Lower Bounds by Birkhoff Interpolation

TL;DR

This paper establishes near-optimal lower bounds on representing univariate polynomials as sums of powers of linear polynomials, using a novel Birkhoff interpolation approach that improves upon previous methods.

Contribution

Introduces a new lower bound technique based on Birkhoff interpolation, achieving bounds of order d for polynomial representations, surpassing prior sqrt(d) bounds.

Findings

Lower bounds of order d for polynomial sums of powers

New method based on Birkhoff interpolation

Improved bounds over previous Wronskian-based results

Abstract

In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order $\Omega$($\sqrt$ d), and were obtained from arguments based on Wronskian determinants and "shifted derivatives." We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as "lacunary polynomial interpolation").