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 ⋅ x 2 e 2 ⋅ ⋯ ⋅ x n e n ), where r ∈ Q and all exponents e i are either 0 or 1.For example 2 ( x 1 x 2 x 5 ) 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 + … 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 ≤ 0.I have came across this notion recently, and I am not at all an expert. I have the following questionQUESTION Can a system of polynomial inequalities of max-degree 1 have a solution in the reals R but none in rationals Q? Any example?

Question

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        ⋅      x    2            e      2        ⋅  ⋯  ⋅      x    n            e      n        ), where   r  ∈  Q and all exponents       e    i   are either 0 or 1.For example   2  (      x    1        x    2        x    5    ) 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    +  …      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  ≤  0.I have came across this notion recently, and I am not at all an expert. I have the following questionQUESTION Can a system of polynomial inequalities of max-degree 1 have a solution in the reals R but none in rationals   Q? Any example?

ladaroh · Accepted Answer

The system      {                                        x            1                    −                      x            2                    =          0                                                  x            1                                x            2                    −          2          =          0                        has solution set  {  (      2    ,      2    )  ,    (  −      2    ,  −      2    )  }so the system has real solution pairs   (      x    1    ,      x    2    ), but no rational solution pairs.