Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients

TL;DR

This paper establishes upper bounds on the ratios of solutions' magnitudes for certain integer coefficient linear systems, proving the bounds are sharp and confirming a related conjecture.

Contribution

It provides the first tight bounds on solution magnitudes for these systems and verifies a conjecture by A. Tyszka.

Findings

Bound |x_j|/|x_i| ≤ k^{n-1} for solutions

Bounds are proven to be sharp

Confirmed Tyszka's conjecture

Abstract

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some positive integer $k$. We show that if the system has a nontrivial solution then there exists a nontrivial solution $\x=(x_1,...,x_n)\trans$ such that $\frac{|x_j|}{|x_i|}\le k^{n-1}$ for each $i,j$ satisfying $x_ix_j\ne 0$. This inequality is sharp. We also prove a conjecture of A. Tyszka related to our results.