On a dense winding of the 2-dimensional torus

An important role in the solution of a class of optimal control problems is played by a certain polynomial of degree $2(n-1)$ of special form with integer coefficients. The linear independence of a family of $k$ special roots of this polynomial over $\mathbb{Q}$ implies the existence of a solution of the original problem with optimal control in the form of a dense winding of a $k$-dimensional Clifford torus, which is traversed in finite time. In this paper, it is proved that for every integer $n>3$ one can take $k$ to be equal to $2$.
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