Representing integers by multilinear polynomials

TL;DR

This paper establishes conditions under which a specific class of multilinear homogeneous polynomials with integer coefficients can represent any integer value, and provides a finite-step method to find such integer vectors.

Contribution

It introduces sufficient conditions for representing all integers by multilinear polynomials and ensures a finite-step algorithm to find the representing vectors.

Findings

Conditions guarantee integer representation for all integers

Finite-step method to find integer solutions

Applicable to polynomials with degree up to number of variables

Abstract

Let $F(\boldsymbol x)$ be a homogeneous polynomial in $n \ge 1$ variables of degree $1 \leq d \leq n$ 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 $\boldsymbol a$ such that $F(\boldsymbol a) = b$. The conditions provided also guarantee that the vector $\boldsymbol a$ can be found in a finite number of steps.