On zeros of multilinear polynomials

TL;DR

This paper investigates the existence of bounds for zeros of multilinear polynomials over number fields, providing new height inequalities and extending classical results like Siegel's lemma to a multilinear context.

Contribution

It introduces new search bounds for zeros of multilinear polynomials, including systems and homogeneous cases, using novel height inequalities and technical conditions.

Findings

Existence of height bounds for zeros of polynomial systems

Bounds for zeros outside prescribed algebraic sets

Height inequalities of independent interest

Abstract

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.