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Real polynomial iterative roots in the case of nonmonotonicity height \(\geqslant 2\). (English) Zbl 1259.68246

Summary: It is known that a strictly piecewise monotone function with nonmonotonicity height \(\geqslant 2\) on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of “equal to”. Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of “less than” for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height \(\geqslant 2\) which have quadratic polynomial iterative roots of order 2 and answer the question.

MSC:

68W30 Symbolic computation and algebraic computation
39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
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