content_image/317261.jpg content_image/317262.jpg content_image/317260.jpg
the determinant of any non-singular q × q submatrix of G ( y ). Then ¯ α = 0 if and only if y = 0 is not a root of some such determinant and ¯ α > 0 otherwise. In other words, ¯ α = 0 if and only if G (0) is of full row rank. 

Select some matrix ˜ G ( y )$_{l}$ for which ¯ k$_{det(}$$_{˜}$ G ( y )$_{l}$) = ¯ α. Note that then (11) implies that the limit: 

<img src='content_image/317261.jpg'>

is a polynomial in y on R p that is distinct from zero almost everywhere. 

For the matrix of polynomials G ( y ) of rank q and any non-singular q × q matrix S, for the polynomial matrix SG ( y ) there is some α = ( α$_{1}$, ..., α$_{p}$ ) such that 

<img src='content_image/317262.jpg'>

exists and is a finite non-zero polynomial matrix, ¯ G ( y ) . Indeed, define α$_{i}$ = min { j ¯ k$_{{}$$_{SG}$$_{(}$$_{y}$$_{)}$$_{}}$$_{ij}$ } , where { SG ( y ) }$_{ij}$ denotes the polynomial that is the ij − th element of the matrix SG ( y ) . From (11) existence of the limit matrix follows. 

Lemma 4.1. Suppose that there exists a = ( α$_{1}$, ..., α$_{q}$ ) with α$_{i}$ ≥ 0 and a non-singular q × q matrix S such that the limit matrix: 

<img src='content_image/317260.jpg'>

is a finite non-zero matrix. Then for ¯ α for which (13) holds we get q ∑ i =1 α$_{i}$ ≤ ¯ α. ¯ G ( y ) is non-singular if and only if 

