Fedoryuk values and stability of global H\"{o}lderian error bounds for polynomial functions

TL;DR

This paper investigates the stability of global H"olderian error bounds for polynomial functions, providing criteria, explicit formulas involving Fedoryuk values, and classifications of stability types under perturbations.

Contribution

It introduces criteria and explicit formulas for the existence and stability of global H"olderian error bounds for polynomial sublevel sets, using Fedoryuk values.

Findings

Criteria for existence of global H"olderian error bounds.

Explicit formulas for the set of thresholds with error bounds.

Classification of stability types for error bounds.

Abstract

Let $f$ be a polynomial function of $n$ variables. In this paper, we study stability of global H\"{o}lderian error bound for a nonempty sublevel set $[f \le t]$ under a perturbation of $t$. In this paper, we give: * Criteria for the existence of a global H\"{o}lderian error bound of $[f \le t]$; * Formulas for computing explicitly the set $$H(f) := \{ t \in \mathbb{R}: [f \le t]\ \text{has a global H\"{o}lderian error bound}\}$$ via some Fedoryuk values of $f$ and definition of threshold for the existence of global H\"{o}lderian error bound of $f$; * Definition of all types of stability of global H\"{o}lderian error bound of $[f \le t]$.

Equations

$$H(f):=\{t\in \mathbb{R}:[f\le t]\text{has a global Ho¨lderian error bound}\}$$

$$[f\le t]:=\{x\in {\mathbb{R}}^n|f(x)\le t\}$$

$$[f(x)-t{]}_{+}^{\alpha }+[f(x)-t{]}_{+}^{\beta }\ge c\mathrm{dist}(x,[f\le t])\text{for all}x\in {\mathbb{R}}^n.$$

$$H(f):=\{t\in \mathbb{R}:[f\le t]\text{has a global Ho¨lderian error bound}\}.$$

$$F(f):=\{t\in \mathbb{R}:\exists \{x^k\}\subset {\mathbb{R}}^n,\Vert x^k\Vert \to \infty ,\Vert \nabla f(x^k)\Vert \to 0,f(x^k)\to t\}.$$

$$\text{Either}H(f)=[h(f),+\infty )\setminus F^1(f)\text{or}H(f)=(h(f),+\infty )\setminus F^1(f).$$

$$\Vert x^k\Vert$$

$$f(x^k)>t,f(x^k)$$

$$\exists \delta >0\text{s.t.}\mathrm{dist}(x^k,[f\le t])$$

$$\Vert x^k\Vert$$

$$\exists M\in \mathbb{R}:t<f(x^k)$$

$$\mathrm{dist}(x^k,[f\le t])$$

$$[f(x)-t{]}_{+}^{\alpha }+[f(x)-t{]}_{+}^{\beta }\ge c\mathrm{dist}(x,[f\le t])\text{for all}x\in {\mathbb{R}}^n.$$

$$F^1(f)$$

$$F^2(f)$$

$$h(f)=\{\begin{array}{l}\inf f\text{ if }{F}^2(f)=\{\inf f\}\text{or}{F}^2(f)=\varnothing ,\\ +\infty \text{ if }{F}^2(f)=\mathbb{R},\\ \sup \{t\in \mathbb{R}:t\in F^2(f)\}\text{ if }{F}^2(f)\ne \varnothing \text{and}{F}^2(f)\ne \mathbb{R}.\end{array}$$

$$\text{either}{F}^2(f)$$

$$\text{or}{F}^2(f)$$

$$f(x_1,\dots ,x_n)=a_0{x}_n^d+a_1(x_1,\dots ,x_{n-1})x_n^{d-1}+\cdots +a_d(x_1,\dots ,x_{n-1})(\ast ),$$

$$\Vert x^k\Vert$$

$$f(x^k)>t,$$

$$\mathrm{dist}(x^k,[f\le t])$$

$$\text{and}\{x^k\}$$

$$\Vert x^k\Vert$$

$$t<f(x^k)$$

$$\mathrm{dist}(x^k,[f\le t])$$

$$\text{and}\{x^k\}$$

$$P^1(f)$$

$$P^2(f)$$

$$P(f)$$

$$[f(x)-t{]}_{+}^{{\alpha }_1}+[f(x)-t{]}_{+}^{{\beta }_1}\ge c_1\mathrm{dist}(x,[f\le t]),$$

$$[f(x)-t{]}_{+}^{{\alpha }_1}+[f(x)-t{]}_{+}^{{\beta }_1}+[f(x)-t{]}_{+}^{\frac{1}{d}}\ge c\mathrm{dist}(x,[f\le t]),$$

$$\psi (\tau ):=\{\begin{array}{ll}0& \text{if}[f(x)-t{]}_{+}=\tau \text{is empty}\\ \underset{[f(x)-t{]}_{+}=\tau ,x\in V_1}{\sup }\mathrm{dist}(x,[f\le t])& \text{if}[f(x)-t{]}_{+}=\tau \text{is not empty}\end{array}.$$

$$\psi (\tau )=a_0{\tau }^{\tilde{\alpha }}+o({\tau }^{\tilde{\alpha }})\text{as}\tau \to 0,\text{where}{a}_0>0.$$

$$\psi (\tau )=b_0{\tau }^{\tilde{\beta }}+o({\tau }^{\tilde{\beta }})\text{as}\tau \to +\infty ,\text{where}{b}_0>0.$$

$$[f(x)-t{]}_{+}^{\frac{1}{\tilde{\alpha }}}\ge \frac{a_0}{2}\mathrm{dist}(x,[f\le t]),$$

Full text

[TABLE]

via some special Fedoryuk values of $f$. Then, based on these formulas, we can determine the stability type of a global Hölderian error bound of $[f\leq t]$ for any value $t\in\mathbb{R}$.

1. Introduction

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a polynomial function. For $t\in\mathbb{R}$, put

[TABLE]

and $[a]_{+}:=\max\{0,a\}$.

Definition 1.1**.**

[Ha] We say that the nonempty set $[f\leq t]$ has a global Hölderian error bound (GHEB for short) if there exist $\alpha,\beta,c>0$ such that

[TABLE]

Note that, if $\alpha=\beta=1$, then (1) becomes a global Lipschitzian error bound for $[f\leq t]$.

The existence of error bounds have many important applications, including sensitivity analysis, convergence analysis in optimization problems, variational inequalities… After the earliest work by Hoffman ([Hoff]) and extended paper of Robinson ([Ro]), the study of error bounds has received rising awareness in many papers of mathematical programming in recent years, see [LL, WP, LS, Y, LiG1, LiG2, Ha, Ng, LMP, DHP] (for the case of polynomial functions) and [Hoff, Ro, M, AC, LiW, K, KL, P, LP, Luo, Jo, NZ, CM, LTW, I, BNPS, DL] (for non-polynomial cases). The reader is referred to survey papers [LP, P, Az, I] and the references therein for the theory and applications of error bounds.

Studying the stability of error bounds under perturbation is fundamental and hard problem. It has been investigated recently in the works of Daniel, Luo-Tseng, Deng, Ngai-Kruger-Théra, Kruger-Ngai-Théra, Kruger-López-Théra,… (see [Da, LT, D, NKT, KNT, KLT]).

In this paper, we study stability of a global Hölderian error bound for the set $[f\leq t]$ under a perturbation of $t$, i.e. the perturbation of $f$ by a constant term. The following questions arise

Suppose that $[f\leq t]$ has a GHEB, when does there exist an open interval $I(t)\subset\mathbb{R},t\in I(t)$, such that for any $t^{\prime}\in I(t)$, $[f\leq t^{\prime}]$ has also a GHEB? 2. 2.

Suppose that $[f\leq t]$ does not have GHEB, when does there exist an open interval $I(t)\subset\mathbb{R},t\in I(t)$, such that for any $t^{\prime}\in I(t)$, $[f\leq t^{\prime}]$ also does not have GHEB? 3. 3.

Are there other types of stability which are different from types in questions 1 and 2?

To classify the stability types of GHEB, our idea is computing the set

[TABLE]

It turns out that the set $H(f)$ can be determined via some speacial values of the Fedoryuk set of $f$.

According [KOS], the Fedoryuk set $F(f)$ of a polynomial $f$ is defined by

[TABLE]

We will show that there exists a value $h(f)\in F(f)\cup\{\pm\infty\}$, which will be called the threshold of global Hölderian error bounds of $f$ and a subset $F^{1}(f)$ of $F(f)$, such that

[TABLE]

Since $F^{1}(f)$ is a semialgebraic subset of $\mathbb{R}$, this formula allows us answer the questions 1 and 2. Moreover, we can discover some other types of stability which are different from the types in questions 1-2 and give the list of all possible types of stability.

The paper is organized as follows. In Section 2, we give two different formulas for computing the set $H(f)$. The first formula is based on criterion for the existence of GHEB for $[f\leq t]$, given in [Ha]. The second formula follows from a new criterion for the existence of global Hölderian error bounds. In Section 3, the relationship between $H(f)$ and the set of Fedoryuk values of $f$ will be established. In Section 4, we use the formulas of $H(f)$ and relationship between $H(f)$ and $F(f)$ to study our problems. It turns out that $F(f)$ is a semialgebraic subset of $\mathbb{R}$, hence $F(f)$ is either empty, or a finite set or a disjoint of finite number of points and intervals. Therefore, it is convenient to consider each of these cases separately.

In Subsection 4.1, we consider the case $F(f)=\emptyset$. In this case, $H(f)=(\inf f,+\infty)$ or $H(f)=[\inf f,+\infty)$ (Theorem 4.1). Therefore, there are two stability types of GHEB if $H(f)=[\inf f,+\infty)$. Namely, any point $t$ of $(\inf f,+\infty)$ is y-stable, by this we mean that $t\in H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset H(f)$. Besides, $t=\inf f$ is y-right stable, by this we mean that $t\in H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset H(f)$ and $(t-\epsilon,t)\cap H(f)=\emptyset$. Note that, for almost every polynomial $f$, $F(f)=\emptyset$. Hence, $H(f)=(\inf f,+\infty)$ or $[\inf f,+\infty)$ if $f$ is generic (Remark 4.1).

In Subsection 4.2, we consider the case when $F(f)$ is a non-empty finite set. In this case, we show that

• •

$H(f)\neq\emptyset$ (Proposition 4.1);

• •

Beside of y-stable type and y-right stable, there are at most 4 other stability types of GHEB. We have

**Case A: **

If $h(f)=-\infty$, then there are 2 types

1. **(i): **

$t$ is y-stable. 2. **(ii): **

$t$ is a n-isolated point: $t\in\mathbb{R}\setminus H(f)$ and for $\epsilon>0$ sufficiently small, $(t-\epsilon,t)\cup(t,t+\epsilon)\subset H(f)$.

**Case B: **

If $h(f)$ is a finite value, then there are 5 types for all $t\in[\inf f,+\infty)$

1. **1.: **

$t$ is y-stable; 2. **2.: **

$t$ is y-right stable; 3. **1’.: **

$t$ is n-stable: $t\in[\inf f,+\infty)\setminus H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset[\inf f,+\infty)\setminus H(f)$; 4. **2’.: **

$t$ is n-right stable: $t\in[\inf f,+\infty)\setminus H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset[\inf f,+\infty)\setminus H(f)$ and $(t-\epsilon,t)\cap([\inf f,+\infty)\setminus H(f))=\emptyset$; 5. **3’.: **

$t$ is n-left stable: $t\in[\inf f,+\infty)\setminus H(f)$ and there exists $\epsilon>0$ such that $(t-\epsilon,t]\subset[\inf f,+\infty)\setminus H(f)$ and $(t,t+\epsilon)\cap H(f)\neq\emptyset$; 6. **4’.: **

$t$ is a n-isolated point;

Note that:

• **–: **

If $t$ is y-right stable or $t$ is n-left stable, then it is necessarily that $t=h(f)$;

• **–: **

If $t$ is n-right stable, then it is necessarily that $t=\inf f<h(f)$ and $f^{-1}(\inf f)\neq\emptyset$.

• •

We can determine the type of stability of any $t\in[\inf f,+\infty)$ (Theorem 4.3);

• •

We give an estimation of the number of connected components of $H(f)$ (Theorem 4.4);

In Subsection 4.3, we consider the case when $\#F(f)=+\infty$. In this case

• •

Any value $t$ of $[\inf f,+\infty)$ belongs to one of the following types

$t$ is y-stable; 2. 2.

$t$ is y-right stable; 3. 3.

$t$ is y-left stable: $t\in H(f)$ and there exists $\epsilon>0$ such that $(t-\epsilon,t]\subset H(f)$ and $(t,t+\epsilon)\cap H(f)=\emptyset$; 4. 4.

$t$ is an y-isolated point: $t\in H(f)$ and for $\epsilon>0$ sufficiently small, $(t-\epsilon,t)\cup(t,t+\epsilon)\subset(\inf f,+\infty)\setminus H(f)$; 5. 1’.

$t$ is n-stable; 6. 2’.

$t$ is n-right stable; 7. 3’.

$t$ is n-left stable; 8. 4’.

$t$ is an n-isolated point.

• •

We can determine the type of stability of any $t\in[\inf f,+\infty)$ (Theorem 4.5).

We conclude with some examples which illustrates some types of stability.

2. The set $H(f)$

2.1. The first formula of $H(f)$

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a polynomial function and $t\in\mathbb{R}$.

Definition 2.1** ([DHN, Ha]).**

We say that

1. (i)

A sequence $\{x^{k}\}\subset\mathbb{R}^{n}$ is the first type of $[f\leq t]$ if

[TABLE] 2. (ii)

A sequence $\{x^{k}\}\subset\mathbb{R}^{n}$ is the second type of $[f\leq t]$ if

[TABLE]

Theorem 2.1** ([Ha]).**

The following statements are equivalent:

1. (i)

There are no sequences of the first or second types of $[f\leq t]$. 2. (ii)

$[f\leq t]$* has a GHEB, i.e. there exist $\alpha,\beta,c>0$ such that*

[TABLE]

Put

[TABLE]

Definition 2.2**.**

Put

[TABLE]

We call $h(f)$ the threshold of global Hölderian error bounds of $f$.

Theorem 2.2** (The first formula of $H(f)$).**

We have

1. (i)

If $h(f)=\inf f$, then $H(f)=[\inf f,+\infty)\setminus F^{1}(f)$; 2. (ii)

If $h(f)=+\infty$, then $H(f)=\emptyset$; 3. (iii)

If $h(f)\in F^{2}(f)$, then $H(f)=(h(f),+\infty)\setminus F^{1}(f)$; 4. (iv)

If $h(f)\notin F^{2}(f)$, then $H(f)=[h(f),+\infty)\setminus F^{1}(f)$.

Proof.

Clearly, if $t\in F^{2}(f)$ and $\inf f\leq t^{\prime}\leq t$, then $t^{\prime}\in F^{2}(f)$. Hence,

[TABLE]

Therefore, Theorem 2.2 follows from Theorem 2.1. ∎

2.2. A new criterion of the existence of a GHEB of $[f\leq t]$ and the second formula of $H(f)$

Let $d$ be the degree of a polynomial $f$. By a linear change of coordinates, we can put $f$ in the form

[TABLE]

where $a_{0}\neq 0$ and $a_{i}(x_{1},\dots,x_{n-1})$ are polynomials in $(x_{1},\dots,x_{n-1})$, where degrees $\deg a_{i}\leq i,i=1,\dots,d$.

Put $V_{1}=\{x\in\mathbb{R}:\dfrac{\partial f}{\partial x_{n}}(x)=0\}$.

Definition 2.3**.**

We say that

1. (i)

A sequence $\{x^{k}\}$ is of the first type of $[f\leq t]$ w.r.t $V_{1}$ if

[TABLE] 2. (ii)

A sequence $\{x^{k}\}$ is of the second type of $[f\leq t]$ w.r.t $V_{1}$ if

[TABLE]

Let $f$ be of the form $(*)$. Put

[TABLE]

Theorem 2.3**.**

Let $f$ be of the form $(*)$. Then the following statements are equivalent

1. (i)

There are no sequences of the first or second types of $[f\leq t]$ w.r.t $V_{1}$; 2. (ii)

$\exists\alpha_{1},\beta_{1},c>0$* such that*

[TABLE]

for all $x\in V_{1}$; 3. (iii)

$\exists\alpha_{1},\beta_{1},c>0$* such that*

[TABLE]

for all $x\in\mathbb{R}^{n}$; 4. (iv)

$[f\leq t]$* has a global Hölderian error bound.*

Proof.

We will prove that $(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(iv)\Rightarrow(i)$.

Proof of $(i)\Rightarrow(ii):$

For $\tau>0$, put

[TABLE]

By (i), $\psi(\tau)$ is well defined on $[0,+\infty)$. Moreover, it follows from Tarski-Seidenberg theorem (see, for example, [BCR, C, HP]), $\psi(\tau)$ is a semialgebraic function.

To prove (ii), it is important to know the behavior of $\psi(\tau)$, as $\tau\to 0$ or $\tau\to+\infty$. We distinguish 4 possibilities

1. (a)

$\psi(\tau)\equiv 0$ for $\tau$ sufficiently small and $\psi(\tau)\equiv 0$ for $\tau$ sufficiently large; 2. (b)

$\psi(\tau)\equiv 0$ for $\tau$ sufficiently small and $\psi(\tau)\not\equiv 0$ for $\tau$ sufficiently large; 3. (c)

$\psi(\tau)\not\equiv 0$ for $\tau$ sufficiently small and $\psi(\tau)\equiv 0$ for $\tau$ sufficiently large; 4. (d)

$\psi(\tau)\not\equiv 0$ both for $\tau$ sufficiently small and $\tau$ sufficiently large.

We will prove (i) $\Rightarrow$ (ii) for the case (d) because the proofs of other cases are similar.

In this case, since $\psi(\tau)$ is semialgebraic and $\psi(\tau)\not\equiv 0$ for any $\tau\in[0,+\infty)$, we have

[TABLE]

and

[TABLE]

Clearly, $\tilde{\alpha}>0$. It follows from (2) that there exists $\delta>0$ such that

[TABLE]

for $x\in\{x\in V_{1}:[f(x)-t]_{+}\leq\delta\}$.

It follows from (3) that there exists $\Delta>0$ sufficiently large, such that for any $x\in\{x\in V_{1}:[f(x)-t]_{+}\geq\Delta\}$. We have

[TABLE]

if $\tilde{\beta}\leq 0$ and

[TABLE]

if $\tilde{\beta}>0$.

Since, by (i), there are no sequences of the second type, the function $\operatornamewithlimits{dist}(x,[f\leq t])$ is bounded on the set

[TABLE]

This fact, together with (4), (5) and (6), give the proof of (i) $\Rightarrow$ (ii).

Proof of (ii) $\Rightarrow$ (iii):

The proof is based on the following classical result

Lemma** (van der Corput, [G]).**

Let $u(\tau)$ be a real valued $C^{d}$-function, $d\in\mathbb{N}$, that satisfies $|u^{(d)}(\tau)|\geq 1$ for all $\tau\in\mathbb{R}$. Then the following estimate is valid for all $\epsilon>0$:

[TABLE]

Suppose that we have (ii). Then

• •

If $x\in[f\leq t]$, then $\operatornamewithlimits{dist}(x,[f\leq t])=0$ and (iii) holds automatically.

• •

If $x\in V_{1}$, then (iii) follows from (ii).

Assume that $x\notin[f\leq t]\cup V_{1}$.

Clearly

• •

(ii) holds if and only if there exists $c>0$ such that

[TABLE]

for all $x\in V_{1}$.

• •

(iii) holds if and only if there exists $c>0$

[TABLE]

for all $x\in\mathbb{R}^{n}$.

Let $x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R},x^{\prime}=(x_{1},\dots,x_{n-1})$. We put

[TABLE]

and

[TABLE]

Since $u_{x^{\prime}}^{(d)}(\tau)=1$, it follows from the van der Corput Lemma that there exists a constant $c>0$, independent of $x$ such that

[TABLE]

Clearly, $\Sigma(x^{\prime})\neq\emptyset$ and $\Sigma(x^{\prime})\neq\mathbb{R}$. Since $\Sigma(x^{\prime})$ is a closed semi-algebraic subset of $\mathbb{R}$, we have

[TABLE]

where $a_{i},b_{i},c_{j}\in\mathbb{R},i=1,\dots,m;j=1,\dots,s$, and

[TABLE]

Firstly, we see that $x_{n}\neq c_{j},\forall j=1,\dots,s$. In fact, since $c_{j}$ is an isolated point of $\Sigma(x^{\prime})$, $c_{j}$ is a local extremum of $u_{x^{\prime}}(\tau)$. Hence,

[TABLE]

or $\dfrac{\partial f}{\partial x_{n}}(x^{\prime},c_{j})=0$ i.e. $(x^{\prime},c_{j})\in V_{1}$, while by assumption, $x=(x^{\prime},x_{n})\notin V_{1}$. Thus, $x_{n}\in\{a_{i},b_{i};i=1,\dots,m\}$.

Without loss of generality, we may assume that $x_{n}=a_{1}$. Since $|u_{x^{\prime}}(a_{1})|=|u_{x^{\prime}}(b_{1})|$, we distinguish two cases

• •

If $u_{x^{\prime}}(a_{1})=-u_{x^{\prime}}(b_{1})$, then there exists $\tau_{1}\in[a_{1},b_{1}]$ such that $u_{x^{\prime}}(\tau_{1})=0$, which means that $f(x^{\prime},\tau_{1})=t$ or $(x^{\prime},\tau_{1})\in f^{-1}(t)\subset[f\leq t]$. Hence

[TABLE]

Then, by (9), (iii) holds.

• •

If $u_{x^{\prime}}(a_{1})=u_{x^{\prime}}(b_{1})$, then, by Rolle’s Theorem, there exists $\tau_{2}\in[a_{1},b_{1}]$ such that

[TABLE]

which means that $(x^{\prime},\tau_{2})\in V_{1}$. Applying (7), there exists $c_{1}>0$ such that

[TABLE]

Moreover, since $\tau_{2}\in\Sigma(x^{\prime})$, we have

[TABLE]

Let $P(x^{\prime},\tau_{2})$ be the point of $[f\leq t]$ such that

[TABLE]

We have

[TABLE]

Now:

• –

If $\max\{\operatornamewithlimits{dist}(x,(x^{\prime},\tau_{2})),\operatornamewithlimits{dist}((x^{\prime},\tau_{2}),P(x^{\prime},\tau_{2}))\}=\operatornamewithlimits{dist}(x,(x^{\prime},\tau_{2}))$, then

[TABLE]

• –

If $\max\{\operatornamewithlimits{dist}(x,(x^{\prime},\tau_{2})),\operatornamewithlimits{dist}((x^{\prime},\tau_{2}),P(x^{\prime},\tau_{2}))\}=\operatornamewithlimits{dist}((x^{\prime},\tau_{2}),P(x^{\prime},\tau_{2}))$, then

[TABLE]

Then (iii) follows from (10).

Hence, the implication (ii) $\Rightarrow$ (iii) is proved.

Proof of (iii) $\Rightarrow$ (iv):

Clearly, if (iii) holds, then there are no sequences of the first or second types of $[f\leq t]$. Hence, by Theorem 2.1, (iv) holds.

The proof of (iv) $\Rightarrow$ (i) is straightforward. ∎

Proposition 2.1**.**

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a polynomial function and $A:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a linear isomorphism. Then we have

[TABLE]

Proof.

Let $y=Ax$ and put $g=f\circ A$.

Firstly, we prove that $t_{0}\in H(g)\Rightarrow t_{0}\in H(f)$.

We have $f(y)=f(A\circ A^{-1}(y))=g(A^{-1}(y))$. This implies that

[TABLE]

Since $t_{0}\in H(g)$, then there exists $\alpha,\beta,c>0$ such that

[TABLE]

Suppose that $\operatornamewithlimits{dist}(A^{-1}(y),[g\leq t_{0}])=\|A^{-1}(y)-x_{0}\|$, where $g(x_{0})=t_{0}$ or $f(A(x_{0}))=t_{0}$. Since $y_{0}=Ax_{0}$ and $A$ is a linear isomorphism, we have $f(y_{0})=t_{0}$ and there exists $c^{\prime}>0$ such that

[TABLE]

It follows that

[TABLE]

Combining (11), (12) and above fact, we have

[TABLE]

i.e., $t_{0}\in H(f)$. The claim $t_{0}\in H(f)\Rightarrow t_{0}\in H(g)$ is proved similarly. ∎

We have the following theorem

Theorem 2.4** (The second formula of $H(f)$).**

Let $f$ be a polynomial of the form $(*)$. Then we have

1. (i)

$h(f)=\sup\{t\in\mathbb{R}:t\in P^{2}(f)\}$; 2. (ii)

If $h(f)=\inf f$, then $H(f)=[\inf f,+\infty)\setminus P^{1}(f)$; 3. (iii)

If $h(f)=+\infty$, then $H(f)=\emptyset$; 4. (iv)

If $h(f)\in\mathbb{R}$ and $h(f)\in P^{2}(f)$, then $H(f)=(h(f),+\infty)\setminus P^{1}(f)$; 5. (v)

If $h(f)\in\mathbb{R}$ and $h(f)\notin P^{2}(f)$, then $H(f)=[h(f),+\infty)\setminus P^{1}(f)$.

3. The relationship between $H(f)$ and Fedoryuk values

The relationship between Fedoryuk values and the existence of global Hölderian error bounds is well-known and has been explored in many previous works, see, for example, [Az, CM, LP, Ha, I]. In this section, we will establish this relationship by proving that $h(f)\in F(f)\cup\{\pm\infty\}$ and $F^{1}(f)\subset F(f)$. We recall

Definition 3.1**.**

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a polynomial function. The set of Fedoryuk values of $f$ is defined by

[TABLE]

Moreover, we have

Lemma 3.1**.**

$F(f)$* is a semialgebraic subset of $\mathbb{R}$.*

Remark 3.1**.**

It follows from Lemma 3.1 that either $F(f)$ is empty or $F(f)$ is finite set or $F(f)$ is a union of finitely many points and intervals.

Note that $F(f)$ can be an infinite set, for example (see [Par]), if $f(x,y,z)=x+x^{2}y+x^{4}yz$, then $F(f)=\mathbb{R}$ and $F(f^{2})=(0,+\infty)$ (see also [KOS] and [Sch]).

To prove the lemma, it is more convenient to use the logical formulation of the Tarski-Seidenberg Theorem. Let us to recall it.

A first-order formula is obtained as follows recursively (see, for example, [BCR, C, HP])

1. (1)

If $f\in\mathbb{R}[X_{1},\dots,X_{n}]$, then $f=0$ and $f>0$ are first-order formulas (with free variables $X=(X_{1},\dots,X_{n})$) and $\{x\in\mathbb{R}^{n}|f(x)=0\}$ and $\{x\in\mathbb{R}^{n}|f(x)>0\}$ are respectively the subsets of $\mathbb{R}^{n}$ such that the formulas $f=0$ and $f>0$ hold. 2. (2)

If $\Phi$ and $\Psi$ are first-order formulas, then $\Phi\vee\Psi$ (conjunction), $\Phi\wedge\Psi$ (disjunction) and $\lnot\Phi$ (negation) are also first-order formulas. 3. (3)

If $\Phi$ is a formula and $X$ is a variable ranging over $\mathbb{R}$, then $\exists X\Phi$ and $\forall X\Phi$ are first-order formulas.

Theorem** (Logical formulation of the Tarski–Seidenberg Theorem [BCR, C, HP]).**

If $\Phi(X_{1},\dots,X_{n})$ is a first-order formula, then the set

[TABLE]

is semialgebraic.

Proof of Lemma 3.1.

We have

[TABLE]

It follows from above that the set $F(f)$ can be determined by a first-order formula, hence by the Tarski-Seidenberg Theorem, it is a semialgebraic subset of $\mathbb{R}$. ∎

The following proposition is contained implicitly in [Ha, Proof of Theorem B].

Proposition 3.1**.**

$F^{1}(f)\subset F(f)$.

Proof.

Put $X=\{x\in\mathbb{R}^{n}:f(x)\geq t\}$. By the metric induced from that of $\mathbb{R}^{n}$, $X$ is a complete metric space and the function $f:X\to\mathbb{R}$ is bounded from below. Let $t\in F^{1}(f)$ and $\{x^{k}\}$ be a sequence of the first type of $[f\leq t]$:

[TABLE]

Let $\epsilon_{k}=f(x^{k})-t$. Then $\epsilon_{k}>0$ and $\epsilon_{k}\to 0$ as $k\to+\infty$. Set $\lambda_{k}=\sqrt{\epsilon_{k}}$. By the Ekeland’s Variational Principle ([E]), there exists a sequence $\{y^{k}\}\subset X$ such that

[TABLE]

and for any $x\in X,x\neq y^{k}$, we have

[TABLE]

Since $\operatornamewithlimits{dist}(y^{k},x^{k})\leq\lambda_{k}=\sqrt{\epsilon_{k}}\to 0$ and $\operatornamewithlimits{dist}(x^{k},[f\leq t])\geq\delta>0$, the ball $B(y^{k},\delta/2)=\{x\in\mathbb{R}^{n}:\operatornamewithlimits{dist}(y^{k},x)\leq\delta/2\}$ is contained in $X$. Then, inequality (13) implies that

[TABLE]

holds true for every $u\in\mathbb{R}^{n},\|u\|=1$ and $\tau\in[0,\delta/2)$. This gives us

[TABLE]

Putting $u=-\dfrac{\nabla f(y^{k})}{\|\nabla f(y^{k})\|}$, we get $\|\nabla f(y^{k})\|\leq\sqrt{\epsilon_{k}}\to 0$.

Clearly $f(y^{k})\to t$. Therefore $t\in F(f)$. ∎

Proposition 3.2**.**

If there is a sequence of the second type of $[f\leq t]$:

[TABLE]

then there exists a sequence $\{y^{k}\}$ of the second type of $[f\leq t]$:

[TABLE]

with additional conditions

[TABLE]

In particular, the segment $[t,M]$ contains at least one point of F(f).

Proof.

Put $X=\{x\in\mathbb{R}^{n}:f(x)\geq t\},\epsilon_{k}=f(x^{k})-t$ and $\lambda_{k}=\dfrac{1}{2}\operatornamewithlimits{dist}(x^{k},[f\leq t])$.

As in the proof of Proposition 3.1, we can find a sequence $\{y^{k}\}\subset X$ such that

[TABLE]

Since

[TABLE]

we have $\operatornamewithlimits{dist}(y^{k},[f\leq t])\to+\infty$. The proposition is proved. ∎

Proposition 3.3**.**

If $h(f)\neq-\infty$ and $\#F(f)<+\infty$, then $h(f)\in F(f)$.

Proof.

Assume that $h(f)\neq-\infty$. By contradiction, suppose that $h(f)\notin F(f)$. Hence, either $F(f)=\emptyset$ or $F(f)$ is a non-empty finite set.

By definition of $h(f)$, $[f\leq h(f)-\epsilon]$ has a sequence of second type. Hence, it follows from Proposition 3.2, $F(f)\neq\emptyset$. Thus, $F(f)$ is a non-empty finite set. Then, for any $\epsilon>0$ sufficiently small, we have $[h(f)-\epsilon,h(f)]\cap F(f)=\emptyset$ and $h(f)-\epsilon\in F^{2}(f)$.

Let $\{x^{k}\}$ be a sequence of the second type of $[f\leq h(f)-\epsilon]$:

[TABLE]

By Proposition 3.2, we may assume that $\|\nabla f(x^{k})\|\to 0$ and there exists $t_{1}\in F(f)\cap[h(f)-\epsilon,M]$ and $t_{1}=\lim\limits_{k\to\infty}f(x^{k})$.

Let $\delta_{1}>0$ such that $t_{1}-\delta_{1}\notin F(f)$ and $t_{1}-\delta_{1}>h(f)$. Since $f(x^{k})\to t_{1}$, we can assume that $f(x^{k})>t_{1}-\delta_{1}$ for all $k$. Let $y^{k}$ be the point of $[f\leq t_{1}-\delta_{1}]$ such that $\operatornamewithlimits{dist}(x^{k},[f\leq t_{1}-\delta_{1}])=\|x^{k}-y^{k}\|$. Clearly, $y^{k}\in f^{-1}(t_{1}-\delta_{1})$.

Claim: $\{y^{k}\}$ is a sequence of second type of $[f\leq h(f)-\epsilon]$.

Proof of Claim.

Since $t_{1}-\delta_{1}>h(f),t_{1}-\delta_{1}\notin F^{2}(f)$. Hence, for some $A>0$, we have $\|x^{k}-y^{k}\|\leq A<+\infty$ for all $k$.

Let $z^{k}$ be the point of $[f\leq h(f)-\epsilon]$ such that $\operatornamewithlimits{dist}(y^{k},[f\leq h(f)-\epsilon])=\|y^{k}-z^{k}\|$. We have

[TABLE]

This shows that $\operatornamewithlimits{dist}(y^{k},[f\leq h(f)-\epsilon])\to+\infty$ and the claim is proved. ∎

Since $\{y^{k}\}$ is a sequence of the second type of $[f\leq h(f)-\epsilon]$ and $f(y^{k})=t_{1}-\delta_{1}\notin F(f)$, by Proposition 3.2, there exists $t_{2}\in[h(f)-\epsilon,t_{1}-\delta_{1}]\cap F(f)$. Choose $\delta_{2}$ such that $t_{1}-\delta_{2}>h(f)$ and $t_{2}-\delta_{2}\notin F(f)$. Similarly as in the proof of Claim, we can find a sequence of the second type $\{y^{\prime k}\}$ of $[f\leq h(f)-\epsilon]$ such that $f(y^{\prime k})=t_{2}-\delta_{2}$ and $t_{3}\in F(f)$ such that $h(f)-\epsilon<t_{3}<t_{2}$.

Making this process iteratively, we see that the interval $[h(f)-\epsilon,M]$ contains a infinite number of points in $F(f)$, which is a contradiction. ∎

4. Types of stability of global Hölderian error bounds

We will distinguish 3 cases.

4.1. Case 1 - $F(f)=\emptyset$

Theorem 4.1**.**

If $F(f)=\emptyset$ then $H(f)=(\inf f,+\infty)$ or $H(f)=[\inf f,+\infty)$.

Proof.

Assume that $F(f)=\emptyset$. Then by Proposition 3.1, $F^{1}(f)=\emptyset$. Moreover, it follows from Proposition 3.2 that $F^{2}(f)$ is also empty.

Hence, by Theorem 2.1, $H(f)=(\inf f,+\infty)\setminus(F^{1}(f)\cup F^{2}(f))=(\inf f,+\infty)$ or $H(f)=[\inf f,+\infty)\setminus(F^{1}(f)\cup F^{2}(f))=[\inf f,+\infty)$. ∎

Definition 4.1**.**

Let $t\in\mathbb{R}$.

$t$ is called y-stable if $t\in H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset H(f)$; 2. 2.

$t$ is called y-right stable if $t\in H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset H(f)$ and $(t-\epsilon,t)\cap H(f)=\emptyset$.

Corollary 4.1**.**

If $F(f)=\emptyset$, then we have two cases

If $H(f)=(\inf f,+\infty)$, then there is only one type of stability of GHEB. Namely, for all $t\in(\inf f,+\infty)$, $t$ is y-stable. 2. 2.

If $H(f)=[\inf f,+\infty)$, then then there are two stability types of GHEB. Namely, for all $t\in(\inf f,+\infty)$, $t$ is y-stable and for $t=\inf f$, $t$ is y-right stable.

Remark 4.1**.**

We recall here results of [Ha] about the role that Newton polyhedron plays in studying GHEB’s.

Let $f(x)=\sum a_{\alpha}x^{\alpha}$ be a polynomial in $n$ variables. Put $supp(f)=\{\alpha\in(\mathbb{N}\cup\{0\})^{n}:a_{\alpha}\neq 0\}$ and denote $\Gamma_{f}$ the convex hull in $\mathbb{R}^{n}$ of the set $\{(0,0,\dots,0)\}\cup supp(f)$. Following [Kou] we call $\Gamma_{f}$ the Newton polyhedron at infinity of $f$.

Let $\Delta$ be a face (of any dimension) of $\Gamma_{f}$, set:

[TABLE]

Definition** ([Kou]).**

We say that a polynomial $f$ is nondegenerate with respect to its Newton boundary at infinity (nondegenerate for short), if for every face $\Delta$ of $\Gamma_{f}$ not containing the origin, the system

[TABLE]

has no solution in $(\mathbb{R}\setminus\{0\})^{n}$.**

Definition**.**

A polynomial $f(x)=\sum a_{\alpha}x^{\alpha}$ in $n$ variables is said to be convenient if for every $i$, there exists a monomial of $f$ of the form $x_{i}^{\alpha_{i}},\alpha_{i}>0$, with a non-zero coefficient.**

Theorem 4.2** ([Ha]).**

If $f$ is convenient and nondegenerate w.r.t. its Newton polyhedron at infinity, then there exist $r,\delta>0$ such that

[TABLE]

In particular, $F(f)=\emptyset$.

Let $\mathbb{R}[x_{1},\dots,x_{n}]$ denote the ring of polynomials in $n$ variables over $\mathbb{R}$.

For $g\in\mathbb{R}[x_{1},\dots,x_{n}]$, as before, $\Gamma_{g}$ denotes the Newton polyhedron at infinity of $g$. Let $f\in\mathbb{R}[x_{1},\dots,x_{n}]$ be a convenient polynomial.

Put $\Gamma:=\Gamma_{f}$ and

[TABLE]

The set $\mathcal{A}_{\Gamma}$ can be identified to the space $\mathbb{R}^{m}$, where $m$ is the number of integer points of $\Gamma$.

Put \mathcal{B}_{\Gamma}=\{h\in\mathcal{A}_{\Gamma}:\Gamma_{h}=\Gamma\ \text{and h is nondegenerate}\}. According to [Kou], $\mathcal{B}_{\Gamma}$ is an open and dense subset of $\mathcal{A}_{\Gamma}$. Hence, Theorem 4.1 and 4.2 show that if $f$ is a generic polynomial, then $H(f)=(\inf f,+\infty)$ or $H(f)=[\inf f,+\infty)$. By Corollary 4.1, any value $t\in(\inf f,+\infty)$ is y-stable and $t=\inf f$ is y-right stable where $H(f)=[\inf f,+\infty)$

4.2. Case 2 - $F(f)$ is non-empty finite set

Proposition 4.1**.**

If $\#F(f)<+\infty$, then $H(f)\neq\emptyset$.

Proof.

By contradiction, assume that $H(f)=\emptyset$. Since $\#F(f)<+\infty$, we have $\#F^{1}(f)<+\infty$ (Proposition 3.1). Then, it follows from the first formula that $H(f)=\emptyset$ if and only if $h(f)=+\infty$ but the later is impossible, since we have

Claim: If $h(f)=+\infty$, then $\#F(f)=+\infty$.

Proof of Claim.

Take $t_{1}\in\mathbb{R}$, since $h(f)=+\infty$, $[f\leq t_{1}]$ has a sequence of the second type. By Proposition 3.2, there exists $M_{1}>t_{1}$ and $a_{1}\in[t_{1},M_{1}]\cap F(f)$. Take $t_{2}$ such that $M_{1}<t_{2}$, then $[f\leq t_{2}]$ has a sequence of the second type. Hence, there exists $M_{2}>t_{2}$ and $a_{2}$ such that $a_{2}\in[t_{2},M_{2}]\cap F(f)$. Continuing this way, we find an infinite sequence $a_{1},a_{2},a_{3},\dots$ of $F(f)$. Therefore, $\#F(f)=+\infty$. ∎

∎

Now, we classify the stability types of GHEB in the case when $F(f)$ is a non-empty finite set.

Definition 4.2**.**

Let $t\in\mathbb{R}$.

Recall that $t$ is called y-stable if $t\in H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset H(f)$; 2. 2.

Recall that $t$ is called y-right stable if $t\in H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset H(f)$ and $(t-\epsilon,t)\cap H(f)=\emptyset$; 3. 1’.

$t$ is called n-stable if $t\in[\inf f,+\infty)\setminus H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset\mathbb{R}\setminus H(f)$; 4. 2’.

$t$ is called n-right stable if $t\in[\inf f,+\infty)\setminus H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset[\inf f,+\infty)\setminus H(f)$ and $(t-\epsilon,t)\cap([\inf f,+\infty)\setminus H(f))=\emptyset$; 5. 3’.

$t$ is called n-left stable if $t\in[\inf f,+\infty)\setminus H(f)$ and there exists $\epsilon>0$ such that $(t-\epsilon,t]\subset[\inf f,+\infty)\setminus H(f)$ and $(t,t+\epsilon)\cap H(f)\neq\emptyset$; 6. 4’.

$t$ is called n-isolated if $t\in\mathbb{R}\setminus H(f)$ and for $\epsilon>0$ sufficiently small, $(t-\epsilon,t)\cup(t,t+\epsilon)\subset H(f)$.

It follows from the first formula that

Theorem 4.3**.**

Let $F(f)$ be a non-empty finite set and $t\in[\inf f,+\infty)$. Then, $t$ is one of the following types

**Case A: **

If $h(f)=-\infty$, then

1. **(i): **

$t$* is y-stable if and only if $t\notin F^{1}(f)$.* 2. **(ii): **

$t$* is a n-isolated point if and only if $t\in F^{1}(f)$.*

**Case B: **

If $h(f)$ is a finite value, then

1. **1.: **

$t$* is y-stable if and only if $t>h(f)$ and $t\notin F^{1}(f)$;* 2. **2.: **

$t$* is y-right stable if and only if $t=h(f)$ and $h(f)\in H(f)$;* 3. **1’.: **

$t$* is n-stable if and only if $\inf f<t<h(f)$;* 4. **2’.: **

$t$* is n-right stable if and only if $t=\inf f<h(f)$ and $f^{-1}(\inf f)\neq\emptyset$;* 5. **3’.: **

$t$* is n-left stable if and only if $t=h(f)$ and $h(f)\notin H(f)$;* 6. **4’.: **

$t$* is a n-isolated point if and only if $t>h(f)$ and $t\in F^{1}(f)$.*

Remark 4.2**.**

Here, if we have item 2, then we does not have item 3’ and vice versa.

Now, to complete this subsection, we add an estimation of the number of connected components of $H(f)$ for the case $\#F(f)<+\infty$.

Let us denote $C(S)$ the number of connected components of $S\subset\mathbb{R}^{n}$, we have the following result

Theorem 4.4**.**

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be an any polynomial of degree $d$. Then, if $\#F(f)<+\infty$, we have

[TABLE]

Proof.

Put

[TABLE]

Since $\#F(f)<+\infty$, we have $\#F_{\mathbb{C}}(f)<+\infty$. Then, according to Theorem 1.1 of [Je], we have

[TABLE]

Hence, it follows from the first formula that

[TABLE]

∎

4.3. Case 3 - $F(f)$ is an infinite set

In this case, the following lemma tells us that the set $H(f)$ has still very simple structure

Lemma 4.1**.**

$H(f)$* is a semialgebraic subset of $\mathbb{R}$.*

Using the first formula for $H(f)$ (Theorem 2.2), it is enough to show that $F^{1}(f)$ is semialgebraic.

Proof of Lemma 4.1.

We have

[TABLE]

[TABLE]

It follows from (a) and (b) that the set $F^{1}(f)$ can be determined by a first-order formula, hence it is a semialgebraic subset of $\mathbb{R}$. ∎

Since $H(f)$ is a semialgebraic subset of $\mathbb{R}$, we have

Corollary 4.2**.**

If $H(f)\neq\emptyset$ and $H(f)\neq\mathbb{R}$, then it is a union of finitely many points and intervals.

By Corollary 4.2, we have to consider three cases

1. (a)

$H(f)=\mathbb{R}$; 2. (b)

$H(f)=\emptyset$; 3. (c)

$H(f)$ is a non-empty proper semialgebraic subset of $\mathbb{R}$.

• •

In the case (a), we have only one stable type: $t$ is y-stable for all $t\in\mathbb{R}$;

• •

In the case (b), we have only one stable type: $t$ is n-stable for all $t\in\mathbb{R}$;

• •

In the case (c), $H(f)$ is a disjoint union of the sets of the following types:

[TABLE]

Where

1. (1)

$I_{(a^{1}_{i},a^{2}_{i})}=\emptyset$ or $I_{(a^{1}_{i},a^{2}_{i})}={(a^{1}_{i},a^{2}_{i})},i=1,\dots,p$; 2. (2)

$I_{[b^{1}_{j},b^{2}_{j})}=\emptyset$ or $I_{[b^{1}_{j},b^{2}_{j})}={[b^{1}_{j},b^{2}_{j})},j=1,\dots,q$; 3. (3)

$I_{(c^{1}_{k},c^{2}_{k}]}=\emptyset$ or $I_{(c^{1}_{k},c^{2}_{k}]}={(c^{1}_{k},c^{2}_{k}]},k=1,\dots,r$; 4. (4)

$I_{[d^{1}_{l},d^{2}_{l}]}=\emptyset$ or $I_{[d^{1}_{l},d^{2}_{l}]}={[d^{1}_{l},d^{2}_{l}]},l=1,\dots,s$; 5. (5)

$A(m)=\emptyset$ or $A(m)=\{e_{1},\dots,e_{m}\}$, where $e_{1},\dots,e_{m}$ are isolated points; 6. (6)

$I_{-\infty}=\emptyset$ or $I_{-\infty}=(-\infty,a]$ or $I_{-\infty}=(-\infty,a)$, where $a\in\mathbb{R}$; 7. (7)

$I_{+\infty}=\emptyset$ or $I_{+\infty}=[b,+\infty)$ or $I_{+\infty}=(b,+\infty)$, where $b\in\mathbb{R}$.

Similarly, $\mathbb{R}\setminus H(f)$ is a disjoint union of the sets of the following types:

[TABLE]

We have the following definition

Definition 4.3**.**

Let $t\in\mathbb{R}$.

Recall that $t$ is said to be y-stable if $t\in H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset H(f)$; 2. 2.

Recall that $t$ is said to be y-right stable if $t\in H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset H(f)$ and $(t-\epsilon,t)\cap H(f)=\emptyset$; 3. 3.

$t$ is said to be y-left stable if $t\in H(f)$ and there exists $\epsilon>0$ such that $(t-\epsilon,t]\subset H(f)$ and $(t,t+\epsilon)\cap H(f)=\emptyset$; 4. 4.

$t$ is said to be y-isolated if $t\in H(f)$ and for $\epsilon>0$ sufficiently small, $(t-\epsilon,t)\cup(t,t+\epsilon)\subset\mathbb{R}\setminus H(f)$; 5. 1’.

Recall that $t$ is called n-stable if $t\in[\inf f,+\infty)\setminus H(f)$ and there exists an open interval $I(t)$ such that $t\in I(t)\subset[\inf f,+\infty)\setminus H(f)$; 6. 2’.

Recall that $t$ is called n-right stable if $t\in[\inf f,+\infty)\setminus H(f)$ and there exists $\epsilon>0$ such that $[t,t+\epsilon)\subset[\inf f,+\infty)\setminus H(f)$ and $(t-\epsilon,t)\cap([\inf f,+\infty)\setminus H(f))\neq\emptyset$; 7. 3’.

Recall that $t$ is called n-left stable if $t\in[\inf f,+\infty)\setminus H(f)$ and there exists $\epsilon>0$ such that $(t-\epsilon,t]\subset[\inf f,+\infty)\setminus H(f)$ and $(t,t+\epsilon)\cap H(f)=\emptyset$; 8. 4’.

Recall that $t$ is called n-isolated if $t\in[\inf f,+\infty)\setminus H(f)$ and for $\epsilon>0$ sufficiently small, $(t-\epsilon,t)\cup(t,t+\epsilon)\subset H(f)$.

Using the first formula of $H(f)$, we have

Theorem 4.5**.**

Let $H(f)$ be of the form (c) and $t\in[\inf f,+\infty)$. Then we have

$t$* is y-stable if and only if $t$ is an interior point of the sets*

[TABLE] 2. 2.

$t$* is y-right stable if and only if we have $t=b^{1}_{j}$ or $t=d^{1}_{l}$ or $t=b$ (where $I_{+\infty}=[b,+\infty)$);* 3. 3.

$t$* is y-left stable if and only if we have $t=c^{2}_{k}$ or $t=d^{2}_{l}$ or $t=a$ (where $I_{-\infty}=(-\infty,a]$);* 4. 4.

$t$* is an y-isolated point if and only if $t\in A(m)$.* 5. 1’.

$t$* is n-stable if and only if $t$ is an interior point of the set:*

[TABLE] 6. 2’.

$t$* is n-right stable if and only if we have $t=b^{\prime 1}_{j}$ or $t=d^{\prime 1}_{l}$ or $t=b^{\prime}$ (where $I^{\prime}_{+\infty}=[b^{\prime},+\infty)$);* 7. 3’.

$t$* is n-left stable if and only if we have $t=c^{\prime 2}_{k}$ or $t=d^{\prime 2}_{l}$ or $t=a^{\prime}$ (where $I^{\prime}_{-\infty}=(-\infty,a^{\prime}]$);* 8. 4’.

$t$* is an n-isolated point if and only if $t\in A^{\prime}(m^{\prime})$.*

Remark 4.3**.**

In the above list, we collect all types of stability that could theoretically exist. The problem of deciding when this or that type really appears, seems to be very difficult.

We finish our paper by considering the following simple example

Example 4.1**.**

Let $f(x,y)=(y^{2}-1)^{2}+(xy-1)^{2}$ ([HT]). Clearly, $f$ is of the form $(*)$.

We have $\dfrac{\partial f}{\partial y}=4y^{3}+2x^{2}y-4y-2x=2(2y^{3}+x^{2}y-2y-x)$. Hence, the roots of $\dfrac{\partial f}{\partial y}=0$ are:

[TABLE]

We have

[TABLE]

Hence $P(f)=\{1\}$.

It is not difficult to show that

• •

$F^{2}(f)=[0,1)$, hence $h(f)=1$;

• •

and $F^{1}(f)=\emptyset$.

Therefore, by the second formula, $H(f)=[1,+\infty)$. In this example, for any $t\in[0,+\infty)$:

• •

If $t\in(1,+\infty)$, then $t$ is y-stable;

• •

If $t=1$, then $t$ is y-right stable;

• •

If $t\in(0,1)$, then $t$ is n-stable;

• •

If $t=0$, then $t$ is n-right stable.

Acknowledgments

This research was partially supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam; Grant numbers 101.04-2017.12 of the first author and 101.04-2019.302 of the second author.