Around a multivariate Schmidt–Spitzer theorem

Given an arbitrary complex-valued infinite matrix A=(aij)A=(aij), i=1,…,∞i=1,…,∞; j=1,…,∞j=1,…,∞ and a positive integer n   we introduce a naturally associated polynomial basis BABA of C[x0,…,xn]C[x0,…,xn]. We discuss some properties of the locus of common zeros of all polynomials in BABA having a given degree m  ; the latter locus can be interpreted as the spectrum of the m×(m+n)m×(m+n)-submatrix of AA formed by its m   first rows and by the (m+n)(m+n) first columns. We initiate the study of the asymptotic of these spectra when m→∞m→∞ in the case when AA 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 asymptotic for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases BABA and multivariate orthogonal polynomials.