Sum of powers

Provide the most elegant proof to the following claim you can produce. Suppose z_1,\ldots,z_s and w_1,\ldots,w_r are nonzero complex numbers and for every positive integer n it is true that z_1^n +\cdots+z_s^n = w_1^n+ \cdots+w_r^n. then r=s and the w_i are a reordering of the z_j.

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Sum of powers

One thought on “Sum of powers

  1. Given z_0\in\Bbb{C^{\times}}, the assignment n\mapsto z_0^n defines a complex character of the additive semigroup \Bbb{N}, which we will denote \hat{z_0}. The set of such characters is linearly independent, and the hypotheses of the problem imply \hat{z_1}+\dots+ \hat{z_s}- \hat{w_1}-\dots- \hat{w_r}=0. This linear combination is then trivial and so the result follows.

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