An explicit univariate and radical parametrization of the septic proper Zolotarev polynomials in power form
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The problem of determining an explicit one-parameter power form representation of the proper n-th degree Zolotarev polynomials on [-1,1] can be traced back to P. L. Chebyshev. It turned out to be complicated, even for small values of n. Such a representation was known to A. A. Markov (1889) for n=2 and n=3. But already for n=4 it seems that nobody really believed that an explicit form can be found. As a matter of fact it was, by V. A. Markov in 1892, as A. Shadrin put it in 2004. The next higher degrees, n=5 and n=6, were resolved only recently, by G. Grasegger and N. Th. Vo (2017) respectively by the present authors (2019). In this paper we settle the case n=7 using symbolic computation. The parametrization for the degrees n∈{2,3,4} is a rational one, whereas for n∈{5,6,7} it is a radical one. However, the case n=7 among the radical parametrizations requires special attention, since it is not a simple radical one.
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