The {\L}ojasiewicz exponent for weighted homogeneous polynomials of two real variables

TL;DR

This paper determines the exact {a}ojasiewicz exponent for isolated weighted homogeneous polynomials in two real variables based on their weights, providing precise mathematical characterization.

Contribution

It provides a formula for the {a}ojasiewicz exponent of weighted homogeneous polynomials in two real variables, linking it explicitly to the polynomial's weights.

Findings

Exact {a}ojasiewicz exponent values derived for the class of polynomials.

Explicit relationship established between weights and exponent.

Enhances understanding of singularity behavior in real polynomial functions.

Abstract

The purpose of this paper is to give the exact value of the {\L}ojasiewicz exponent for an isolated weighted homogeneous polynomials of two real variaibles in terms of its weights.

Equations

$$L(f):=\inf \{\lambda >0:\mid \text{grad}f\mid \ge \text{const.}\mid x{\mid }^{\lambda }\text{ near zero }\},$$

$$L(f)=\sup \left\{\frac{\text{ord}(\text{grad}f(\phi (t)))}{\text{ord}(\phi (t))}:0\ne \phi (t)\in \mathbb{R}\{t{\}}^n,\phi (0)=0\right\},$$

$$L(f)=\underset{i=1}{\overset{n}{\max }}(\frac{d}{w_i}-1).$$

$${\alpha }_1{w}_1+\cdots +{\alpha }_n{w}_n=d.$$

$$S_r=\{x\in {\mathbb{R}}^n:\rho (x)=r\},r>0.$$

$$t\cdot x=(t^{w_1}{x}_1,\dots ,t^{w_n}{x}_n)$$

$$⟨{\rho }^{w_i}\frac{\partial }{\partial x_i},{\rho }^{w_j}\frac{\partial }{\partial x_j}⟩={\delta }_{i,j}:=\{\begin{array}{ll}1& \text{ if }i=j\\ 0& \text{ if }i\ne j\end{array}$$

$$f(x)=H_d(x)+H_{d+1}(x)+\cdots ;H_d\ne 0.$$

$$\Vert {\text{grad}}_{w}f(x){\Vert }_w\lesssim {\rho }^{d_w(f)}(x),$$

$$\Vert {\text{grad}}_w{H}_j(\frac{1}{\rho (x)}\cdot x){\Vert }_w=\frac{\Vert {\text{grad}}_w{H}_j(x){\Vert }_w}{\rho (x{)}^j}\lesssim 1.$$

$$\parallel {\text{grad}}_{w}f(x){\parallel }_w\lesssim \sum _{j\ge d_w(f)}\parallel {\text{grad}}_w{H}_j(x){\parallel }_w\lesssim {\rho }^{d_w(f)}(x).$$

$$\Vert {\text{grad}}_{w}f(x){\Vert }_w\gtrsim \rho (x{)}^d.$$

$$\Vert {\text{grad}}_{w}f(x){\Vert }_w={\left(\sum _{i=1}^n|{\rho }^{w_i}(x)\frac{\partial f}{\partial z_i}(x){|}^2\right)}^{\frac{1}{2}}\gtrsim 1,\forall x\in S_r.$$

$$\Vert {\text{grad}}_{w}f(\frac{r}{\rho (x)}\cdot x){\Vert }_w=r^d\frac{\Vert {\text{grad}}_{w}f(x){\Vert }_w}{\rho (x{)}^d}\gtrsim 1.$$

$$L(f)=\{\begin{array}{ll}\frac{d-w_1}{w_2}& \text{ if }\{\frac{\partial f}{\partial x_1}(x_1,x_2)=0\}\subset \{x_2=0\},\\ \frac{d}{w_2}-1& \text{ if not.}\end{array}$$

$$\Vert {\text{grad}}_{w}f(x){\Vert }_w\sim \rho (x{)}^d.$$

$$L(f)=\frac{d}{w_2}-1=\frac{d-w_1}{w_2}.$$

$$L(f)\ge \frac{\text{ord}(\partial f(\phi (t))}{\text{ord}(\phi (t))}=\frac{d-w_1}{w_2}.$$

$$f(x_1,x_2)=\phi (x_1,x_2)\text{ or }f(x_1,x_2)=x_2\phi (x_1,x_2).$$

$$\parallel \partial f\parallel \gtrsim {\rho }^{d-w_1}\gtrsim (\mid x_1{\mid }^{1/w_1}+\mid x_2{\mid }^{1/w_2}{)}^{d-w_1}\gtrsim \parallel (x_1,x_2){\parallel }^{\frac{d-w_1}{w_2}}.$$

$$\parallel \partial f{\parallel }^2=\mid x_2{\partial }_{x_1}\phi {\mid }^2+\mid \phi +x_2{\partial }_{x_2}\phi {\mid }^2$$

$$\phi (x_1,x_2)=\frac{w_1}{d-w_2}{x}_1{\partial }_{x_1}\phi (x_1,x_2)+\frac{w_2}{d-w_2}{x}_2{\partial }_{x_2}\phi (x_1,x_2).$$

$$\parallel \partial f{\parallel }^2=\parallel (\frac{w_1}{d-w_2}{x}_1,x_2){\parallel }^2\mid {\partial }_{x_1}\phi {\mid }^2+2\frac{w_{1}d}{d-w_2}{x}_1{x}_2{\partial }_{x_1}\phi {\partial }_{x_2}\phi +\mid \frac{d}{d-w_2}{x}_2{\partial }_{x_2}\phi {\mid }^2.$$

$$d_w(({\partial }_{x_1}\phi {)}^2)=2(d-w_2-w_1)<d-w_2-w_1+d-w_2-w_2=d_w({\partial }_{x_1}\phi {\partial }_{x_2}\phi )<d_w(({\partial }_{x_2}\phi {)}^2).$$

$$\parallel \partial f\parallel \gtrsim \parallel (x_1,x_2)\parallel {\rho }^{d-w_2-w_1}=\parallel (x_1,x_2)\parallel (\mid x_1{\mid }^{1/w_1}+\mid x_2{\mid }^{1/w_2}{)}^{d-w_2-w_1}\gtrsim \parallel (x_1,x_2){\parallel }^{\frac{d-w_1}{w_2}}.$$

$$L(f)\ge \frac{\text{ord}(\partial f(\phi (t))}{\text{ord}(\phi (t))}\ge \frac{d-w_2}{w_2}.$$

$${\rho }^{w_2}\parallel \partial f\parallel \gtrsim \Vert {\text{grad}}_{w}f(x){\Vert }_w\gtrsim \rho (x{)}^d.$$

$$\parallel \partial f\parallel \gtrsim {\rho }^{d-w_2}=(\mid x_1{\mid }^{1/w_1}+\mid x_2{\mid }^{1/w_2}{)}^{d-w_2}\gtrsim \parallel (x_1,x_2){\parallel }^{\frac{d-w_2}{w_2}}.$$

$$f(x,y)=x^3+x{y}^6+y^9,$$

$$f(x,y)=y(x^5+x{y}^{12}+y^{15}),$$

Full text

Let $f\colon(\mathbb{R}^{n},0)\rightarrow(\mathbb{R},0)$ be a analytic function. The Łojasiewicz exponent $L(f)$ of $f$ is by definition

[TABLE]

It is well known (see [3, 8]) that the Łojasiewicz exponent can be calculated by means of analytic paths

[TABLE]

where $\text{ord}(\phi):=\inf_{i}\{\text{ord}(\phi_{i})\}$ for $\phi\in{\mathbb{R}\{t\}}^{n}$. By definition, we put $\text{ord}(0)=+\infty$. It is also known that $L(f)<+\infty$ if and only if $f$ has an isolated singularity at the origin. The computation or estimation of the Łojasiewicz Exponent is a quite interesting problem not only in geometric analysis but also in singularity theory. For example, Kuiper-kuo theorem ([6, 7]) proved that for any integer $r$ greater than $L(f)$, $f$ is a $C^{0}$-sufficient, $r$-jet, i.e., adding to the function $f$ monomials of order greater than $L(f)$ does note change its topological type. J. Bochnak and S. Łojasiewicz [2] showed that $C^{0}$-sufficiency degree of $f$ (i.e., the minimal integer $r$ such that f is $C^{0}$-sufficient, $r$-jet) is equal to $[L(f)]+1$, where $[L(f)]$ denotes integral part of $L(f)$.

[TABLE]

Estimates of the Łojasiewicz Exponent for weighted homogeneous isolated singularities in the real cases are in a recent paper by Haraux and Pham [5].

Motivated by the above observation, we are looking to establish the Łojasiewicz Exponent for the classes of weighted homogeneous polynomials of two real variables in terms of the weights. To prove the main result (Theorem 3 below), we recall the notion of weighted homogenous filtration introduced by Paunescu in [9]. By using it and the generalized Euler identity, we can compute the Łojasiewicz exponent of weighted homogeneous polynomials of two real variables.

Let $\varphi,\,\,\psi\colon(\mathbb{R}^{n},0)\to\mathbb{R}$ be two function germs. We say that $\varphi(x)\lesssim\psi(x)$ if there exists a positive constant $C>0$ and an open neighborhood $U$ of the origin in $\mathbb{R}^{n}$ such that $\varphi(x)\leq C\;\psi(x)$, for all $x\in U$. We write $\varphi(x)\sim\psi(x)$ if $\varphi(x)\lesssim\psi(x)$ and $\psi(x)\lesssim\varphi(x)$. Finally, $|\varphi(x)|\ll|\psi(x)|$ (when $x$ tends to $x_{0}$) means $\lim_{x\to x_{0}}\frac{\varphi(x)}{\psi(x)}=0$.

1. Weighted homogeneous filtration, main results

Let $\mathbb{N}$ be the set of nonnegative integers and $\mathcal{O}_{n}$ denote the ring of analytic function germs $f\colon(\mathbb{R}^{n},0)\to(\mathbb{R},0)$.

From now, we shall fix a system of positive integers $w=(w_{1},\dots,w_{n})\in(\mathbb{N}-\{0\})^{n}$, the weights of variables $x_{i},$ $w(x_{i})=w_{i},$ $1\leq i\leq n$, and a positive integer $d$, then a polynomial $f\in\mathbb{R}[x_{1},\dots,x_{n}]$ is called weighted homogeneous of degree $d$ with respect the weight $w=(w_{1},\dots,w_{n})$ (or type $(d;w)$) if $f$ may be written as a sum of monomials $x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$ with

[TABLE]

We say that an analytic function $f\colon(\mathbb{R}^{n},0)\to(\mathbb{R},0)$ is non-degenerate if $\{\frac{\partial f}{\partial_{x_{1}}}(x)=\cdots=\frac{\partial f}{\partial_{x_{n}}}(x)=0\}=\{0\}$ as germs at the origin of $\mathbb{R}^{n}$.

We may introduce (see [9]) the function $\rho(x)=\left(|x_{1}|^{\frac{2}{w_{1}}}+\cdots+|x_{n}|^{\frac{2}{w_{n}}}\right)^{\frac{1}{2}}$. We also consider the spheres associated to this $\rho$

[TABLE]

Here $\cdot$ means the weighted action, with respect to the $\mathbb{R}^{*}$ action defined below

[TABLE]

Definition 1**.**

Using $\rho$, we define a singular Riemannian metric on $\mathbb{R}^{n}$ by the following bilinear form

[TABLE]

We will denote by $\text{grad}_{w}$ and $\parallel\;\parallel_{w}$, the corresponding gradient and norm associated with this Riemannian metric (for more details about these see [9]).

Let $f\in\mathcal{O}_{n}$. We denote the Taylor expansion of $f$ at the origin by $\sum c_{\nu}x^{\nu}$. Setting $H_{j}(x)=\sum c_{\nu}x^{\nu}$ where the sum is taken over $n$ with $<w,\nu>=w_{1}\nu_{1}+\cdots+w_{n}\nu_{n}=j$, we can write the weighted homogeneous Taylor expansion $f$

[TABLE]

We call $d$ the weighted degree of $f$ and $H_{d}$ the weighted initial form of $f$ about the weight. Furthermore, for any $f\in\mathcal{O}_{n}$ we get

[TABLE]

where $d_{w}(f)$ denotes the degree of $f$ with respect to $w$. Indeed, as all nonzero $x$, we find $\frac{1}{\rho(x)}\cdot x\in S_{1}$, moreover, we have $\frac{\partial H_{j}}{\partial x_{i}}$ is zero or a weighted homogeneous polynomial of degree $d-w_{j}$, then we obtain

[TABLE]

Therefor,

[TABLE]

Proposition 2**.**

Let $f\in\mathcal{O}_{n}$ be a weighted homogeneous isolated singularity of type $(d;w)$ at $0\in\mathbb{R}^{n}$. Then

[TABLE]

Proof.

Since $f$ has only isolated singularity at the origin, then for small values of $r$ we have

[TABLE]

On the other hand, $\frac{\partial f}{\partial x_{i}}$ is weighted homogeneous of degree $d-w_{i}$ for $i=1,\dots,n$ and also, $\frac{r}{\rho(x)}\cdot x\in S_{r}$ for all nonzero $x$. Thus, by (1.4) we obtain

[TABLE]

This completes the proof of the proposition. ∎

The main result of this paper is the following:

Theorem 3**.**

Let $f\colon(\mathbb{R}^{2},0)\to(\mathbb{R},0)$ be non-degenerate weighted homogeneous polynomial of type $(d;w_{1},w_{2})$ such that $w_{1}\geq w_{2}$. Then

[TABLE]

2. Proof of Theorem 3

We first note that in the case where $w_{1}=w_{2}$ (i.e, homogenous filtration), so we can find from the (1.2) and (1.3) that

[TABLE]

Butt $\|\text{grad}_{w}f(x)\|_{w}\sim\parallel x\parallel\;\parallel\partial f\parallel$ and $\rho(x)^{d}\sim\parallel x\parallel^{\frac{d}{w_{1}}}=\parallel x\parallel^{\frac{d}{w_{2}}}$. Hence, we get

[TABLE]

From now, we suppose that $w_{1}>w_{2}$. There are two cases to be considered.

Case 1. In this case, we suppose that $\{\frac{\partial f}{\partial x_{1}}(x_{1},x_{2})=0\}\subset\{x_{2}=0\}$, take an analytic path $\varphi(t)=(t^{w_{1}},t^{w_{2}})=t\cdot(1,1)$, then from (0.1) we get

[TABLE]

Since $f$ defining an isolated singularity at the origin $0\in\mathbb{R}^{n}$, there exist the terms

$\varphi\colon(\mathbb{R}^{2},0)\to(\mathbb{R},0)$ with the origin is an isolated zero of $\partial_{x}\varphi$ i.e., $\{\partial_{x_{1}}\varphi=0\}=\{0\}$ and

[TABLE]

For the case where the origin is an isolated zero of $\partial_{x_{1}}f$, since $\partial_{x_{1}}f$ $w$-form of degree $d-w_{1}$, it follow from (1.4) that

[TABLE]

Hence $L(f)\leq\frac{d-w_{1}}{w_{2}}$.

This ends the proof of Theorem 3 in the case where the origin is an isolated zero of $\partial_{x_{1}}f$.

From now we suppose that $f(x_{1},x_{2})=x_{2}\varphi(x_{1},x_{2})$, it is easy to see that $\varphi$ is weighted homogeneous of degree $d-w_{2}$ (type $(d-w_{2};w_{1},w_{2}))$, we have that

[TABLE]

Moreover, it follows from the generalized Euler identity that

[TABLE]

Then, we obtain that

[TABLE]

But, it follows from $w_{1}>w_{2}$ that

[TABLE]

Therefore , by the origin is an isolated zero of $\partial_{x_{1}}\varphi$ we get that

[TABLE]

Hence $L(f)\leq\frac{d-w_{1}}{w_{2}}$.

This ends the proof of Theorem 3 in the first case.

Case 2. In this case, we suppose that $\{\frac{\partial f}{\partial x_{1}}(x_{1},x_{2})=0\}\nsubseteq\{x_{2}=0\}$. Let $a=(a_{1},a_{2})\in\{\frac{\partial f}{\partial x_{1}}(x_{1},x_{2})=0\}$ and $a_{2}\neq 0$, take an analytic path $\varphi(t)=(t^{w_{1}}a_{1},t^{w_{2}}2_{2})=t\cdot a$, then from (0.1) we get

[TABLE]

On the other hand, by the proposition 2, we obtain that

[TABLE]

Then

[TABLE]

Hence $L(f)\leq\frac{d-w_{2}}{w_{2}}$.

This ends the proof of Theorem 3.

We conclude with several examples.

Example 4**.**

Let

[TABLE]

$f$* is weighted homogenous of type $(9;3,1)$ defining an isolated singularity, since $\{\partial_{x}f=0\}=\{0\}$, then by theorem 3. we get $L(f)=6$. Also, for $g(x,y)=x^{3}-xy^{6}+y^{9}$ can be seen as weighted homogenous of the same type, but $\{\partial_{x}g=0\}\nsubseteq\{y=0\}$, hence by theorem 3 we get $L(g)=8$.*

Example 5**.**

Let

[TABLE]

$f$* is weighted homogenous of type $(16;3,1)$ defining an isolated singularity, since $\{\partial_{x}f=0\}\subset\{y=0\}$, then by theorem 3 we get $L(f)=13$. Also, for $g(x,y)=y(x^{5}-xy^{12}+y^{15})$ can be seen as weighted homogenous of the same type, but $\{\partial_{x}g=0\}\nsubseteq\{y=0\}$, so by theorem 3 we obtain $L(g)=15$.*