Lower Bounds by Birkhoff Interpolation

Ignacio Garcia-Marco (1) , Pascal Koiran (1, 2)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 MC2 - Modèles de calcul, Complexité, Combinatoire

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 Ω(√ 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").

Mots clés: lower bounds; arithmetic circuits; polynomial interpolation

Domaines: Complexité [cs.CC]

Ignacio Garcia-Marco, Pascal Koiran. Lower Bounds by Birkhoff Interpolation. Journal of Complexity, 2017. ⟨ensl-01172296⟩