Hyper-simultaneous equation

Question

I am new here and of course I would like to be specific. I've found a general solution to the following system:

If $a$, $b$ and $c$ are given such that: $$ x+y+z=a $$ $$ x²+y²+z²=b $$ $$ xⁿ+yⁿ+zⁿ=c $$

Where $n=8$.

$x$, $y$ and $z$ could be devised using the method I have derived.

I just want to basically ask if this kind of thing is publishable. Thanks for your reply in anticipation.

Answer 1

I'm doubtful that this would be publishable unless the method is particularly neat or generalizable; it's not hard to find a method using matrices. Explicitly:

If a 3 by 3 matrix $A$ has eigenvalues $x, y, z$, then $A^8$ has eigenvalues $x^8, y^8, z^8$. Therefore, $tr(A^8) = x^8 + y^8 + z^8$. We can choose $A$ to be the companion matrix for a polynomial $x^3 - l x^2 - mx - n$. Then

$$x + y + z = tr(A) = l$$

$$x^2 + y^2 + z^2 = tr(A^2) = l^2 + 2m$$

$$x^8 + y^8 + z^8 = tr(A^8) = l^8 + 8 l^6 m + 20 l^4 m^2 + 16 l^2 m^3 + 2 m^4 + 8 l^5 n + 32 l^3 m n + 24 l m^2 n + 12 l^2 n^2 + 8 m n^2$$

So we set $l = a, m = \frac{b - a^2}{2}$, and solve the third equation to find $n$ (it's quadratic in $n$, so the solution won't be too difficult). Then $x, y, z$ are the eigenvalues of $A$, which can be determined by solving the above cubic.

Answer 2

See Newton-Girard's formulas for a more complete explanation of this technique.

Let: $$ S_n =\sum_{k=1}^l x_i ^n $$ $$ \sigma_1 = S_1 $$ $$ \sigma_2 = \sum_{i\neq j} x_i x_j $$ $$ \sigma_3 = \sum_{i\neq j \neq k} x_i x_j x_k $$

Then: $$ S_2= \sigma_1^2-2\sigma_2 $$

$$ S_n = \sigma_1 S_{n-1}-\sigma_2S_{n-2}+\sigma_3 S_{n-3}-\cdots $$

Thus in your case, we have: $$ a=\sigma_1 $$ $$ b= S_2 $$ $$ c = S_8 $$

Note that if we have $\sigma_1$, $\sigma_2$ and $\sigma_3$, then $x$, $y$ and $z$ are the roots of the equation:

$$ \lambda^3-\sigma_1 \lambda^2+\sigma_2 \lambda-\sigma_3=0 $$

Then $\sigma_2$ can be computed almost directly:

$$ \sigma_2 = \frac{a^2-b}{2} $$

It remains to find $\sigma_3$, but it can be computed be recurring the formula until $S_8$ is written as a function of $\sigma_1$, $\sigma_2$ and $S_2$. It will likely yield an equation similar to the one @user44191 presented.

Basically, if you have an analytic method that does not incur into solving a third or higher degree polynomial, you might have something interesting in hand.