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Suppose that I have the following system, $$ \begin{equation} \left\{\begin{array}{lcl} a + b + c + d & = & \alpha \\ a^2 + b^2 + c^2 + d^2 & = & \beta \\ a^3 + b^3 + c^3 + d^3& = & \gamma \\ a^4 + b^4 + c^4 + d^4& = & \theta \end{array}\right. \end{equation} $$ in which I know that $\alpha, \beta, \gamma$ are positive, $\alpha=1$, $\beta<1$, $\gamma<1$, and $\theta<1$. If I know the values of $\beta$, $\gamma$, and $\theta$, how can I find $a, b, c, d$? Can someone tell me if there is a method to solve this? Which system of equations is this one? Does this have a specific name?

Odds'n'Ends
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  • Dots are not an existing analytical method to define anything, sorry! Is this supposed to be a finite set of equations with finitely many summands, or what? It's your problem, so you have to tell us what kind of system of equations this is. –  Jun 11 '20 at 19:54
  • I reframed the problem. I hope it become more clear. – Odds'n'Ends Jun 11 '20 at 22:47
  • I believe that questions equivalent to this have been asked and answered here many times. See, for example, https://math.stackexchange.com/questions/402856/finding-the-fraction-fraca5b5c5d5a6b6c6d6-when-knowing-the – Gerry Myerson Jun 11 '20 at 23:36
  • Thanks Gerry! I have found a good hint in this link!! – Odds'n'Ends Jun 11 '20 at 23:56

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