On the linear independence of shifted powers

Pascal Koiran (1, 2) , Timothée Pecatte (1) , Ignacio García-Marco (3)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 MC2 - Modèles de calcul, Complexité, Combinatoire
3 I2M - Institut de Mathématiques de Marseille

We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by $F$. In particular, we give simple criteria ensuring that the dimension of the span of $F$ is at least $c.|F|$ for some absolute constant $c<1$. We also propose conjectures implying the linear independence of the elements of $F$. These conjectures are known to be true for the field of real numbers, but not for the field of complex numbers. The verification of these conjectures for complex polynomials directly imply new lower bounds in algebraic complexity.

Keywords: wronskian; linear independence; shifted power; Birkhoff interpolation; real and complex polynomial; Waring rank; shifted differential equation

Domains: Commutative Algebra [math.AC]; Computational Complexity [cs.CC]

Pascal Koiran, Timothée Pecatte, Ignacio García-Marco. On the linear independence of shifted powers. Journal of Complexity, 2018, 45, pp.67-82. ⟨hal-01520724v2⟩