Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach

This paper formulates the problem of determining the degrees of all polynomial entries in a minimal polynomial basis of a polynomial matrix by using only the degree structure of the given matrix. Using a block Toeplitz matrix structure corresponding to the given polynomial matrix, it is shown that the degrees of the elements in its minimal polynomial basis, in the generic case, depend only on the degree structure of the given matrix, and not explicitly on the polynomial coefficients. The genericity assumption ensures that numerical issues in determination of the nullspace do not arise like for a specific case. Instead of the generic case, when dealing with a specific polynomial matrix, this method gives an upper bound on the degree structure of its minimal polynomial basis. The Toeplitz structure effectively reduces the problem of the computation of a right annihilator of a polynomial matrix to a relatively simpler, equivalent problem of computing an annihilator of a constant matrix.