Khazanov, V. B. On some properties of polynomial bases of subspaces over the field of rational functions in several variables. (English. Russian original) Zbl 1071.15001 J. Math. Sci., New York 121, No. 4, 2538-2545 (2004); translation from Zap. Nauchn. Semin. POMI 284, 177-191 (2002). Summary: Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Cited in 3 Documents MSC: 15A03 Vector spaces, linear dependence, rank, lineability 15A54 Matrices over function rings in one or more variables Keywords:multiparameter rational vectors; polynomial bases; regular spectrum; singular spectra × Cite Format Result Cite Review PDF Full Text: DOI