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}.

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Classical Lojasiewicz inequalities

Let {f : \mathbb{R}^n \rightarrow \mathbb{R}} be a real analytic function with {f(0) = 0}. Let {V := \{x \in \mathbb{R}^n | f(x) = 0\}} and {K} be a compact subset in {\mathbb{R}^n}. Then the (classical) \L ojasiewicz inequality asserts that:

  • There exist {c > 0, \alpha > 0} such that

    \displaystyle |f(x)| \ge cd(x, V)^\alpha\quad \text{for}\ x \in K.\ \ \ \ \ (1)

\noindent Let {f : \mathbb{R}^n \rightarrow \mathbb{R}} be a real analytic function with {f(0) = 0} and {\nabla f(0) = 0}. The \L ojasiewicz gradient inequality asserts that:

  • There exist {C > 0, \rho \in [0, 1)} and a neighbourhood {U} of {0} such that

    \displaystyle \|\nabla f(x)\| \ge C|f(x)|^\rho\quad \text{for}\ x \in U.\ \ \ \ \ (2)

As a consequence, in (1), the order of zero of an analytic function is finite, and if {f (x)} is close to {0} then {x} is close to the zero set of {f}. However, if {K} is not compact, the latter is not always true and the inequality (1) does not always hold. The inequality (2) is similar to (1), it is not true in the case of K is non-compact.

Two inequalities (1) and (2) have some special cases. For example, in the inequality (1), if {f} has only isolated zero, i.e. {V = f^{-1}(0) = \{(0, 0, \dots, 0)\}}, this implies {d(x, V) = \|x\|}. Hence, we have

\displaystyle |f(x)| \ge C\|x\|^\alpha, \text{for}\ x\in K.

On the other hand, being different from (2), we have another inequality:

\displaystyle \|\nabla f(x)\| \ge c\|x\|^\beta, \text{for}\ x \in U.

There are some relations between {\alpha, \beta} and {\rho} in complex case and real cases…