ca.analysis and odes - 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 $F:mathbb R^nto mathbb R^n$. Assume also that $|DF(X)^{-1}|le C_1$, that $max_{Yin B(X, delta)}|D^2F(Y)|le C_2$, and that $C_1C_2deltalefrac 12$. Then $F(B(X,delta))supset B(F(X),frac{delta}{2C_1})$.

2) Take $n=2ell-1$ and consider the mapping $F:mathbb R^nto mathbb R^n$ 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 $fracdelta{2A^n}ge C^{-n}$ with some absolute $C>2$.

6) In particular, it contains two points with the difference $(0,0,dots,0,D^{-ell},D^{-(ell+1)},dots,D^{-(2ell-1)})$ with some absolute $D>1$, which is equivalent to what we need.