# A positive polynomial not reaching its infimum

A positive real polynomial function of one variable is always having a minimum.

This is not true for polynomial functions of two variables or more.
Consider the polynomial function:
$P(x,y)=x^2+(x \cdot y -1)^2$

$$P$$ is obviously positive. Its infimum is $$0$$ as $$\lim\limits_{t \to + \infty} P(\frac{1}{t},t) = 0$$. However $$P$$ never takes $$0$$ as a value as this implies the incompatible equations:
$\left\{ \begin{array}{l} x=0\\ x \cdot y =1\\ \end{array} \right.$

See the picture below to have a clue on $$P$$ graph.