Extremal and Approximative Properties of Simple Partial Fractions

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPFs) as suggested by Dolzhenko. Many solved and unsolved extremal problems, related to SPFs, are traced back to works of Boole, Macintyre, Fuchs, Marstrand, Gorin, Gonchar, and Dolzhenko. Now many authors systematically develop methods for approximation and interpolation by SPFs and their modifications. Simultaneously, related problems, being of independent interest, arise for SPFs: obtaining inequalities of different metrics, estimation of derivatives, separation of singularities, etc. In introduction to this survey, we systematize some of these problems. In themain part, we formulate principal results and outline methods to prove them whenever possible.