Consider a complex degree-$(n-1)$ polynomial $p(z) = \sum\limits_{i=0}^{n-1} a_i z^i$.
- Given a number $0 < m < 2n$ of positions in the complex plane with absolute value requirements, i.e. $|p(z_j)| \overset{!}{=} b_j$ (with $b_j \geq 0$), is there a practical algorithm to find the coefficients $a_i$ such that $p(z)$ satisfies those requirements? In other words, is there a way to solve a complex polynomial interpolation problem based only on given absolute values, leaving the argument (angle) of the polynomial completely arbitrary at any point?
- How big can $m$ be for such an algorithm? In other words, how many degrees of freedom are gained by only specifying the absolute value instead of a "full" complex number consisting of an absolute value and an argument (angle)?
Considering the simple case of $n = 2$, it turns out to be rather trivial to find $m = 3$ conditions $|p(z_j)| \overset{!}{=} b_j$ that no polynomial of degree $1$ can satisfy. Hence, $m$ can not be greater than $n$ in the general case. There are however some (not so rare) examples of conditions which can be satisfied.
For $m \leq n$, the polynomial can easily be found by choosing a random argument (angle) for each $b_j$ and then solving the resulting system of linear equations. Since a polynomial only exists in some special cases when $m > n$, an algorithm to find that polynomial for arbitrary conditions cannot exist. There still might be a way to determine the space of conditions that yield a polynomial and a corresponding algorithm, though.