Diversification improves interpolation

TL;DR

This paper introduces improved algorithms for polynomial interpolation over finite fields and complex numbers, utilizing randomization and diversity techniques to enhance efficiency and stability without relying on root finding.

Contribution

It presents a novel randomized interpolation algorithm that reduces evaluations over large finite fields and offers the first provably stable numerical interpolation method for sparse complex polynomials.

Findings

Fewer evaluations needed for finite field interpolation

First stable algorithm for numerical sparse polynomial interpolation

Demonstrated practical efficiency through trial implementations

Abstract

We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the best-known algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations. Using related techniques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algorithm (in the sense of relative error) for this problem, at the cost of modestly more evaluations. A key new technique is a randomization which makes all coefficients of the unknown polynomial distinguishable, producing what we call a diverse polynomial. Another departure from most previous approaches is that our algorithms do not rely on root finding as a subroutine. We show how these improvements affect the practical performance with trial implementations.