On formulation of limitary spectrum of banded toeplitz matrices

A problem on the formulation of the limit set for the sequences of increasing Toeplitz matrices eigenvalues (of the limitary spectrum) with the chosen symbol is solved. The limitary spectrum of the banded Toeplitz matrices is described as a semialgebraic set, viz a solution set of some system of algebraic equations and inequalities. In the solution to the problem, the multivariate resultant technique is used. An algorithm of the limitary spectrum determination is formulated for the particular case. An example of the limitary spectrum computation for the specific symbol is considered. These results check the classical results by F. Spitzer and P. Schmidt on the description of the limitary spectrum of the non-selfadjoint Toeplitz matrices. It should be pointed out that these classical results are significantly existence theorems, and the problem on the effective computation of limitary spectrum by F. Spitzer and P. Schmidt is not considered.