  Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero
roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with
at most t+1 monomial terms. We prove, using an unified approach based on
Vandermonde determinants, that Bm(t,L)<=t^2 Bm(t,K) for any local field L with
a non-archimedean valuation v such that v(n)=0 for all non-zero integer n and
residue field K, and that Bm(t,K)<=(t^2-t+1)(p^f-1) for any finite extension
K/Qp with residual class degree f and ramification index e, assuming that
p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound
Bm(t,K)>=(2t-1)(p^f-1), which gives the sharp estimation Bm(2,K)=3(p^f-1) for
trinomials when p>2+e.
