Abstract
A polynomial with exactly two critical values is called a generalized Chebyshev polynomial (or Shabat polynomial). A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials f and g are called Z-homotopic if there exists a family pα, α \( \epsilon \) [0, 1], where p0 = f, p1 = g, and pα is a Zolotarev polynomial if α \( \epsilon \) (0, 1). As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work, we prove some necessary geometric conditions for the existence of Z-homotopy of plane trees, describe Z-homotopy for trees with five and six edges, and study one interesting example in the class of trees with seven edges.
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References
S. Lando and A. Zvonkin, Graphs on Surfaces and Their Applications, Springer, Berlin (2004).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 6, pp. 161–170, 2013.
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Kochetkov, Y.Y. Chebyshev Polynomials, Zolotarev Polynomials, and Plane Trees. J Math Sci 209, 275–281 (2015). https://doi.org/10.1007/s10958-015-2502-6
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DOI: https://doi.org/10.1007/s10958-015-2502-6