Chebyshev Polynomials, Zolotarev Polynomials, and Plane Trees
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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.
KeywordsPlane Tree Chebyshev Polynomial Inverse Image Continuous Family Monodromy Group
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