**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 1if and only if is a proper mapping.

Remark 1Recall 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 .