Robust polynomial regression up to the information theoretic limit

We consider the problem of robust polynomial regression, where one receives samples (xi, yi) that are usually within σ of a polynomial y = p(x), but have a ρ chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate p only when ρ < frac1log d. We give an algorithm that works for the entire feasible range of ρ < 1/2, while simultaneously improving other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a 1.09 approximation is impossible even with infinitely many samples.