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 $\mathbf{a}_1,\ldots, \mathbf{a}_N$ 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 $|{f(\mathbf{a}_i) - \beta_i} \leq \sfrac{1}{2^t}$ for all $i$, and has a running time of $\left((Nm + d^m)(s + t)\right)^{1 + o(1)}$ 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 $f(\mathbf{a}_1), \ldots, f(\mathbf{a}_N)$, in time $\left((Nm + d^m)(s + c)\right)^{1 + o(1)}$ for all $m$ and all sufficiently large $d$.

Our results also naturally extend to the case when the input is over the field of real or complex numbers under an appropriate standard model of representation of field elements in such fields.

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 [Proc. 62nd 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 and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans [Proc. 63rd 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.

Fixing some references, and a few cosmetic changes

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 $\mathbf{a}_1,\ldots, \mathbf{a}_N$ 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 $|{f(\mathbf{a}_i) - \beta_i} \leq \sfrac{1}{2^t}$ for all $i$, and has a running time of $\left((Nm + d^m)(s + t)\right)^{1 + o(1)}$ 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 $f(\mathbf{a}_1), \ldots, f(\mathbf{a}_N)$, in time $\left((Nm + d^m)(s + c)\right)^{1 + o(1)}$ for all $m$ and all sufficiently large $d$.

Our results also naturally extend to the case when the input is over the field of real or complex numbers under an appropriate standard model of representation of field elements in such fields.

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 [Proc. 62nd 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 and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans [Proc. 63rd 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.