# On the linear independence of shifted powers

## TL;DR

This paper establishes criteria for the linear independence of shifted powers, providing bounds on the span dimension and proposing conjectures that hold over real numbers but remain open over complex numbers.

## Contribution

It introduces broadly applicable criteria for linear independence of shifted powers and proposes conjectures that differentiate between real and complex fields.

## Key findings

- Criteria ensuring span dimension at least 0.5|F|
- Conjectures on linear independence over reals and complex numbers
- Results applicable to finite families of shifted powers

## Abstract

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.