input $F$ outputs a specific value $\alpha$. Our approach gives an algorithm which for a given $GF[2]$ multivariate polynomial $p$ and given $\varepsilon >0$ approximates the number of zeros of $p$ within a multiplicative factor $1+ \varepsilon$. The algorithm runs in time $exp(exp(O({\sqrt{log(n/\varepsilon)}))$, where $n$ is the size of the circuit. We also obtain an algorithm which given a DNF formula $F$ and $\varepsilon >0$ approximates the number of satisfying assignments of $F$ within a factor of $1+\varepsilon$ and runs in time $exp(O(log(n/\varepsilon))^4))$.