Fast Multipoint-Evaluation of Bivariate Polynomials

TL;DR

This paper presents a novel algorithm for efficiently evaluating bivariate polynomials at multiple points with sub-quadratic total complexity, significantly improving computational efficiency for large-scale polynomial evaluations.

Contribution

It introduces a new method for multipoint evaluation of bivariate polynomials with sublinear amortized cost per point, extending univariate techniques to two variables.

Findings

Total evaluation complexity is O(n^{2.667}) operations.

Amortized cost per point is O(N^{0.334}) where N is the input size.

Method significantly reduces computational effort for large polynomial evaluations.

Abstract

We generalize univariate multipoint evaluation of polynomials of degree n at sublinear amortized cost per point. More precisely, it is shown how to evaluate a bivariate polynomial p of maximum degree less than n, specified by its n^2 coefficients, simultaneously at n^2 given points using a total of O(n^{2.667}) arithmetic operations. In terms of the input size N being quadratic in n, this amounts to an amortized cost of O(N^{0.334}) per point.