TITLE: Simultaneous Equations Involving Power Sums
QUESTION [7 upvotes]: Let $\ell$ be a positive integer greater than 1. The problem is to find a set of $n$ real positive numbers $x_i$ and $n+1$ numbers $y_i$ such that
$$\sum_{i=1}^n x_i^k= \sum_{i=1}^{n+1} y_i^k$$
for $k=\ell,\cdots,2\ell-1$. These $2n+1$ numbers need to be upper/lower bounded by a constant independent of $\ell$ [thus $x_i,y_i=\Theta(1)$] and also I suspect that it is possible to do so with just $n=\ell$ or $n=O(\ell)$. [$\ell$ equations with $2\ell$ unknowns, why not!] An existential proof suffices but a constructive proof or a recipe would be really nice.
For me it is useful to find a bounded from below solution that scales in polynomially in the following sense: There exist positive $c$, and $s$ such that $c\le x_i,y_i$ and
$$\sum_i x_i+\sum_i y_i=O(\ell^s)$$. 
The problem is related to a follow up on this paper of mine: arxiv:0908.1526 .

REPLY [11 votes]: Actually Darsh gave an almost full solution. Let me fill in the minor technical details.
1) We need the following quantitative form of the inverse function theorem. Suppose that  $F:\mathbb R^n\to \mathbb R^n$. Assume also that $\|DF(X)^{-1}\|\le C_1$, that $\max_{Y\in B(X, \delta)}\|D^2F(Y)\|\le C_2$, and that $C_1C_2\delta\le\frac 12$. Then $F(B(X,\delta))\supset B(F(X),\frac{\delta}{2C_1})$.  
2) Take $n=2\ell-1$ and consider the mapping $F:\mathbb R^n\to \mathbb R^n$ given by $F(y_1,\dots,y_n)_k=\sum_{j=1}^n y_j^k$ where $k=1,2,\dots,n$. Take $X=(x_1,\dots,x_n)$ where $x_j=\frac{n+j}{n}$ for $j=1,\dots,n$.
3) Note that in $B(X,1)$, we have $\|D^2F\|\le A^n$ for some absolute $A>1$.
4) Note also that $DF(X)^*$ is the linear operator that maps the vector $(c_1,\dots,c_n)$ to the vector $p(x_1),\ldots,p(x_n)$ consisting of the values of the polynomial $p(x)=\sum_{k=1}^n c_k kx^{k-1}$. The inverse operator is given by the standard interpolation formula, which allows us to estimate its norm by $B^n$ with some absolute $B>1$.
5) Thus, taking $\delta=2^{-1}(AB)^{-n}$, we conclude that the image of the ball $B(X,\delta)$ contains a ball of radius $\frac\delta{2A^n}\ge C^{-n}$ with some absolute $C>2$.
6) In particular, it contains two points with the difference $(0,0,\dots,0,D^{-\ell},D^{-(\ell+1)},\dots,D^{-(2\ell-1)})$ with some absolute $D>1$, which is equivalent to what we need.