Do power sums determine the variables?

Question

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.

Answer 1

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, 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.

Answer 2

A related question was studied in the paper Conca, Krattenthaler, Watanabe, "Regular sequences of symmetric polynomials" Rend. Sem. Mat. Univ. Padova 121 (2009), 179-199. The published version seems to be freely available at Numdam.
For integers $a$, $n$, let $p_a(n)=x_1^a+\dotsb+x_n^a$. Then the authors ask: Which integers $n$ and which index sets $A\subset \mathbb{N}$ of size $n$ have the property that the sequence of power sums $p_a(n)$ with $a\in A$ form a regular sequence in the polynomial ring $\mathbb{C}[x_1,\dotsb,x_n]$?
Here are some of their results (numbered as in the paper):

Lemma 2.8: If $p_a(n)$, $a\in A$ is a regular sequence, then $n!$ must divide the product $\prod_{a\in A}a$.

Proposition 2.9: If $A\subset\mathbb{N}$ is a sequence of $n$ consecutive numbers, then $p_a(n)$, $a\in A$ is a regular sequence.

They also have an interesting conjecture, which is still open to my knowledge:

Conjecture 2.10: If $n=3$, and if $A=\{a,b,c\}$ where $\gcd(a,b,c)=1$, then $p_a(3), p_b(3), p_c(3)$ is a regular sequence if and only if $$abc\equiv 0 \mod 6.$$

In relation to the original post: I guess the point is that the sequence $p_a(n)$, $a\in A$ is regular if and only if the subalgebra they generate $\mathbb{C}[p_a(n); a\in A]$ is polynomial and $\mathbb{C}[x_1,\dotsb,x_n]$ is a finite free module over it, which means that the variables are then uniquely determined by the $p_a(n)$, $a\in A$ up to choice of basis.