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.

翻译：我们设计了在有理数、实数和复数域上针对多元多点求值问题的近线性时间数值算法。我们同时考虑了算法的**精确**版本和**近似**版本。算法的输入为：(1) 一个每变量次数为 $d$ 的 $m$ 元多项式 $f$ 的系数，以及 (2) 坐标值均不超过 1 且比特复杂度为 $s$ 的点 $a_1,\ldots, a_N$。 * **近似版本**：若额外给定精度参数 $t$，该算法可计算有理数 $\beta_1,\ldots,\beta_N$，使得对所有 $i$ 满足 $|f(a_i) - \beta_i| \leq \frac{1}{2^t}$，且对于所有 $m$ 及所有足够大的 $d$，运行时间为 $((Nm + d^m)(s + t))^{1 + o(1)}$。 * **精确版本**（适用于有理数域）：若额外给定所有求值结果的比特复杂度上界 $c$，该算法可计算有理数 $f(a_1), \ldots, f(a_N)$，且对于所有 $m$ 及所有足够大的 $d$，运行时间为 $((Nm + d^m)(s + c))^{1 + o(1)}$。 在此工作之前，对任意无限域上的多元多点求值（精确或近似）的近线性时间算法似乎仅对单变量多项式已知，且由 Moroz (FOCS 2021) 在近期工作中发现。本研究将该结果从单变量场景推广至多元场景。然而，我们的算法基于的概念性思路与 Moroz (FOCS 2021) 截然不同，并关键依赖于 Bhargava、Ghosh、Guo、Kumar 和 Umans (FOCS 2022) 在有限域上多元多点求值的近期算法，以及计算数论中有理数重构与快速中国剩余定理的已知高效算法。