  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.
