Leírás
First, the original question is demonstrated: existence of a special
conformal mapping on the complex plane. Then, two more, closely
related problems are mentioned (rational functions with prescribed
critical points or with prescribed critical values) and some earlier
theorems are also recalled. Then, some new results are presented,
including an existence theorem (with Riemann-Roch theorem),
constructive solutions (with Hilbert's Nullstellensatz and Gröbner
bases), some natural connections between critical points and critical
values (without coefficients, using elimination theory) and a related
question leads to applying Stahl's theorem (on analytic continuation
and quadratic differentials). Finally, some pending questions are
mentioned as conjectures.
This talk is based on a joint work with Sergei Kalmykov (SJTU) and
Olivier Sete (TU Berlin) and is in progress.