The \L ojasiewicz exponent at infinity
References: Paper of Krasinski, On the \L ojasiewicz exponent at infinity of polynomial mappings, Acta. Math. Vietnam. Vol. 32, No. 2-3, 2007.
Definition:
Let be a polynomial mapping. The \L ojasiewicz exponent of
at infinity is defined as the best exponent
for which the following inequality holds
for some constant and sufficiently large
.
Note that .
Definition
Let be a polynomial mapping and
be unbounded set. The \L ojasiewicz exponent of
at infinity on
is defined as the best exponent
for which the following inequality holds
for some constant and sufficiently large
in
. This exponent is denoted by
:
For example
- We have
.
- And
where
.
\L ojasiewicz exponent has relationship with properness of mappings
Theorem 1
if and only if
is a proper mapping.
Remark 1 Recall the properness of map.
is called proper mapping if
sastisfies the following property:
is a compact set if
is a compact set. This fact is equivalent to
.