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(}$$_{˜}$ G ( y )$_{l}$) = ¯ α. Note that then (11) implies that the limit: is a polynomial in y on R p that is distinct from zero almost everywhere. For the matrix of polynomials G ( y ) of rank q and any non-singular q × q matrix S, for the polynomial matrix SG ( y ) there is some α = ( α$_{1}$, ..., α$_{p}$ ) such that exists and is a finite non-zero polynomial matrix, ¯ G ( y ) . Indeed, define α$_{i}$ = min { j ¯ k$_{{}$$_{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: 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 17 ¯ G ( y ) = lim λ $_{→∞}$diag ( λ $^{α$_{i}$}$) SG ( y/λ ) (15) lim λ $_{→∞}$λ ¯ α det ( ˜ G$_{l}$ ( y/λ ) ) (13) lim λ $_{→∞}$diag ( λ $^{α$_{i}$}$) SG ( y/λ ) (14)