Abstract
AbstractLet X be an irreducible complex affine variety of dimension greater than one and let $$f:X \rightarrow \mathbb {C}^m$$
f
:
X
→
C
m
be a polynomial mapping. Let $${|}*{|}$$
|
∗
|
be a semialgebraic norm on $$\mathbb {C}^m.$$
C
m
.
Then for R large enough the sets $$f^{-1}(B_R), f^{-1}(S_R), X{\setminus } f^{-1}(B_R)$$
f
-
1
(
B
R
)
,
f
-
1
(
S
R
)
,
X
\
f
-
1
(
B
R
)
are all connected, where $$B_R=\{ z\in \mathbb {C}^m : |z|\le R\}$$
B
R
=
{
z
∈
C
m
:
|
z
|
≤
R
}
and $$S_R=\{ z\in \mathbb {C}^m : |z| = R\}.$$
S
R
=
{
z
∈
C
m
:
|
z
|
=
R
}
.
As an application we show that if F is a counterexample to the Jacobian Conjecture, then the non-properness set of F has a non-trivial link at infinity.
Publisher
Springer Science and Business Media LLC
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