Linear systems of diophantine equations

TL;DR

This paper presents a method to construct a basis for solutions of linear Diophantine systems over principal ideal domains by leveraging invariant factors and elementary divisors, improving basis computation efficiency.

Contribution

It introduces a procedure to explicitly construct a basis of the solution module from a known submodule basis using invariant factors, applicable to matrices over principal ideal domains.

Findings

Constructed a basis of the solution module from a submodule basis.

Determined invariant factors of the quotient module.

Provided an explicit basis construction method for Diophantine solutions.

Abstract

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace~$L$ in $R^n$ is a free $R$-module of rank~$f=n-r$. We construct a free submodule $M$ of $L$ of rank~$f$ naturally associated to $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$, and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.