(not treated in class) Determine the minimum of $x^{2}+y^{2}+z^{2}$ under the condition $x^{3}+y^{3}+z^{3}-3 x y z=1$.
We calculate that $S_{3}=\sigma_{1} S_{2}-\sigma_{1} \sigma_{2}+3 \sigma_{3}$, so $S_{3}-3 \sigma_{3}=\sigma_{1}\left(S_{2}-\frac{3 S_{2}-\sigma_{1}^{2}}{2}\right)$. We must therefore minimize $S_{2}$ under the condition that $\sigma_{1}\left(3 S_{2}-\sigma_{1}^{2}\right)=2$.

Now, $3 S_{2}=\frac{2}{\sigma_{1}}+\sigma_{1}^{2} \geqslant 3$ by the AM-GM inequality (or by a function study), so $S_{2} \geqslant 1$. The bound is achieved when $\sigma_{1}=1$ (for example, when $x=1$ and $y=z=0$).