Consider the set $\mathcal{S} \subseteq \mathbb{R}^n$, defined by the following $m > n \geq 4 $ linear inequalities:
$$\begin{cases} a_{1,1}x_1 + a_{1,2}x_2 + \ldots + a_{1,n} x_n \leq b_1\\ ~\vdots \\ a_{m,1}x_1 + a_{m,2}x_2 + \ldots + a_{m,n} x_n \leq b_m, \end{cases} $$
where $a_{i,j}$ and $b_i$ are integer (i.e. $\in \mathbb{Z}$) constants, and $x_j$ are unknown variables.
As a side remark, $a_{i,j}$ can be represented as a $m$-by-$n$ matrix $A$, $b_{i}$ can be represented by a $m$-by-$1$ vector and $x_j$ can be represented by a $n$-by-$1$ vector.
Is it possible to establish in an algebraic way if the set $\mathcal{S}$ is empty? In other words, are there properties that $A$ and $b$ must fulfill in order $\mathcal{S} = \emptyset$?
It is easy to answer to this question when $n = 2$ or $n=3$ by using geometrical arguments. But what can we do for $n \geq 4$?