On some properties of polynomial bases of subspaces over the field of rational functions in several variables

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 in multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis for which the associated basis multiparameter polynomial matrix has no finite regular spectrum, is refuted on an example. Some consequences of this fact are indicated. Simpler proofs of some properties of singular spectra of the basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented.