# Markov’s inequality and polynomial mappings

## Abstract

Markov’s inequality is a certain estimate for the norm of the derivative of a polynomial in terms of the degree and the norm of this polynomial. It has many interesting applications in approximation theory, constructive function theory and in analysis (for instance, to Sobolev inequalities or Whitney-type extension problems). One of the purposes of this paper is to give a solution to an old problem, studied among others by Baran and Pleśniak, and concerning the invariance of Markov’s inequality under polynomial mappings (polynomial images). We also address the issue of preserving Markov’s inequality when taking polynomial preimages. Lastly, we give a sufficient condition for a subset of a Markov set to be a Markov set.

## 1 Introduction

**Theorem 1.1**

*P*is a polynomial of one variable, then

From the point of view of applications, it is important that the constant \((\deg P)^2\) in Markov’s inequality grows not too fast (that is, polynomially) with respect to the degree of the polynomial *P*. This is the reason why the concept of a Markov set is widely investigated.

**Definition 1.2**

Obviously, if \(\emptyset \ne E_1,\ldots , E_p\subset \mathbb {C}^N\) are compact sets satisfying Markov’s inequality, then the union \(E_1\cup \cdots \cup E_p\) satisfies Markov’s inequality as well. In general, this is no longer so for the intersection \(E_1\cap \cdots \cap E_p\).

It is straightforward to show that the Cartesian product of Markov sets is a Markov set. More precisely, if \(\emptyset \ne E_j\subset \mathbb {C}^{N_j}\) (\(N_j\in \mathbb {N}\)) is a compact set satisfying Definition 1.2 with \(\varepsilon _j, C_j>0\) (\(j=1,\ldots , p\)), then \(E_1\times \cdots \times E_p\subset \mathbb {C}^{N_1+\cdots +N_p}\) satisfies this definition with \(\varepsilon :=\max \{\varepsilon _1,\ldots ,\varepsilon _p\}\) and \(C:=\max \{C_1,\ldots ,C_p\}\).

In Sect. 5, we give a sufficient condition for a subset of a Markov set to be a Markov set—see Theorem 5.1 and Corollary 5.2.

Let \(\emptyset \ne E\subset \mathbb {C}\) be a compact set such that, for each connected component

*K*of*E*, we have \(\text {diam}(K)\ge \eta \) with some \(\eta >0\) being independent of*K*. Then*E*is a Markov set—see Lemma 3.1 in [56] and Sect. 3.- By Theorem 3.1 in [43], each compact UPC set satisfies Markov’s inequality. Recall that a set \(E\subset \mathbb {R}^N\) is UPC (uniformly polynomially cuspidal) if there exist \(\upsilon ,\, \theta > 0\) and \(d\in \mathbb {N}\) such that, for each \(x \in \overline{E}\), we can choose a polynomial map \(\displaystyle S_x : \mathbb {R} \longrightarrow \mathbb {R}^N\) with \(\deg S_x\le d\) satisfying the following conditions:Note that a UPC set is in particular fat, that is \(\overline{E} = \overline{\mathrm{Int} E}\). In [43, 44, 46], some large classes of UPC sets (and hence Markov sets) are given. These classes include for example all compact, fat and semialgebraic subsets of \(\mathbb {R}^N\) (see Sect. 2 for the definition).
\(S_x (0) = x\),

\(\displaystyle \mathrm{dist} \big (S_x (t),\, \mathbb {R}^N {\setminus } E\big ) \ge \theta t^{\upsilon }\) for each \(t \in [0,\, 1]\).

**Theorem 1.3**

(Baran, Pleśniak) Let \(E\subset \mathbb {R}^N\) be a compact UPC set. Suppose that \(h:\mathbb {R}^N\longrightarrow \mathbb {R}^N\) is a polynomial map such that \(\mathrm{{Jac}}\,h(\zeta )\ne 0\) for each \(\zeta \in \mathrm{{Int}}E\). Then *h*(*E*) satisfies Markov’s inequality.

Since each compact UPC set satisfies Markov’s inequality, the Baran–Pleśniak theorem says that, under a certain assumption on a Markov set \(E\subset \mathbb {R}^N\) and under a certain assumption on a polynomial map \(h:\mathbb {R}^N\longrightarrow \mathbb {R}^N\), the image *h*(*E*) also satisfies Markov’s inequality.

Very strong UPC assumption on the Markov set

*E*is superfluous.- The assumption that \(\mathrm{{Jac}}\,h(\zeta )\ne 0\) for each \(\zeta \in \mathrm{{Int}}E\) can be replaced by much weaker assumption that \(h:\mathbb {R}^N\longrightarrow \mathbb {R}^{N'}\) andMoreover, the latter assumption is the weakest possible condition on the polynomial map$$\begin{aligned} \mathrm{{rank}}\,h:= \max \left\{ \mathrm{{rank}}\,d_{\zeta } h:\, \zeta \in \mathbb {R}^N\right\} =N'. \end{aligned}$$
*h*that must be assumed (see Lemma 2.3).

**Theorem 1.4**

*h*(

*E*) also satisfies Markov’s inequality.

It is worth noting that there is a holomorphic version of Theorem 1.3 in [3], which reads as follows. Suppose that \( E\subset \mathbb {C}^N\) is a compact, polynomially convex set satisfying Markov’s inequality. If \(h:U \longrightarrow \mathbb {C}^N\) is a holomorphic map in a neighbourhood *U* of *E* such that *h*(*E*) is nonpluripolar and \(\mathrm{{Jac}}\,h(\zeta )\ne 0\) for each \(\zeta \in E\), then *h*(*E*) also satisfies Markov’s inequality. (The notion of a polynomially convex set and the notion of a nonpluripolar set are defined in Sect. 3.)

In connection with Theorem 1.4, the following question naturally arises.

**Question 1.5**

Suppose that \(\emptyset \ne E\subset \mathbb {K}^{N'}\) is a compact set satisfying Markov’s inequality and \(g:\mathbb {K}^N\longrightarrow \mathbb {K}^{N'}\) is a polynomial map (\(N,N'\in \mathbb {N}\)). Under what conditions is it true that \(g^{-1}(E)\) satisfies Markov’s inequality?

The precise answer is not known to us. However, we will address this issue in Sects. 3 and 4. In particular, we will give some specific examples to show a variety of situations that we encounter exploring this problem. Eventually, we will give a result (Theorem 3.8) being a partial answer to Question 1.5.

## 2 A proof of Theorem 1.4

We will need the notion of a semialgebraic set and the notion of a semialgebraic map.

**Definition 2.1**

**Definition 2.2**

A map \(f: A \longrightarrow \mathbb {R}^{N'}\), where \(A\subset \mathbb {R}^N\), is said to be semialgebraic if its graph is a semialgebraic subset of \(\mathbb {R}^{N+N'}\).

All semialgebraic sets constitute the simplest polynomially bounded o-minimal structure (see [59, 60] for the definition and properties of o-minimal structures). However, the knowledge of o-minimal structures is not necessary to follow the present paper. Whenever we say “a set (a map) definable in a polynomially bounded o-minimal structure”, the reader who is not familiar with the basic notions of o-minimality can just think of a semialgebraic set (map).

Before going to the proof of Theorem 1.4, it is worth noting that the assumption that \(\mathrm{{rank}}\,h=N'\) is necessary in this theorem, as is seen by the following lemma.

**Lemma 2.3**

Suppose that \(h:\mathbb {K}^N\longrightarrow \mathbb {K}^{N'}\) is a polynomial map such that \(\mathrm{{rank}}\,h<N'\) (\(N,N'\in \mathbb {N}\)). Then for each compact set \(\emptyset \ne E\subset \mathbb {K}^N\) the image *h*(*E*) does not satisfy Markov’s inequality.

*Proof*

By Sard’s theorem, the set \(h(\mathbb {K}^N)\) has Lebesgue measure zero.

^{1}and \(P|_{h(\mathbb {R}^N)}\equiv 0\). Take a point \(a\in h(E)\). For each \(w\in \mathbb {R}^{N'}\), we have

*h*(

*E*) does not satisfy Markov’s inequality.

Case 2:\(\mathbb {K}=\mathbb {C}\). By Chevalley’s theorem, the set \(h(\mathbb {C}^N)\) is constructible (see [41, pp. 393–396], for the definition and details). Moreover, \(\overline{h(\mathbb {C}^N)}\ne \mathbb {C}^{N'}\)^{2} and \(\overline{h(\mathbb {C}^N)}\) is a complex algebraic set (see [41, p. 394]), that is the set of common zeros of some collection of complex polynomials. In particular, there exists \(P\in \mathbb {C}[w_1, \ldots ,w_{N'}]\) such that \(P\not \equiv 0\) and \(P|_{h(\mathbb {C}^N)}\equiv 0\). Arguing as in Case 1 we see that *h*(*E*) does not satisfy Markov’s inequality.^{3}\(\square \)

We will try to keep the exposition as self-contained as possible. It should be stressed, however, that our proof of Theorem 1.4 is influenced by ideas from the original proof of Theorem 1.3 by Baran and Pleśniak.

*Proof of Theorem*1.4. Clearly, it suffices to consider the case \(\mathbb {K}=\mathbb {C}\). Put

*T*is (complex) algebraic and nowhere dense (see [41, p. 158]), it follows that \(\chi ^{-1}(A)\subset \mathbb {R}^{2N}\) is open, semialgebraic and \(\overline{\chi ^{-1}(A)}=\chi ^{-1}(\overline{I})\). Consequently, by Corollary 6.6 in [43], \(\chi ^{-1}(A)\) is UPC. Therefore there exist \(\upsilon ,\, \theta > 0\) and \(d\in \mathbb {N}\) such that, for each \(x \in \overline{\chi ^{-1}(A)}\), we can choose a polynomial map \(\displaystyle S_x : \mathbb {R} \longrightarrow \mathbb {R}^{2N}\) satisfying the following conditions:

- (i)
\(\deg S_x\le d\),

- (ii)
\(S_x (0) = x\),

- (iii)
\(\displaystyle \mathrm{dist} \big (S_x (t),\, \mathbb {R}^{2N} {\setminus } \chi ^{-1}(A)\big ) \ge \theta t^{\upsilon }\) for each \(t \in [0,\, 1]\).

*h*(

*E*) satisfies Markov’s inequality.

*Q*,

*l*,

*a*as above. First, we will show that, for each \(\zeta \in \mathbb {C}^N\) and each \(j\in \{1,\ldots , N\}\),

^{4}and the estimates (6) and (8).

^{5}

## 3 Markov’s inequality and polynomial preimages

In this section, we will look at Markov’s inequality from the point of view of polynomial preimages.

*E*and \(\Phi _E\le \Phi _K\) provided that \(\emptyset \ne K\subset E\) and

*K*is compact. However, except for some very special cases, no explicit expression for \(\Phi _E\) is known.

^{6}functions \(\phi \) in \(\mathbb {C}^N\) satisfying the condition

If \(\emptyset \ne E\subset \mathbb {C}^N\) is a compact set and \(\Phi _{E}\) is continuous at every point of *E*, then \(\Phi _E\) is continuous in \(\mathbb {C}^N\), in other words, the set *E* is *L*-regular (cf. Proposition 6.1 in [55] or Corollary 5.1.4 in [34]).

**Definition 3.1**

We will also need the notion of a pluripolar set.

**Definition 3.2**

- (i)
For each point \(a\in A\), there exists an open neighbourhood

*U*of*a*such that \(A\cap U\subset \{z\in U:\, u(z)=-\infty \}\) for some plurisubharmonic function \(u: U\longrightarrow [-\infty ,+\infty )\). - (ii)
There exists a plurisubharmonic function \(\psi \) in \(\mathbb {C}^N\) such that \(A \subset \{z\in \mathbb {C}^N:\, \psi (z)=-\infty \}\).

The situation is quite different if we consider the complex case (\(\mathbb {K}=\mathbb {C}\)). But also in this case the claim that the polynomial preimage of a Markov set is a Markov set is still far from being valid.

*Example 3.3*

The Shilov boundary of the open polydisc \(\mathbb {D}_N:=\{z\in \mathbb {C}^N:\, |z|<1\}\)

^{7}is its skeleton, that is the set \(\{z\in \mathbb {C}^N:\, |z_1|=\cdots =|z_N|=1\}\) (see [53, p. 22]).- The closed polydisk \( \overline{\mathbb {D}}_N\) satisfies Markov’s inequality. Indeed, for \(N=1\), this the content of Bernstein’s theorem: for each complex polynomial
*Q*of one variable,where \(\mathbb {D}:=\mathbb {D}_1\) (see [10, p. 233]). For \(N>1\), it is enough to use the fact that the Cartesian product of Markov sets is a Markov set (see Sect.1).$$\begin{aligned} {\Vert Q'\Vert }_{\overline{\mathbb {D}}}\le (\deg Q) \, {\Vert Q\Vert }_{\overline{\mathbb {D}}}, \end{aligned}$$

*E*satisfies Markov’s inequality. However, the set \(g^{-1}(E)=\Gamma \cup \left\{ \left( \alpha , {\beta }/{\alpha } \right) \right\} \) does not satisfy Markov’s inequality. Indeed, suppose otherwise and take \(\varepsilon , C\) of Definition 1.2. Then, for polynomials \(\Psi _n\in \mathbb {C}[w_1,w_2]\) (\(n\in \mathbb {N}\)) defined by \(\Psi _n (w_1,w_2):=(\beta -\alpha w_2)w_2^n\), we have

The situation described in the above example is particular, because the set \(g^{-1}(E)\) is not *L*-regular.^{8} This is no longer the case in the next example (Example 3.6).

It will be convenient to state beforehand, for easy reference, two results. The first one gives a sufficient and necessary condition for a bounded set \(A \subset \mathbb {R}^2\) definable in some polynomially bounded o-minimal structure to be UPC (cf. [44], Theorem B).

**Theorem 3.4**

*A*is UPC.*A*is fat and, for each \(a\in \overline{A}\), \(\rho >0\) and any connected component*S*of the set \(\mathrm{Int} A\cap {B}(a,\,\rho )\) such that \(a\in \overline{S}\), there is a polynomial arc \(\gamma :(0,\,1) \longrightarrow S\) such that \(\displaystyle \lim \limits _{t \rightarrow 0} \gamma (t) = a\), where \({B}(a,\,\rho )=\{x\in \mathbb {R}^2 :\, |x-a|<\rho \}\).

The second result is a special case of the (semi)analytic accessibility criterion due to Pleśniak (cf. [47]).^{9}

**Theorem 3.5**

Let \(K\subset \mathbb {K}^N\) be a compact set. Suppose that there exists a polynomial mapping \(\gamma : \mathbb {K}\longrightarrow \mathbb {K}^N\) such that \(\gamma ((0,\,1])\subset \mathrm{Int}K\). Then *K* is *L*-regular at \(\gamma (0)\), i.e., \(\Phi _K\) is continuous at \(\gamma (0)\).

*Example 3.6*

\(f>0\) in \((0,\,R]\),

\(\displaystyle \lim _{t\rightarrow 0}\frac{f(t)}{t^r}=0\) for each \(r>0\),

there exists \(R_0\in (0,\,R]\) such that

*f*is nondecreasing in \([0,\,R_0]\),there exists \(R_1\in [0,\,R_0)\) such that \(f|_{[R_1,\,R]}\) is definable in a certain polynomially bounded o-minimal structure (for simplicity, \(f|_{[R_1,\,R]}\) can be thought of as a semialgebraic map).

*E*satisfies Markov’s inequality,\(F^{-1}(E)\) is

*L*-regular,\(F^{-1}(E)\) does not satisfy Markov’s inequality for \(\mathbb {K}=\mathbb {R}\) but does satisfy Markov’s inequality for \(\mathbb {K}=\mathbb {C}\).

*x*and the midpoint of the diagonals of \(H_x\). Since \(H_x\subset E\), it follows that, for each \(t\in [0,\,1]\),

^{10}Therefore, there exist \(\upsilon ,\, \theta > 0\) and \(d\in \mathbb {N}\) such that, for each \(x \in E_2\), we can choose a polynomial map \(\displaystyle \tilde{S}_x : \mathbb {R} \longrightarrow \mathbb {R}^2\) with \(\deg \tilde{S}_x\le d\) satisfying the following conditions:

\(\tilde{S}_x (0) = x\),

\(\displaystyle \mathrm{dist} \big (\tilde{S}_x (t),\, \mathbb {R}^2 {\setminus } E_2\big ) \ge \theta t^{\upsilon }\) for each \(t \in [0,\, 1]\).

*L*-regular.

^{11}Suppose, to derive a contradiction, that \(F^{-1}(E)\) is a Markov set. In particular, there exist \(\varepsilon , C>0\) such that, for each polynomial \(P\in \mathbb {C}[w_1,w_2]\),

*E*is UPC, it has the HCP property: there exist \(M_1, \mu >0\) such that, for each \(z\in E_{(1)}\),

*L*-regular. \(\square \)

The previous examples may suggest that a compact, *L*-regular set, which is the inverse image of a Markov set under a complex (i.e., holomorphic) polynomial map, is also a Markov set. This claim is however not valid.

*Example 3.7*

*E*of Example 3.6, we can show that \(D_1, D_2\) satisfy Markov’s inequality. Moreover, put

*L*-regular and does not satisfy Markov’s inequality. On the other hand, the set

*D*, as the Cartesian product of the Markov sets, is a Markov set. \(\square \)

After giving the above examples illustrating various situations which occur naturally when we consider Markov’s inequality in the context of polynomial preimages, we conclude this section with the statement of the following result, to be proved in the next section.

**Theorem 3.8**

\(N=N'\),

\(g^{-1}(E)\) has the HCP property and, in particular, is a Markov set.

*E*:

*E*is polynomially convex. For example, each compact subset of \(\mathbb {R}^N\) is polynomially convex in \(\mathbb {C}^N\) (cf. Lemma 5.4.1 in [34]).

## 4 A proof of Theorem 3.8

For the convenience of the reader we recall first the relevant notions and results from [41].

**Definition 4.1**

**Definition 4.2**

*U*of

*a*and there exist holomorphic functions \(\xi _1,\ldots ,\xi _k:U\longrightarrow \mathbb {C}\) such that

**Definition 4.3**

*U*of

*a*and there exist holomorphic functions \(\xi _1,\ldots ,\xi _k:U\longrightarrow \mathbb {C}\) such that

The subsequent proofs make use of the following two results.

**Theorem 4.4**

*f*(

*B*) is a countable union of submanifolds of dimension \(\le m\).

*Proof*

See [41, p. 254]. \(\square \)

**Theorem 4.5**

Every compact analytic subset of \(\mathbb {C}^N\) is finite.

*Proof*

See [41, p. 235]. \(\square \)

Before proceeding with the proof of Theorem 3.8, let us also state the following lemma.

**Lemma 4.6**

Assume that \(f:W\longrightarrow \mathbb {C}^{N'}\) is a holomorphic mapping, where \(W\subset \mathbb {C}^N\) is open (\(N,N'\in \mathbb {N}\)). Suppose that a set \( A \subset f(W)\) is nonpluripolar. Then \(f^{-1}(A)\) is nonpluripolar as well.

*Proof*

We will consider two cases.

Case 1:\(N<N'\). Obviously, \(\mathrm{{rank}}\,d_{w} f\le N\) for each \(w\in W\). By Theorem 4.4, *f*(*W*) is a countable union of submanifolds of dimension \(\le N\). In particular, the set *f*(*W*) (and hence *A*) is pluripolar, which is a contradiction. The case \(N<N'\) cannot therefore occur.

*B*is an analytic subset of

*W*. As in Case 1, we show via Theorem 4.4 that

*f*(

*B*) is pluripolar. In particular, the set \(A\cap f(W_0)\) is nonpluripolar.

*u*in \(\mathbb {C}^N\). Note that, for each \(y\in \Omega _{a_l}\),

^{12}\(\square \)

*A*in \(\Omega \) is defined as follows:

*Proof of Theorem*3.8. We will consider three cases.

Case 2:\(N<N'\). It follows from Theorem 4.4 that *g*(*U*) is a countable union of submanifolds of dimension \(\le N\). In particular, *g*(*U*) (and hence *E*) is pluripolar, which is a contradiction. The case \(N<N'\) cannot therefore occur.

^{13}Note that there exists \(\epsilon >0\) such that

With regard to Theorem 3.8, we have the following remark.

*Remark 4.7*

In Theorem 3.8, even if \(U=\mathbb {C}^N\) and \(g:\mathbb {C}^N\longrightarrow \mathbb {C}^{N}\) is a polynomial map, the assumption that \(\hat{E}\subset g(U)\) and \(g^{-1}(\hat{E})\) is compact cannot be replaced by the assumption that \(E\subset g(U)\) and \(g^{-1}(E)\) is compact and *L*-regular.

*Proof*

\(K\subset \left( 0,\,\frac{1}{2}\right] \times (1,\,2]\),

*K*is*L*-regular,*K*does not satisfy Markov’s inequality.

*f*.)

^{14}It is well known that, for each \(u\in \mathbb {C}\),

*E*has the HCP property: \(\Phi _{E}(z)\le 1+\mathrm{dist}(z,\,E)\) for each \(z\in \mathbb {C}^2\),\(g^{-1}(E)= \Gamma \cup K\) is

*L*-regular, because \(\Gamma \) and*K*are*L*-regular.

*K*are also polynomially convex. Therefore, by Kallin’s separation lemma (cf. [33, p. 302]), we obtain the polynomial convexity of the set \({\overline{\mathbb {D}}}_2\cup K\). On account of Corollary 5.2, we get a contradiction, because

*K*is not a Markov set. \(\square \)

## 5 Subsets of Markov sets

In this section, we will prove the following result announced in Introduction.

**Theorem 5.1**

Let \(E\subset \mathbb {C}^N\) be a compact and polynomially convex set satisfying Markov’s inequality. Assume that \(K\subset E\) is compact, nonpluripolar and open in *E*. Then *K* is a Markov set. Furthermore, if *E* satisfies Markov’s inequality with an exponent \(\varepsilon >0\) (see Definition 1.2), then *K* satisfies Markov’s inequality with the exponent \(\varepsilon \) as well.

*Proof*

*E*. Therefore, for each polynomial \(P\in \mathbb {C}[z_1,\ldots ,z_N]\) and each \(\alpha \in \mathbb {N}_0^N\),

*Q*,

*a*as above and take \(b\in K\) such that \(|a-b|=\mathrm{dist}(a,\,K)\). Clearly, \(a\in \mathcal {Z}\cap K_{\lambda }\) and

*K*satisfies Markov’s inequality with the exponent \(\varepsilon \). \(\square \)

**Corollary 5.2**

- 1.
*E*satisfies Markov’s inequality with the exponent \(\varepsilon \). - 2.
For each \(j\le p\), the set \(E_j\) satisfies Markov’s inequality with the exponent \(\varepsilon \).

We conclude this section with the following example concerning Corollary 5.2.

*Example 5.3*

*Q*of one variable,

*E*satisfies Markov’s inequality. Moreover,

*E*is not polynomially convex. In Corollary 5.2, the assumption that

*E*is polynomially convex is therefore relevant even if \(N=1\). \(\square \)

## Footnotes

- 1.
Because otherwise \(\emptyset \ne H_j\subset \mathrm{{Int}\,h}(\mathbb {R}^N)\) for some \(j\le s\).

- 2.
- 3.
In fact, we could first consider the case \(\mathbb {K}=\mathbb {C}\) and then notice that the real case follows from the complex case.

- 4.If \(B=[b_{ij}]\) is a \(q\times q\) matrix of complex numbers, then$$\begin{aligned} |\det B|^2 \le \prod _{i=1}^q \sum _{j=1}^q |b_{ij}|^2. \end{aligned}$$
- 5.
*Schur’s inequality*: For each polynomial*R*of one variable,—see [10, p. 233], where this inequality is stated for real polynomials. If however \(R\in \mathbb {C}[\tau ]\) and \(R=R_1+iR_2\) with \(R_1,R_2\in \mathbb {R}[\tau ]\), then for \(\tau _0\in [-1,\,1]\) such that \({\Vert R\Vert }_{[-1,\,1]}=|R(\tau _0)|\), we have$$\begin{aligned} {\Vert R\Vert }_{[-1,\,1]}\le \left( \deg R+1\right) {\big \Vert \sqrt{1-\tau ^2}R(\tau ) \big \Vert }_{[-1,\,1]} \qquad \qquad (\lozenge ) \end{aligned}$$which proves \((\lozenge )\).$$\begin{aligned} |R(\tau _0)|^2\le & {} {\Vert R_1(\tau _0)R_1+ R_2(\tau _0)R_2 \Vert }_{[-1,\,1]}\\\le & {} \left( \deg R+1\right) {\big \Vert \sqrt{1-\tau ^2} \big ( R_1(\tau _0)R_1(\tau )+ R_2(\tau _0)R_2(\tau )\big ) \big \Vert }_{[-1,\,1]}\\\le & {} |R(\tau _0)|\left( \deg R+1\right) {\big \Vert \sqrt{1-\tau ^2}R(\tau ) \big \Vert }_{[-1,\,1]}, \end{aligned}$$ - 6.
See [34] for the definition and basic properties of plurisubharmonic functions.

- 7.That is, a closed set \(S\subset \partial \mathbb {D}_N\) such that:
- (i)for each continuous function \(f: \overline{\mathbb {D}}_N\longrightarrow \mathbb {C}\), holomorphic in \(\mathbb {D}_N\),$$\begin{aligned}{\Vert f\Vert }_{\overline{\mathbb {D}}_N}={\Vert f\Vert }_{S}, \end{aligned}$$
- (ii)
any closed set \(\tilde{S}\subset \partial \mathbb {D}_N\) satisfying (i) contains

*S*.

- (i)
- 8.For the polynomial \(Q(w_1,w_2):=w_2\), we have \({\Vert Q\Vert }_{\Gamma }=1\) and \(\left| Q\left( \alpha ,{\beta }/{\alpha } \right) \right| >1\). Consequently, \(\Phi _{\Gamma }\left( \alpha , {\beta }/{\alpha } \right) >1\) and combining this with Corollary 5.2.5 in [34] we obtain(recall that \(\phi ^*\) denotes the upper semicontinuous regularization of \(\phi \)). Since \(\Phi _{g^{-1}(E)}^*\left( \alpha , {\beta }/{\alpha } \right) >\Phi _{g^{-1}(E)}\left( \alpha , {\beta }/{\alpha } \right) \), it follows that \(\Phi _{g^{-1}(E)}\) is not continuous at the point \((\alpha , \beta /\alpha )\).$$\begin{aligned} \Phi _{g^{-1}(E)}^*\left( \alpha , {\beta }/{\alpha } \right) = \Phi _{\Gamma }^*\left( \alpha , {\beta }/{\alpha } \right) \ge \Phi _{\Gamma }\left( \alpha , {\beta }/{\alpha } \right) >1 = \Phi _{g^{-1}(E)}\left( \alpha , {\beta }/{\alpha } \right) \end{aligned}$$
- 9.
An alternative proof can also be found in [45, Corollary 2.8]

- 10.
The assumption that \(f|_{[R_1,\,R]}\) is definable in a certain polynomially bounded o-minimal structure is used here to guarantee definability of \(E_2\) in a polynomially bounded o-minimal structure and to guarantee the existence of a polynomial arc \(\gamma :(0,\,1) \longrightarrow \mathrm{{Int}}E\) such that \(\displaystyle \lim \limits _{t \rightarrow 0} \gamma (t) = (R,f(R))\). An explicit example of such an arc is \(\gamma : (0,\,1)\ni t\longmapsto \big (R-(\eta t)^m, f(R)-\eta t\big )\in \mathbb {R}^2\), where \(\eta >0\) is sufficiently small and \(m\in \mathbb {N}\) is sufficiently large, which follows from the definition of a polynomially bounded o-minimal structure.

- 11.
The only problem here is to see that there exists a polynomial arc \(\varphi :(0,\,1) \longrightarrow \mathrm{{Int}}F^{-1}(E)\) such that \(\displaystyle \lim \limits _{t \rightarrow 0} \varphi (t) = \big (R,\sqrt{f(R)}\big )\). However, this immediately follows from the assumption that \(f|_{[R_1,\,R]}\) is definable in a certain polynomially bounded o-minimal structure.

- 12.
Recall that pluripolar sets have Lebesgue measure zero.

- 13.
Recall that \(K_{(\lambda )}:=\{w\in \mathbb {C}^N:\, \mathrm{dist}(w,\,K) \le \lambda \}\) and \( K_{\lambda }:=\{w\in \mathbb {C}^N:\, \mathrm{dist}(w,\,K) < \lambda \}. \)

- 14.
Recall that \(\mathbb {D}_N:=\{z\in \mathbb {C}^N:\, |z|<1\}\).

## Notes

### Acknowledgments

I am very grateful to the referee for the comments and suggestions which improved the exposition.

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