# The \L ojasiewicz exponent at infinity

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 ${F = (F_1, \dots, F_m) : \mathbb{C}^n \rightarrow \mathbb{C}^m}$ be a polynomial mapping. The \L ojasiewicz exponent of ${F}$ at infinity is defined as the best exponent ${\nu}$ for which the following inequality holds

$\displaystyle |F(z)| \ge C|z|^\nu,$

for some constant ${C > 0}$ and sufficiently large ${|z|}$.

$\displaystyle \mathcal{L}_{\infty}(F) := \sup\{\nu \in \mathbb{R}| \exists C >0, R>0 : \forall z \in \mathbb{C}^n, |z| \ge R, |F(z)| \ge C|z|^\nu\}.$

Note that ${\mathcal{L}_{\infty}(F) \in \mathbb{R} \cup \{-\infty\}}$.

Definition

Let ${F = (F_1, \dots, F_m): \mathbb{C}^n \rightarrow \mathbb{C}^m}$ be a polynomial mapping and ${S \subset \mathbb{C}^n}$ be unbounded set. The \L ojasiewicz exponent of ${F}$ at infinity on ${S}$ is defined as the best exponent ${\nu}$ for which the following inequality holds

$\displaystyle |F(z)| \ge C|z|^\nu$

for some constant ${C > 0}$ and sufficiently large ${|z|}$ in ${S}$. This exponent is denoted by ${\mathcal{L}_{\infty}(F|S)}$:

$\displaystyle \mathcal{L}_{\infty}(F|S) :=\sup\{\nu \in \mathbb{R}| \exists C >0, R>0 : \forall z \in S, |z| \ge R, |F(z)| \ge C|z|^\nu\}.$

For example

${F(x,y) = (x, xy-1): \mathbb{C}^2 \rightarrow \mathbb{C}^2.}$

• We have ${\mathcal{L}_{\infty}(F) = -1}$.
• And ${\mathcal{L}_{\infty}(F|S) = 0}$ where ${S = \{y = 0\}}$.

\L ojasiewicz exponent has relationship with properness of mappings

Theorem 1 ${\mathcal{L}_{\infty}(F) > 0}$ if and only if ${F}$ is a proper mapping.

Remark 1 Recall the properness of map. ${F}$ is called proper mapping if ${F}$ sastisfies the following property: ${F^{-1}(K)}$ is a compact set if ${K}$ is a compact set. This fact is equivalent to ${|x| \rightarrow +\infty \Rightarrow |F(x)| \rightarrow +\infty}$.