Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\poly(\log q, d, p, n)\right) \, ,$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the applications to data structure upper bounds for polynomial evaluation and to an upper bound on the rigidity of Vandermonde matrices.