Lower Bounds by Birkhoff Interpolation

Ignacio Garcia-Marco; Pascal Koiran

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arXiv · 1507.02015

Lower Bounds by Birkhoff Interpolation

Ignacio Garcia-Marco·Pascal Koiran

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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").

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Ignacio Garcia-Marco, Pascal Koiran. 2015-07-08. Lower Bounds by Birkhoff Interpolation. https://arxiv.org/abs/1507.02015

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