Symbolic Description of the Boundary Curves of the 2D Projections of the Unit Ball B_N; Dr. Robert Vajda
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In classical approximation theory the unit ball of the real univariate polynomials, B_n, that is, the set of polynomials of degree at most n with supremum norm less or equal than one on the interval [-1,1] is widely investigated. This set is complicated for larger n's. Chebyshev famous result on the nice sharp bounds of the coefficients of such polynomials can be considered as a result on the one dimensional projections of B_n along the coordinate axis in the standard monomial basis. However, less is known already about the exact 2D projections of B_n. In this talk we report how we explored the boudary curves of some 2D projections with the aid of symbolic and numeric computational tools for small n's. It turns out that the all the boundary points of these curves correspond to unit normed Zolotarev polynomials. Since for n<7 the latter polynomial families admit a univarite parametrization (see e.g. [Grasegger-Vo 2017], [Rack-Vajda 2019], there exact and relatively simple description is possible. Some open problems will be also presented. [Grasegger 2017] G. Grasegger, N. Vo, An Algebraic-Geometric Method for Computing Zolotarev Polynomials, ISSAC '17 Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, 173-180. [Rack-Vajda 2019] An Explicit Univariate and Radical Parametrization of the Sextic Proper Zolotarev Polynomials in Power Form, Dolomites Res. Notes Approx. 12 (2019), 43?50.