Given a linear inequalities, how can I know the structure of the integer points? An simple example is stated as follows: if \begin{align*} y\geq x\quad \text{and}\quad y\leq2x, \end{align*} then all the integer points in this region can be expressed as \begin{align*} a(1,1)+b(1,2), \end{align*} where $a,b$ are integers satisfying $a\geq0$ and $0\leq b\leq a$. Thus we know the integer solution of this linear inequalities.
I wonder the general answer of this question. Let $A$ be a $m\times n$ integral matrix, and $\alpha$ be an $n$-dimensional vector. What about the structure of the integer solution of
$$A\alpha\geq0 .$$
Can all the solutions be expressed as an integer linear combination of finite points like the example above? I also seek for the relative reference.
Any help will be appreciated!:) Thank you!
You are looking for the Integral Volume (somebody calls it Content), of a Polytope defined by its bounding planes. Which may also turn out to be "in-congruent" and thus defining an empty space.
There are some techniques that allow to re-organize / simplify such "set of Diophantine Inequalities /Polytope" and provide some computational simplification, or formulas for the most regular polytopes. But unfortunately not a general "closed" formula (to my knowledge).
You can search on internet: there exist various papers on this topic.
One interesting is e.g. "Computing the Continuous Discretely" - M. Beck, S. Robins.