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Conservative polynomials. (Russian) Zbl 0728.12003

Studies in algebra, Tbilisi, 115-129 (1984).
[For the entire collection see Zbl 0721.00010.]
A monic polynomial f of a complex variable z is called conservative, if \(f(0)=0\) and all critical points of the mapping \(z\to f(z)\) are fixed. Very little is known about these polynomials. There is a series of hypotheses on the number of conservative polynomials of a given degree and on properties of their fixed points. In this paper an elementary method for describing conservative polynomials is proposed which allows one to obtain some conditional assertions in the framework of the mentioned hypotheses. Validity of all these hypotheses is established for conservative polynomials with \(\leq 3\) critical points, with simple critical points as well as for polynomials of degree \(\leq 6\). [See also the following review.]

MSC:

12D99 Real and complex fields
30C10 Polynomials and rational functions of one complex variable
37B99 Topological dynamics