On zeros of multilinear polynomials

We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with respect to height. For a single such polynomial, we prove the existence of search bounds with respect to height for zeros lying outside of a prescribed algebraic set. We also obtain search bounds in the case of homogeneous multilinear polynomials, which are related to a so-called "sparse" version of Siegel's lemma. Among the tools we develop are height inequalities that are of some independent interest.