Representing integers by multilinear polynomials

Abstract

Let $$F(\varvec{x})$$ 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 $$\varvec{a}$$ such that $$F(\varvec{a}) = b$$ . The conditions provided also guarantee that the vector $$\varvec{a}$$ can be found in a finite number of steps.