Help with non-linear system of equations

Question

This system of equations

$$\begin{align} xy+yz+zx & =3 \\ \\ x^4+y^4+z^4 & =3\end{align}$$

How to solve this system of equations?

Any help, Plz.

Thank all

Answer 1

Assuming that you are solving over the reals.

$1 = \sqrt[4]{ \frac{x^4 + y^4 + z^4} {3}} $ $\geq \sqrt { \frac {x^2 + y^2 + z^2 } { 3}} \geq \sqrt{ \frac{ |xy| + |yz| + |xz| } { 3} } \geq \sqrt{ \frac{ xy+yz+zx} { 3}} = 1$

These inequalities are standard.
The first one is power mean applied to fourth powers and second powers.
The second one is $(|x|-|y|)^2+(|y|-|z|)^2+(|z|-|x|)^2 \geq 0$.
The third one follows from the definition of absolute value / triangle inequality.

Hence equality must hold throughout, so $|x|=|y|=|z|=1$.

A further check shows that $xy+yz+zx = 3$ if and only if $x, y, z$ have the same sign. Thus, we have $x=y=z=1, x=y=z=-1$.

Note: As mentioned, the motivation is that the solution set is one where all variables are equal, suggesting that some kind of inequality can be shown to hold, which gives us the equality condition (which is that all variables are equal).

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If you are solving over the complex numbers, I believe there are many solutions.