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Статья

Chebyshev Polynomials, Zolotarev Polynomials, and Plane Trees

Journal of Mathematical Sciences. 2015. Vol. 209. No. 2. P. 275-281.

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(Formula presented.) [0, 1], where p0 = f, p1 = g, and pα is a Zolotarev polynomial if α(Formula presented.) (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. © 2015 Springer Science+Business Media New York