optimization: feasible algorithm to find interpolation of complex polynomials based on absolute values

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, that is, $$| p (z_j) | overset {!} {=} b_j$$ (with $$b_j geq 0$$), there is a numerically stable and feasible algorithm to find the coefficients $$a_i$$ such that $$p (z)$$ satisfies those requirements? How big can $$m$$ be for such an algorithm?

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?