We construct pseudorandom generators with improved seed length for
several classes of tests. First we consider the class of read-once
polynomials over GF(2) in $m$ variables. For error $\e$ we obtain seed
length $\tilde O (\log(m/\e)) \log(1/\e)$, where $\tilde O$ hides lower-order
terms. This is optimal up to the factor ... more >>>

A map $f:[n]^{\ell}\to[n]^{n}$ has locality $d$ if each output symbol
in $[n]=\{1,2,\ldots,n\}$ depends only on $d$ of the $\ell$ input
symbols in $[n]$. We show that the output distribution of a $d$-local
map has statistical distance at least $1-2\cdot\exp(-n/\log^{c^{d}}n)$
from a uniform permutation of $[n]$. This seems to be the ... more >>>

We characterize the size of monotone span programs computing certain "structured" boolean functions by the Nullstellensatz degree of a related unsatisfiable Boolean formula.

This yields the first exponential lower bounds for monotone span programs over arbitrary fields, the first exponential separations between monotone span programs over fields of different ... more >>>