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 .



Simultaneous Equations Involving Power Sums
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.


