5.35 (1) Find the possible minimum value of the polynomial $P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}$.
(2) Prove that this polynomial cannot be expressed as a sum of squares of polynomials in variables $x, y$.
［Solution］（1）By the Arithmetic Mean - Geometric Mean Inequality, we have
$$1+x^{2} y^{4}+x^{4} y^{2} \geqslant 3 x^{2} y^{2}$$

and the equality holds when $x=y=1$. Therefore, we have
$$\begin{aligned}
P(x, y) & =3+\left(1+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}\right) \\
& \geqslant 3
\end{aligned}$$

and when $x=y=1$, $P(x, y)=3$. Hence, the minimum value sought is 3.
（2）Let $p(x, y)=g_{1}^{2}(x, y)+g_{2}^{2}(x, y)+\cdots+g_{n}^{2}(x, y)$, where $g_{i}(x, y)$ are polynomials, $i=1,2, \cdots, n$.

Since $p(x, 0)=p(0, y)=4$, the polynomials $g_{i}(x, y)$ cannot have monomials of the form $a x^{k}$ and $b y^{\prime}$. Thus, the coefficient of the $x^{2} y^{2}$ term in $p(x, y)$ should be positive, which contradicts the given condition. Therefore, $P(x, y)$ cannot be expressed as a sum of squares of polynomials in variables $x, y$.