Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t⩾0. Denote by Bm(t,K) the supremum of the number of roots in K⁎, counted with multiplicities, that can have 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)⩽t2Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that v|Z≠0≡0 and residue field K, and that Bm(t,K)⩽(t2−t+1)(pf−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)(pf−1), which gives the sharp estimation Bm(2,K)=3(pf−1) for trinomials when p>2+e.