Abstract
An important role is played in the solution of a class of optimal control problems by a certain special polynomial of degree with integer coefficients. The linear independence of a family of roots of this polynomial over the field implies the existence of a solution of the original problem with optimal control in the form of an irrational winding of a -dimensional Clifford torus, which is passed in finite time. In the paper, we prove that for one can take an arbitrary positive integer not exceeding for . The apparatus developed in the paper is applied to the systems of Chebyshev-Hermite polynomials and generalized Chebyshev-Laguerre polynomials. It is proved that for such polynomials of degree every subsystem of roots with pairwise distinct squares is linearly independent over the field .Bibliography: 11 titles.
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