I am interested in an optimization problem of the form $$\min_{\boldsymbol z} \max_j \vert f_j(\boldsymbol z) \vert = \min_{\boldsymbol z} \Vert f_j(\boldsymbol z) \Vert_\infty. $$ Here, the optimization/decision variables are $\boldsymbol z \in \mathbb C^{N} $ and $f_i: \mathbb C^N \to \mathbb C$.
The $f_j$ are essentially polynomials in $\lambda_j \in \mathbb C$ $$f_j(\boldsymbol z ) = \prod_{k=1}^N \big(1 - z_k \lambda_j\big).$$
To provide some more context, I am essentially trying to optimize the common roots of complex polynomials. Write the polynomial $p(\lambda) $ with $p(0) = 1$ as $$p(\lambda) = \prod_{k=1}^N \bigg(1 - \frac{ \lambda}{\tilde z_k} \bigg)$$ where $\tilde z_k$ are the roots of the polynomial. For $z_k := 1/\tilde z_k$ you obtain the representation above.
In principle, one has also to enforce that the roots come in complex-conjugated pairs which would give a linear constraint like $$A \boldsymbol z = \boldsymbol 0.$$