  We design nearly-linear time numerical algorithms for the problem of
multivariate multipoint evaluation over the fields of rational, real and
complex numbers. We consider both \emph{exact} and \emph{approximate} versions
of the algorithm. The input to the algorithms are (1) coefficients of an
$m$-variate polynomial $f$ with degree $d$ in each variable, and (2) points
$a_1,..., a_N$ each of whose coordinate has value bounded by one and
bit-complexity $s$.
  * Approximate version: Given additionally an accuracy parameter $t$, the
algorithm computes rational numbers $\beta_1,\ldots, \beta_N$ such that
$|f(a_i) - \beta_i| \leq \frac{1}{2^t}$ for all $i$, and has a running time of
$((Nm + d^m)(s + t))^{1 + o(1)}$ for all $m$ and all sufficiently large $d$.
  * Exact version (when over rationals): Given additionally a bound $c$ on the
bit-complexity of all evaluations, the algorithm computes the rational numbers
$f(a_1), ... , f(a_N)$, in time $((Nm + d^m)(s + c))^{1 + o(1)}$ for all $m$
and all sufficiently large $d$. .
  Prior to this work, a nearly-linear time algorithm for multivariate
multipoint evaluation (exact or approximate) over any infinite field appears to
be known only for the case of univariate polynomials, and was discovered in a
recent work of Moroz (FOCS 2021). In this work, we extend this result from the
univariate to the multivariate setting. However, our algorithm is based on
ideas that seem to be conceptually different from those of Moroz (FOCS 2021)
and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar &
Umans (FOCS 2022) for multivariate multipoint evaluation over finite fields,
and known efficient algorithms for the problems of rational number
reconstruction and fast Chinese remaindering in computational number theory.
