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I am new here and I just want to ask if the following system has a general solution:

If a, b and c are given such that:

$$ x + y + z = a $$ $$ x^2 + y^2 + z^2 = b $$ $$ x^8 + y^8 + z^8 = c $$ Is there a way to find out if there exists a general solution to find x, y and z?

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  • $\begingroup$ Please, try math.stackexchange.com for this kind of questions. $\endgroup$
    – Agustí Roig
    Commented Mar 13, 2019 at 18:21
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    $\begingroup$ @d.t. It was asked on M.SE (math.stackexchange.com/questions/3146823/…), though there, the question was whether a solution to this set of equations would be publishable research (as a note: it seems unlikely to me). The author was referred here. $\endgroup$
    – user44191
    Commented Mar 13, 2019 at 18:40
  • $\begingroup$ The standard approach would be to express the left hand side in terms of the elementary symmetric functions, and solve for these functions. As the first two equations are linear and quadratic, you end up with one equation in one variable. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented Mar 13, 2019 at 21:09

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This is the wrong forum for your question.

The short answer is yes, there is. I assume x,y, and z are real numbers. For complex numbers or other rings, you will need results from another field of study, possible algebraic geometry.

The first two equations represent a plane possibly intersecting a sphere. There are three cases:

  • no intersection. Thus no solution. Stop

  • intersection is a point. Then x=y=z and thus c had better be 3(a/3)^8, otherwise no solution. (b and a have to satisfy a relation for this case to occur.)

  • intersection is a circle. Then use geometry or Lagrange multipliers to find the image of this circle under the map $x^8 + y^8 + z^8$. Since the circle is compact and the map is continuous, the range is a closed interval. If c is in this range there are anywhere from 1 to at least 6 distinct solutions. If c is not in the interval, no solutions.

Regarding the question of publishability (mentioned in a comment), this result would not be considered appropriate for most if not for all research journals. Even if there is novelty in the approach, one would expect the approach to be much more widely applicable to be considered. If it were presented in a pedagogically engaging way, it might serve as an example lesson for undergraduate students. How one would present this solution is a question for another forum.

Gerhard "You Could Blog This Elsewhere" Paseman. 2019.03.13.

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  • $\begingroup$ As a note, I have provided an alternate, algebraic method over at M.SE. $\endgroup$
    – user44191
    Commented Mar 13, 2019 at 19:57

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