TITLE: Roots of lacunary polynomials over a finite field
QUESTION [13 upvotes]: If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots.

Does this fact have any standard name / reference / proof / refinements / extensions?

REPLY [5 votes]: This is just an uncertainty principle for the cyclic group $\mathbb F_q^*$; see, e.g. this paper by Tao.