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}.

On the diffetential of a mapping 2

In the case {\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m} case, there is a linear map, which is “linear approximation” of {\psi}. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If {M} and {N} are smooth manifolds and {\psi \colon M \rightarrow N} is a smooth map then for each {m \in M}, the map

\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N

is defined by

\displaystyle d\psi(v)(f) = v(f \circ \psi)

is called the pushforward. Actually,

\displaystyle d\psi \colon TM \rightarrow TN.

Suppose that {\dim{M} \ge \dim{N}} and {f \colon M \rightarrow N} is a differentiable mapping. We have

Definition 1 The mapping {f} is called a trivial fibration (differentiable) on {N} if there exists a differential manifold {F}, is called fibre of {f}, and a diffeomorphism

\displaystyle \phi \colon M \rightarrow N \times F

such that the following diagram is commutative

sodo

Representation of a linear functional

We review results of Haviland and Riesz on the reperesentation of a linear functional.

Definition 1 Let {X} be a subset of {\mathbb{R}^n} and {C(X)} be algebra of continuous functions on {X}. A positive linear functional on {C(X)} is a linear functional {L} with {L(f) \ge 0} for all {f \in C(X)} such that {f(a) \ge 0, \forall a \in X}.

We recall Haviland’s result in \cite{Marshall2} (also see \cite{Ha1, Ha2}), with {\mathbb{R}[x_1, \dots, x_n]} denotes the ring of real multivariable polynomials:

Theorem 2 (Haviland) For a linear functional {L: \mathbb{R}[x_1, \dots, x_n]} and closed set {K} in {\mathbb{R}^n}, the following are equivalent:

  1. {L} comes from a Borel measure on {K}, i.e., {\exists} a Borel measure {\mu} on {K} such that, {\forall f \in \mathbb{R}[x_1, \dots, x_n], L(f) = \int f d\mu.}
  2. {L(f) \ge 0} holds for all {f \in \mathbb{R}[x_1, \dots, x_n]} such that {f \ge 0} on {K}.

In Haviland’s theorem, a positive linear functional extended from ring of real multivariable polynomials to larger subalgebra and this theorem can be derived as a consequence of the following Riesz Representation Theorem (see \cite[p. 77]{KS}):

Theorem 3 (Riesz Representation Theorem) Let {X} be a locally compact Hausdorff space and let {L: C_c(X) \rightarrow \mathbb{R}} be a positive linear functional. Then there exists a unique Borel measure {\mu} on {X} such that

\displaystyle L(f) = \int f d\mu, \forall f \in C_c(X).

{C_c(X)} is the algebra of continuous functions with compact support.

Riesz Representation Theorem – 1

This paragraph we follow the book of Rudin [1]. On linear functionals, there is special relationship between integration and linear functionals. In {L^1(\mu)}, a vector space, for any positive measure {\mu}, the mapping

\displaystyle f \mapsto \int_X f d\mu.

example, let {C([0,1])} be the set of all continuous functions on the unit interval {I = [0,1]}. Then

\displaystyle \Lambda f = \int_0^1 f(x)dx \quad \ (f \in C([0,1])),

has two properties:

  • {\Lambda(f + g) = \int_0^1[f(x) + g(x)]dx = \int_0^1f(x)dx + \int_0^1g(x)dx = \Lambda(f) + \Lambda(g)}.
  • {\Lambda(c.f) =\int_0^1cf(x)dx = c\int_0^1f(x)dx= c.\Lambda(f)}.

so it is a linear functional on {C([0,1])}. Moreover, this is a positive linear functional: if {f \ge 0} then {\Lambda(f) \ge 0}.

Consider a segment {(a,b) \subset I} and the class of all {f \in C(I)} with {0 \le f(x) \le 1, \forall x \in I} and {f(x) = 0, \forall x \notin (a,b)} or support of {f} is {(a,b) \subset I}. So we get {\Lambda(f) = \int_a^b f(x)dx \le \int_a^bdx = b-a}. It is important that the length of {(a,b)} related to the values of the functional {\Lambda}.

There is an important theorem of F. Riesz, this illustrates to above event

Theorem 1 (F. Riesz) Let X be a closed subset of \mathbb{R}. Then every positive linear functional {\Lambda} on {C(X)} there corresponds a finite positive Borel measure {\mu} on {X} such that

\displaystyle \Lambda(f) = \int_X fd\mu \quad \ (f \in C(X)).

References

[1] W. Rudin, Real and Complex analysis, Mc.Graw-Hill, 1970.

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…

On a property of holomorphic functions

We can see that a holomorphic functions are the complex functions such that they can express a convergent power series

\displaystyle f(z) = \sum_{i=0}^\infty a_nz^n.

We want to compute {\int\limits_\gamma f(z)dz}. Suppose {\gamma} is closed curve (Jordan curve).

We compute the integral for {\gamma = \mathbb{S}^1}. Firstly, consider {\int_{\mathbb{S}^1}z^ndz, n \ge 1}. Change the variables, {z = e^{2\pi it}, t \in [0,1]}, so

\displaystyle \int_{\mathbb{S}^1}z^ndz = \int_0^1e^{2n\pi it}2 \pi i\cdot e^{2\pi it}dt = \int_0^1e^{2(n+1)\pi it}2 \pi idt = \dfrac{e^{2(n+1)\pi it}}{n+1}\bigg|_0^1 = 0.

This computation, we think that it is important, because it implies a classical result
Theorem 1 (Cauchy integral theorem) Let {f : \mathbb{C} \rightarrow \mathbb{C}} be a holomorphic function on {\mathbb{C}}. Then

\displaystyle \int_\gamma f(z) dz= 0,

with {\gamma} is closed Jordan curve.

Of course it has modulo an any closed Jordan curve: If \gamma_1 and \gamma_2 are two closed Jordan curve with a common fixed point then  \displaystyle \int_{\gamma_1} f(z)dz = \int_{\gamma_2} f(z) dz,

Một ví dụ về mặt chính quy

BT. Cho hàm f(x,y,z)=z^{2}. CMR 0  không là giá trị chính qui của hàm f thế nhưng f^{-1}(0) vẫn là mặt chính qui.

Lời giải. 

Ta có ma trận (f_x \ f_y \ f_z) = (0 \ 0 \ 2z) suy biến tức rankA < 1 khi và chỉ khi z=0, do đó tại điểm (x, y, 0), \forall x, y \in \mathbb{R}, ma trận trên suy biến và do đó là điểm kì dị. Mà f(x,y,0)=0 nên 0 là giá trị tới hạn, hay ko phải giá trị chính qui. Nhưng f^{-1}(0) là mặt phẳng Oxy, đây là mặt trơn nên chính qui.