Definition [Monomial of max-degree 1]. Given $n$ variables ${x_1, ... ,x_n}$, a multivariate monomial of max-degree 1 is an expression of the form: $r(x_1^{e_1} \cdot x_2^{e_2} \cdot \dots \cdot x_n^{e_n})$, where $r\in \mathbb{Q}$ and all exponents $e_i$ are either $0$ or $1$.
For example $2(x_1x_2x_5)$ is a monomial of max-degree 1, but $3x_1^2$ is not.
Definition [Polynomial of max-degree 1]. A polynomial of max-degree 1 is a sum $f = m_1 + \dots m_k$ of multivariate monomials of max-degree 1.
For example: $2x_1x_2 + 3x_1x_3$ is a Polynomial of max-degree 1.
Definition [System of Polynomial inequalities of max-degree 1]. A system of polynomial inequalities is a finite conjunction of inequalities of the form $f\geq 0$ or $f=0$ or $f\leq 0$.
I have came across this notion recently, and I am not at all an expert. I have the following question
QUESTION Can a system of polynomial inequalities of max-degree 1 have a solution in the reals $\mathbb{R}$ but none in rationals $\mathbb{Q}$? Any example?
Or it is perhaps the case that if a solution exists, then there is always a solution in the rationals? If so, any reference?
Thanks!
The system $$ \begin{cases} x_1-x_2 = 0\\[4pt] x_1x_2-2 = 0 \end{cases} $$ has solution set $$ \{(\sqrt{2},\sqrt{2}),\;(-\sqrt{2},-\sqrt{2})\} $$ so the system has real solution pairs $(x_1,x_2)$, but no rational solution pairs.