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Consider a polynomial h ( y ) = Σ n k $_{=0}$h$_{k}$ ( y ) with homogeneous polynomial terms of order k : 

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Denote by ¯ k$_{h}$ the lowest order of homogeneous polynomial entering into polynomial h ( y ) : 

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Note that 

with all r$_{l}$ < 0 and r$_{l}$ ( y ) polynomial with ¯ k$_{r}$$_{l}$ > ¯ k$_{h}$. 

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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. 

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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 

