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.