Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, but $x^2+px \bmod p^2$ has exponentially many factors! We present the first randomized poly($\deg ... more >>>

Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size $s$, we show that all its factors can be computed by arithmetic branching programs of size $\text{poly}(s)$. Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was ... more >>>

We show that Reed-Solomon codes of dimension $k$ and block length $n$ over any finite field $\mathbb{F}$ can be deterministically list decoded from agreement $\sqrt{(k-1)n}$ in time $\text{poly}(n, \log |\mathbb{F}|)$.

Prior to this work, the list decoding algorithms for Reed-Solomon codes, from the celebrated results of ... more >>>