  Let L be any number field or $\mathfrak{p}$-adic field and consider
F:=(f_1,...,f_k) where f_i is in L[x_1,...,x_n]\{0} for all i and there are
exactly m distinct exponent vectors appearing in f_1,...,f_k. We prove that F
has no more than 1+(cmn(m-1)^2 log m)^n geometrically isolated roots in L^n,
where c is an explicit and effectively computable constant depending only on L.
This gives a significantly sharper arithmetic analogue of Khovanski's Theorem
on Fewnomials and a higher-dimensional generalization of an earlier result of
Hendrik W. Lenstra, Jr. for the case of a single univariate polynomial. We also
present some further refinements of our new bounds and briefly discuss the
complexity of finding isolated rational roots.
