kolutastmr

2022-07-06

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: $r\left({x}_{1}^{{e}_{1}}\cdot {x}_{2}^{{e}_{2}}\cdot \cdots \cdot {x}_{n}^{{e}_{n}}\right)$, where $r\in \mathbb{Q}$ and all exponents ${e}_{i}$ are either 0 or 1.
For example $2\left({x}_{1}{x}_{2}{x}_{5}\right)$ is a monomial of max-degree 1, but $3{x}_{1}^{2}$ 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: $2{x}_{1}{x}_{2}+3{x}_{1}{x}_{3}$ 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 $f=0$ or $f=0$ or $f\le 0$.
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 R but none in rationals $\mathbb{Q}$? Any example?

Expert

The system
$\left\{\begin{array}{l}{x}_{1}-{x}_{2}=0\\ {x}_{1}{x}_{2}-2=0\end{array}$
has solution set
$\left\{\left(\sqrt{2},\sqrt{2}\right),\phantom{\rule{thickmathspace}{0ex}}\left(-\sqrt{2},-\sqrt{2}\right)\right\}$
so the system has real solution pairs $\left({x}_{1},{x}_{2}\right)$, but no rational solution pairs.

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