Optimal synthesis of electrical filters and Riemann surfaces

We describe the usage of Ansatz method for the solution of uniform rational approximation problem which is a part of filter synthesis procedure. With the use of certain algebro- geometric formula the problem is reduced to the solution of small set of transcendental equations on moduli of Riemann surfaces. The complexity of this approach does not depend on the degree of filter, but rather on the number of its stop- and pass- bands. Supported by RSF, project 21-11-00325.