On a Question of Bombieri and Bourgain

(1) an n+1-tuple (ρ, χ1, ...., χn) of nontrivial C×-valued multiplicative characters of k×, each extended to k by the requirement that it vanish at 0 ∈ k. (2) an n+1-tuple (g, f1, ...., fn) of nonzero one-variable k-polynomials, which are adapted to the character list above in the following sense. Whenever α ∈ k is a zero of g (respectively of some fi), then ρα (respectively χ ordα(fi) i ) is nontrivial. We define mi := number of distinct zeroes of fi in k,