We prove that every De Morgan formula with $n$ leaves has a pointwise $1/3$-approximating real polynomial of degree $O(\sqrt n)$ and coefficient $\ell_1$-norm $2^{O(\sqrt n)}$. The standard approximate-degree theorem for formulas gives the same degree bound, but only yields the weaker coefficient estimate $2^{O(\sqrt n\log n)}$.

Our proof constructs, for ... more >>>

We show that under a plausible number-theoretic conjecture, for any constant $s$ there exists an $s$-server private information retrieval (PIR) protocol that on an $n$-bit database requires communication $\exp(O((\log n)^{1/s} (\log \log n)^{1-1/s}))$. Previous constructions attaining the same communication required $2^{O(s)}$ servers. Our number-theoretic conjecture is implied by existing conjectures, ... more >>>

We describe a method that lifts an arbitrary polynomial $f$ with sparsity $s$ to a polynomial that requires a read-once oblivious algebraic program of width $s$ for every variable order. To do so, we introduce a technique for constructing a gadget based on erasure codes over finite fields, where each ... more >>>