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: 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 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.
Best Answer
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:
From what I can see from the MathSciNet review, 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.