Definition [Monomial of max-degree 1]. Given n variables x 1 , . . . ,...

kolutastmr

kolutastmr

Answered

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 ( 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 question
QUESTION Can a system of polynomial inequalities of max-degree 1 have a solution in the reals R but none in rationals Q? Any example?

Answer & Explanation

ladaroh

ladaroh

Expert

2022-07-07Added 11 answers

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.

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