In my analysis research, I came across the following problem. Given $n$ positive real numbers $x\_1,\dots,x\_n$, consider the $n$-many power sums
$$ p\_3 = x\_1^3 + x\_2^3 + \dots + x\_n^3 , $$
$$ p\_5 = x\_1^5 + x\_2^5 + \dots + x\_n^5 , $$
$$ \vdots $$
$$ p\_{2n+1} = x\_1^{2n+1} + x\_2^{2n+1} + \dots + x\_n^{2n+1} . $$
Do the values of the power sums $p\_3,p\_5,\dots,p\_{2n+1}$ uniquely determine $x\_1,\dots,x\_n$ (up to reordering)?


I was wondering if this problem exists in the literature? I know the answer to this problem is "yes" in the case of the first $n$ power sums $p\_1,p\_2,\dots,p\_{n}$ by [Newton's identities](https://en.wikipedia.org/wiki/Newton%27s_identities): the power sums $p\_1,p\_2,\dots,p\_{n}$ determine the elementary symmetric polynomials $e\_1,\dots,e\_n$ via explicit formulas, from which we can construct a degree-$n$ polynomial with roots $x\_1,\dots,x\_n$ and appeal to the fundamental theorem of algebra.


I believe the answer is yes, and I've checked some special cases using a computer. Using analysis techniques I can easily get a local uniqueness statement. Indeed, the function $f:(x\_1,\dots,x\_n) \mapsto (p\_3,p\_5,\dots,p\_{2n+1})$ has a Jacobian matrix $Df$ that looks like a Vandermonde matrix, and this makes it easy to compute its determinant (and minors). In particular, on the simplex $\{ 0 < x\_1 < x\_2 < \dots < x\_n \} \subset\mathbb{R}^n$ the Jacobian matrix $Df$ has nonzero determinant, and so the inverse function theorem tells us that $f$ is locally injective. Moreover, this argument applies to all minors of $Df$, and so $Df$ is a [strictly totally positive](https://en.wikipedia.org/wiki/Totally_positive_matrix) matrix. Such matrices turn out to be diagonalizable with distinct positive eigenvalues, but I haven't been able to conclude that $f$ is globally injective.



Do power sums determine the variables?
Let $\gamma (x) = (x^3, x^5, \dots, x^{2n+1})$. By rearranging both sides (using that all the polynomials have odd degree and therefore are antisymmetric), the problem can be restated as:



> 
> Let $0\le a,b$, $a+b \le 2n$ integers, and $x\_1, \dots x\_a,y\_1, \dots y\_b \in \mathbb{R}\_{\ge 0}$. Then
> 
> 
> $$ \sum\_{i=1}^a \gamma(x\_i) = \sum\_{i=1}^b \gamma(y\_i) $$
> 
> 
> has no nontrival solutions.
> 
> 
> 


From what I can see from the [MathSciNet review](https://mathscinet.ams.org/mathscinet-getitem?mr=309867), that is precisely the content of [1]. I haven't been able to find that paper, but I found it as a reference in [2], which also contains the idea of the proof (in a slightly more general setting) in the second half of Section 3. The key idea is using that the Jacobian matrix is a strictly totally positive matrix, indeed!


[1] J. Steinig, *On some rules of Laguerre's, and systems of equal sums of like powers*. Rend. Mat. (6) 4 (1971), 629–644 (1972).


[2] [S. W. Drury and B. P. Marshall, *Fourier restriction theorems for degenerate curves.* Mathematical Proceedings of the Cambridge Philosophical Society, 101(3), 541-553.](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/fourier-restriction-theorems-for-degenerate-curves/7E17A1219062870146EA1D31A0DD211A)


