Bounds for solution of linear diophantine equations

TL;DR

This paper establishes bounds on solutions to linear Diophantine equations, showing that if solutions exist with nonnegative integers, then there is a solution with each variable bounded by a specific maximum value derived from the matrix minors.

Contribution

The paper provides a new bound for solutions of linear Diophantine equations based on the maximum minors of the augmented matrix, extending previous results on solution size limits.

Findings

Existence of bounded solutions when solutions are nonnegative.

Bound on variables is given by the maximum absolute value of minors.

Applicable to equations with full rank matrices.

Abstract

Given linear diophantine equation Ax=b, rank A=m. Let d be the maximum of absolute values of the mxm minors of the matrix (A | b). It is shown that if M={x : Ax=b, x nonnegative and integer} is nonempty, then there exists x=(x1,...,xn) in M, such that xi does not exceed d (i=1,2,..,n).