CONICET | Buscador de Institutos y Recursos Humanos

Let $K$ be a field and $tgeq 0$. Denote by $B_m(t,K)$ the supremum of the number of roots in $K^ast$, 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 $B_m(t,L)leq t^2 B_m(t,K)$ for any local field $L$ with a non-archimedeanvaluation $v:L oRcup{infty}$ such that $v|_{Z_{ eq 0}}equiv 0$ and residue field $K$, and that $B_m(t,K)leq (t^2-t+1)(p^f-1)$ 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 $B_m(t,K)geq (2t-1)(p^f-1)$, which gives the sharp estimation $B_m(2,K)=3(p^f-1)$ for trinomials when $p>2+e$.