Around multivariate Schmidt-Spitzer theorem

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Given an arbitrary complex-valued infinite matrix A and a positive integer n we introduce a naturally associated polynomial basis Bₐ of C[x0...xn]. We discuss some properties of the locus of common zeros of all polynomials in Bₐ having a given degree m; the latter locus can be interpreted as the spectrum of the m*(m+n)-submatrix of A formed by its m first rows and m+n first columns. We initiate the study of the asymptotics of these spectra when m goes to infinity in the case when A is a banded Toeplitz matrix. In particular, we present and partially prove a conjectural multivariate analog of the well-known Schmidt-Spitzer theorem which describes the spectral asymptotics for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases Bₐ and multivariate orthogonal polynomials.