Fast interpolation of sparse multivariate polynomials

TL;DR

This paper introduces a quasi-optimal algorithm for efficiently interpolating sparse multivariate polynomials with integer coefficients from modular evaluations, advancing the computational methods in polynomial reconstruction.

Contribution

It presents the first quasi-optimal algorithm for sparse interpolation of multivariate polynomials from modular black box evaluations.

Findings

Algorithm achieves quasi-optimal complexity

Effective reconstruction from modular evaluations

Improves computational efficiency in polynomial interpolation

Abstract

Consider a sparse multivariate polynomial f with integer coefficients. Assume that f is represented as a "modular black box polynomial", e.g. via an algorithm to evaluate f at arbitrary integer points, modulo arbitrary positive integers. The problem of sparse interpolation is to recover f in its usual sparse representation, as a sum of coefficients times monomials. For the first time we present a quasi-optimal algorithm for this task.