Representing integers by multilinear polynomials

Abstract

Let $F(\mathbf{x})$ be a homogeneous polynomial in $n\ge 1$ variables of degree $1\le d\le n$ with integer coefficients so that its degree in every variable is equal to $1$. We give some sufficient conditions on $F$ to ensure that for every integer $b$ there exists an integer vector $\mathbf{a}$ such that $F(\mathbf{a})=b$. The conditions provided also guarantee that the vector $\mathbf{a}$ can be found in a finite number of steps.

9 pages, to appear in Research in Number Theory