For a sparse polynomial f(x) of high degree and few terms
over a non-algebraically closed field F, the number of F-rational
roots is often very small. In the case F is the real numbers, this
is the famous Descartes's rule. In the case that F is a finite field,
the situation is much more complicated. In this lecture, we discuss
some recent results and conjectures in this direction, both
theoretical and numerical.