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Let T = (f, g) : C2 ---> C2 be a polynomial map in two variables over the
field of complex numbers C such
that the Jacobian of f=f(x,y) and g =g(x,y) in C[x,y] is a nonzero
constant. The conjecture asserts that C[x,y]=C[f,g]. To prove the
conjecture, it is enough to show that T is injective. In this talk we
discuss some topological property of a surface defined by any counter
example to this conjecture, if it exists.
We call the map T ``standard'' if the highest degree term of f is yn for
some n > 0 and C(t , f, g)=C(x, y), where t= y/x. Precisely, we show the
following: ``There is a polynomial automorphism U of C2 such that TU is
standard.''
A point P=(p: q: 0) at infinity of the complex projective 2-space P2 is
said to be ``quasifinite'' if there is an infinite sequence {Pi} of points
in C2 (subset of P2) which converges to P such that the image {T(Pi)}
converges to a point in C2. Then we have the following:
The conjecture holds if and only if there is no quasifinite
point.
If P is a quasifinite point, then P=(1:0:0) in case T is standard.
We also consider a homotopy type obstacle which occurs by the existence of
such a quasifinite point.
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