Positive polynomial

In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, Let ${\displaystyle p}$ be a polynomial in ${\displaystyle n}$ variables with real coefficients and let ${\displaystyle S}$ be a subset of the ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. We say that:

• ${\displaystyle p}$ is positive on ${\displaystyle S}$ if ${\displaystyle p(x)>0}$ for every ${\displaystyle x}$ in ${\displaystyle S}$.
• ${\displaystyle p}$ is non-negative on ${\displaystyle S}$ if ${\displaystyle p(x)\geq 0}$ for every ${\displaystyle x}$ in ${\displaystyle S}$.