Consider a polynomial h ( y ) = Σ n k $_{=0}$h$_{k}$ ( y ) with homogeneous polynomial terms of order k : Denote by ¯ k$_{h}$ the lowest order of homogeneous polynomial entering into polynomial h ( y ) : Note that with all r$_{l}$ < 0 and r$_{l}$ ( y ) polynomial with ¯ k$_{r}$$_{l}$ > ¯ k$_{h}$. Consider all possible ˜ G ( y )$_{l}$ , with ˜ G ( y )$_{l}$ a q × q submatrix of G ( y ) ; l = 1 , ...L with L = p ! q !( p − q $_{)!}$. Define with the convention ¯ k$_{det(}$$_{˜}$ G ( y )$_{l}$) = + ∞ if det ( ˜ G ( y )$_{l}$ ) is the zero polynomial. Note that for some ˜ G$_{l}$ strict inequality ¯ k$_{det(}$$_{˜}$ G ( y )$_{l}$) > ¯ α may hold as shown in the example below. Example 4.1. G ( y ) = ∂g ∂y ′ for g ( y ) = ( y 2 1 + y 3 $_{3}$, y 2 + y 3 $_{4}$, y 2 1 + y 2 $_{2}$) $^{′}$; then ¯ α = min l ( ¯ k$_{det(}$$_{˜}$ G ( y )$_{l}$$_{)}$) (12) λ ¯ $^{k$_{h}$}$h ( y/λ ) = h ¯ $_{k$_{h}$}$( y ) + Σ λ $^{r$_{l}$}$r$_{l}$ ( y ) , (11) ¯ k$_{h}$ = min 0 ≤ k ≤ n { k : h$_{k}$ ( i$_{1}$, ..., i$_{p}$ ) = 0 for some i$_{1}$ + ... + i$_{p}$ = k } . (10) h$_{k}$ ( y ) = Σ$_{i}$$_{1}$$_{+}$$_{...}$$_{+}$$_{i}$$_{p}$$_{=}$$_{k}$ h$_{k}$ ( i$_{1}$, ..., i$_{p}$ ) y i$_{i}$ 1 ...y i$_{p}$ p . (9) 15