In dealing with some quantum-chemistry issues I have to face the following problem:
*Find $N$ distinct real numbers in the interval $[0,1)$ such that:
their average is $1/3$
the average of their squares is $1/5$
the average of their cubes is $1/7$
$\cdots$
M) the average of their $M$-th powers is $1/(2M + 1)$*
$\cdots$ for a total of $M$ conditions (with $M$ smaller than or equal to $N$).
In fact, this is a square polynomial system for $M = N$ and an underdetermined polynomial system for $M < N$. I know that the case of the square system can be reduced to finding the zeroes of a $M$-th degree polynomial using standard Girard-Newton formulae. However, direct numerical verification shows that for square systems with M > 3 one or more unknowns turn out to have complex (not real) solutions.
On the other hand, underdetermined systems can exhibit real solutions provided that $N$ is sufficiently larger than $M$. I verified this feature in the case ($M = 4, N = 5$) and also in the case ($M = 5, N = 7$).
So, my question is: in the general case of arbitrary M, how large must be N in order to ensure N real solutions? And... how can I numerically face the underdetermined problem?
Thank you in advance!