I have a problem. I need to solve an integral coefficients linear equations with $m$ number of equations and $n$ number of variables, and $m<n$.
The coefficients can be defined by myself, so
1.I wonder how can insure that the linear equations have only one integral solutions.
2.what is the expression of the general solutions?
Thanks.
The basis field is $\mathbb{Q}$. We consider the equation in $x\in\mathbb{Q}^n$: $Ax=b$ where $A\in M_{m,n}(\mathbb{Z}),b\in\mathbb{Z}^m$ are given. Then either there are no solutions OR there is a particular solution $x_0\in \mathbb{Q}^n$ and the set of all solutions is $x_0+\ker(A)$. In the sequel, we assume that we are in the last case.
Case 1. There are no solutions in $\mathbb{Z}^n$; example: $m=1,n=2$ and $2x+2y=1$. This equation admits the solution $(0,1/2)$ but no solutions in integers.
Case 2. There is a solution $x_0\in\mathbb{Z}^n$. Yet, there are an infinity of non-zero $y\in \ker(A)\cap\mathbb{Z}^n$.
Conclusion. Either the solution in integers does not exist OR there is an infinity of such solutions.