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A proof of Proposition (19990109).
Put F(x,z)=z X2(x,x2z)/Y(x,x2z)
and G(x,z)=X(x,x2z)/x.
Then (x,y) is a fixed point of grad W (i.e., X=x and Y=y) if and only if
F(x,z)=G(x,z)=1, with z=y/x2.
Note also that (x,x2z) in the interior of D is equivalent to
x > 0 and z in (0,1).
The assumption that W is a polynomial with terms of total degree no less
than 3 (together with other assumptions implying Y(x,0) <> 0,
and consequently X(x,0) <> 0), implies that
G(0+,z)=0 and G(Infinity,z)=Infinity for z in [0,1].
F(x,0+)=0 for positive x also follows from the third assumption
in the Proposition.
That F(x,1) > 1 follows from the assumption in the Problem 1.
Hence there is a point (x,z) in the domain where F(x,z)=G(x,z)=1
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