Geometry of holomorphic mappings and Hölder continuity of the pluricomplex Green function

Abstract

We provide a solution to a long-standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory. The problem is to characterize the holomorphic maps which preserve Hölder continuity of the pluricomplex Green function associated with a compact subset of ${\mathbb {C}}^N$. We also prove, under mild restrictions, that nondegenerate holomorphic maps preserve Markov’s inequality for polynomials.

1 Introduction

For each compact set $\emptyset \ne K\subset {\mathbb {C}}^N$, the following function

$$\begin{aligned} V_{K}(z):=\sup \left\{ \phi (z):\,\phi \in {\mathcal {L}}({\mathbb {C}}^N),\,\phi \le 0\,\,\mathrm{on}\,\, K\right\} \end{aligned}$$

($z\in {\mathbb {C}}^N$), where ${\mathcal {L}}({\mathbb {C}}^N)$ denotes the class of plurisubharmonic functions $\phi$ in ${\mathbb {C}}^N$ (see [23] for the definition and basic properties of plurisubharmonic functions) satisfying the logarithmic growth condition

$$\begin{aligned} \sup _{z\in {\mathbb {C}}^N} \left[ \phi (z)-\log (1+|z|)\right] <+\infty \,, \end{aligned}$$

is called the pluricomplex Green function of K (with pole at infinity) or the Siciak-Zakharyuta extremal function; see for example [4, 23, 25, 39, 43, 44] and the bibliography therein. Here and subsequently, $|\,\,\,|$ denotes the maximum norm in ${\mathbb {C}}^N$.

If $N=1$, $K\subset {\mathbb {C}}$ is nonpolar (that is, of positive logarithmic capacity), and $K_{\infty }$ denotes the unbounded component of $\overline{{\mathbb {C}}}{\setminus } K$, then $V_K$ is harmonic in ${\mathbb {C}}{\setminus } K$ and the restriction of $V_K$ to ${K_{\infty }}$ is the Green function of $K_{\infty }$ with pole at infinity; see [44, 7.1 and 7.2].

Definition 1.1

(see [23]) A set $A\subset {\mathbb {C}}^N$ is said to be pluripolar if, for each point $a\in A$, there exists an open neighbourhood U of a such that $A\cap U\subset \{z\in U:\, \varphi (z)=-\infty \}$ for some plurisubharmonic function $\varphi : U\rightarrow [-\infty ,+\infty )$.

If $K\subset {\mathbb {C}}^N$ is a nonpluripolar compact set, then the upper semicontinuous regularization $V_{K}^*(z):=\limsup _{\zeta \rightarrow z}V_K(\zeta )$ of $V_{K}$ is plurisubharmonic in ${\mathbb {C}}^N$ and satisfies the complex Monge-Ampère equation:

$$\begin{aligned} \left( dd^c V_{K}^*\right) ^N=0 \qquad \text { in } {\mathbb {C}}^N{\setminus } K; \end{aligned}$$

see [4, 23, 26] for more details. Moreover, for each compact set $\emptyset \ne K\subset {\mathbb {C}}^N$,

$$\begin{aligned} K \text { is pluripolar } \iff V_{K}^*\equiv +\infty \iff V_{K}^* \notin {\mathcal {L}}({\mathbb {C}}^N); \end{aligned}$$

(1.1)

see [44, Corollary 3.9 and Theorem 3.10]).

We should also recall that the pluricomplex Green function is essentially equivalent to the Siciak extremal function. The latter is defined by the formula

$$\begin{aligned} \Phi _{K}(z):=\mathrm{sup}\left\{ {|Q(z)|}^{1/\mathrm{deg}\,Q} :\, Q\in {\mathbb {C}}[z_1,\ldots ,z_N]\,,\,\, \mathrm{deg}\,Q>0\,\, \mathrm{\,and}\,\,\,{\left\| Q\right\| }_{K}\le 1 \right\} \end{aligned}$$

($z\in {\mathbb {C}}^N$), where ${\Vert Q\Vert }_{K} :=\sup _{z\in K} |Q(z)|$. Strictly speaking, for each compact set $\emptyset \ne K\subset {\mathbb {C}}^N$,

$$\begin{aligned} V_{K}=\log \Phi _{K}; \end{aligned}$$

(1.2)

see [23, Theorem 5.1.7].

It may be worth noting that, for each $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$ with ${\Vert Q\Vert }_K>0$ and each $z\in {\mathbb {C}}^N$,

$$\begin{aligned} |Q(z)|\le \big ( \Phi _{K}(z)\big )^{\deg Q}{\Vert Q\Vert }_K. \end{aligned}$$

(1.3)

(We adopt here the convention that, for each $r\ge 0$, $(+\infty )^r:=+\infty$. Moreover, for K being nonpluripolar, the additional assumption that ${\Vert Q\Vert }_K>0$ is superfluous.) This trivial (but useful) estimate is called the Bernstein-Walsh inequality.

The pluricomplex Green function has been used to study various problems in (real and complex) analysis, functional analysis, pluripotential theory, complex dynamics and in approximation theory. From the point of view of applications, the most desirable property of this function is the HCP property.

Definition 1.2

(see [29]) We say that a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$ has the HCP property if there exist $\varpi , \mu >0$ such that, for each $z\in K_{(1)}$,

$$\begin{aligned} V_{K}(z)\le \varpi \left( \mathrm{dist}(z,\,K)\right) ^{\mu }. \end{aligned}$$

(1.4)

In the above definition, and subsequently, we use the following notation: for each set $\emptyset \ne A\subset {\mathbb {C}}^N$ and each $r>0$, we put

$$\begin{aligned} A_{(r)}:=\{z\in {\mathbb {C}}^N:\, \mathrm{dist}(z,\,A) \le r\}\,. \end{aligned}$$

Since $V_K\equiv 0$ in K, the HCP property can be reformulated in the following way: for each $z\in K_{(1)}$ and each $z'\in K$,

$$\begin{aligned} |V_{K}(z)-V_K(z')|\le \varpi |z-z'|^{\mu } \,. \end{aligned}$$

Surprisingly, a simple argument due to Błocki shows that this estimate already implies Hölder continuity of $V_K$ in ${\mathbb {C}}^N$; cf. [46, Proposition 3.5] or [42, Lemma 2.3]. More precisely, for each $\mu >0$, the following two conditions are equivalent:

• There exists $\varpi >0$ such that (1.4) holds.

• There exists ${\tilde{\varpi }}>0$ such that, for all $z,z'\in {\mathbb {C}}^N$,

  $$\begin{aligned} |V_{K}(z)-V_K(z')|\le {\tilde{\varpi }} |z-z'|^{\mu } \,. \end{aligned}$$

Example 1.3

Assume that $K_1,\ldots , K_N$ are nonempty compact subsets of ${\mathbb {C}}$ such that, for each $j\le N$ and each connected component $E_j$ of $K_j$, we have $\text {diam}(E_j)\ge \eta$, the constant $\eta >0$ being independent of j. Set $K:=K_1\times \cdots \times K_N$. By [45, Lemma 3.1], for each $j\le N$ and each $u\in (K_j)_{(1)}$,

$$\begin{aligned} V_{K_j}(u)\le \varpi \sqrt{\mathrm{dist}(u,\,K_j)}\,, \end{aligned}$$

where

$$\begin{aligned} \varpi :=\frac{4}{\eta } \left( 1+\sqrt{1+\frac{\eta }{2}}\right) \,. \end{aligned}$$

Fix $z=(z_1,\ldots , z_N)\in K_{(1)}$. By [23, Theorem 5.1.8],

$$\begin{aligned} \begin{aligned} V_K(z)&= \max \big \{ V_{K_{1}}(z_1), \ldots , V_{K_{N}}(z_N) \big \}\ \\&\le \varpi \max \left\{ \sqrt{\mathrm{dist}(z_1,\,K_1)}, \ldots , \sqrt{\mathrm{dist}(z_N,\,K_N)} \right\} \\&= \varpi \sqrt{\mathrm{dist}(z,\,K)}\,, \end{aligned} \end{aligned}$$

which yields the HCP property for the set K, with the exponent $\mu =1/2$. Let us emphasize that, for some product sets of planar compact sets, this exponent is not the best possible. For example, for a polydisk

$$\begin{aligned} {\mathbb {D}}(a,r): =\big \{ z\in {\mathbb {C}}^N:\, |z-a|\le r\big \} \end{aligned}$$

($a\in {\mathbb {C}}^N, r>0$), the HCP property holds with the exponent $\mu =1$. Indeed, since

$$\begin{aligned} V_{{\mathbb {D}}(a,r)}(z)=\max \left\{ 0,\, \log \frac{|z-a|}{r} \right\} \end{aligned}$$

(see [23, Example 5.1.1]), it follows that

$$\begin{aligned} V_{{\mathbb {D}}(a,r)}(z)\le \frac{ \mathrm{dist}\big (z,\,{\mathbb {D}}(a,r)\big )}{r} \end{aligned}$$

for all $z\in {\mathbb {C}}^N$. On the other hand, however, for the cube $[-1,\,1]^N\subset {\mathbb {R}}^N\subset {\mathbb {C}}^N$, the exponent $\mu =1/2$ is the best possible, which follows from the formula

$$\begin{aligned} V_{[-1,\,1]^N}(z)=\max \left\{ \log |z_1+\sqrt{{z_1}^2-1}|,\ldots ,\log |z_N+\sqrt{{z_N}^2-1}| \right\} \,, \end{aligned}$$

valid for all $z=(z_1,\ldots , z_N)\in {\mathbb {C}}^N$; see [23, Corollary 5.4.5]. For each $j\le N$, the square root is so chosen that $|z_j+\sqrt{{z_j}^2-1}|\ge 1$.

There have been several significant advances in understanding the HCP property for compact subsets of ${\mathbb {C}}$. In particular, we should mention here a very interesting work of Carleson and Totik [16], in which they give a sufficient Wiener-type criterion for a compact set $K\subset {\mathbb {C}}$ to have the HCP property.

Incomparably less has been done so far in the multivariate case (that is, for $N>1$). However, Pawłucki and Pleśniak in a seminal paper [29] give a sufficient geometric condition (UPC condition) for a compact set $K\subset {\mathbb {R}}^N$ to have the HCP property. Furthermore, in [29, 30, 32] large and natural classes of compact sets in ${\mathbb {R}}^N$ satisfying the UPC condition (and hence with the HCP property) are provided. More precisely, these classes consist of all compact, fat (a set E is said to be fat if ${\overline{E}} = \overline{\mathrm{Int} E}$) and definable sets in certain o-minimal structures; see [18] for the definition of an o-minimal structure. Each compact, fat and semianalytic subset of ${\mathbb {R}}^N$ is an explicit example of such a set.

Definition 1.4

(see [5, 27]) Let $\Omega \subset {\mathbb {R}}^N$ be an open set. A set $A\subset \Omega$ is said to be a semianalytic subset of $\Omega$ if, for each point in $\Omega$, we can find a neighbourhood W such that $A\cap W$ is a finite union of sets of the form

$$\begin{aligned} \big \{x \in W :\, \xi (x) = 0,\, \, \xi _1 (x)>0,\, \dots ,\, \xi _m (x) > 0\big \}, \end{aligned}$$

where $\xi ,\, \xi _1,\, \dots ,\, \xi _m$ are real analytic functions in W.

One of the long-standing open problems concerning the HCP property is the following.

Problem 1.5

(Pleśniak, 1988) Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a holomorphic map $(N, N'\in {\mathbb {N}}:=\{1,2,3,\ldots \})$. Assume that a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$ has the HCP property and ${\hat{K}}\subset U$. Under what conditions does it happen that h(K) has the HCP property?

Recall that ${\hat{K}}$ denotes the polynomially convex hull of K:

$$\begin{aligned} {\hat{K}}:=\big \{z\in {\mathbb {C}}^{N} :\,\, |Q(z)|\le {\Vert Q\Vert }_{K}\,\, \text { for each } Q\in {\mathbb {C}}[z_1,\ldots ,z_{N}]\big \}\,. \end{aligned}$$

(We set ${\hat{\emptyset }}:=\emptyset$.) If ${\hat{K}} =K$, then we say that K is polynomially convex. Occasionally, we will write $K^{\widehat{\,\,}}$ instead of ${\hat{K}}$. It is well known that:

• A compact set $K\subset {\mathbb {C}}$ is polynomially convex if and only if ${\mathbb {C}}{{\setminus } K}$ is connected; see [22, Corollary 1.3.2].

• Each compact subset of ${\mathbb {R}}^N$ is polynomially convex in ${\mathbb {C}}^N$; see [23, Lemma 5.4.1].

Remark 1.6

Note that, in Problem 1.5, the assumption that ${\hat{K}}\subset U$ is quite natural. Indeed, consider the simplest example: take $a\in {\mathbb {C}}$ with $|a|>1$, and put

$$\begin{aligned} U:={\mathbb {C}}{\setminus }\{0\}\,,\qquad h:U\ni z\mapsto \frac{1}{z}\in {\mathbb {C}}\,,\qquad K:=\{|z|=1\}\cup \left\{ \frac{1}{a} \right\} \subset {\mathbb {C}}\,. \end{aligned}$$

Moreover, for each $n\in {\mathbb {N}}$, set

$$\begin{aligned} Q_n(z):=z^n(z-a)\,, \qquad a_n:=a+\frac{1}{n}\,. \end{aligned}$$

Since ${\hat{K}}=\{|z|\le 1\}$, it follows that

$$\begin{aligned} \Phi _K(z)=\Phi _{{\hat{K}}}(z)=\max \{1, \, |z|\} \end{aligned}$$

for all $z\in {\mathbb {C}}$; see Example 1.3. In particular, the set K has the HCP property. On the other hand,

$$\begin{aligned} \liminf _{n\rightarrow +\infty } \Phi _{h(K)}(a_n)\ge \lim _{n\rightarrow +\infty } \root n+1 \of {\frac{|Q_n(a_n)|}{|a|+1}}=|a|>1=\Phi _{h(K)}(a)\,, \end{aligned}$$

and hence $\Phi _{h(K)}$ (and $V_{h(K)}$) is not even continuous.

Problem 1.5 is well known to specialists in the field and, at least since the 1980s, a number of attempts have been made to give a solution. We should mention here a result due to Pleśniak [37], which reads as follows. Let $h:U\rightarrow {\mathbb {C}}^{N}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a holomorphic map $(N\in {\mathbb {N}})$. Assume that a compact, polynomially convex set $\emptyset \ne K\subset U$ has the HCP property and h is nonsingular (see Definition 1.7) on K. Then h(K) has the HCP property as well. To my knowledge, except for this result of Pleśniak, which goes back to 1988, there has been no satisfactory progress on Problem 1.5. In this paper, we prove Theorem 1.8, which gives a complete solution of this problem. Before we state it, however, we set up terminology.

Definition 1.7

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a holomorphic map $(N, N'\in {\mathbb {N}})$.

• We say that h is nondegenerate if, for each connected component $U_{\iota }$ of U, there exists $\zeta _{\iota }\in U_{\iota }$ such that $\mathrm{{rank}}\,d_{{\zeta }_{\iota }} h=N'$.

• Let $K\subset U$. We say that h is nonsingular on K if $N=N'$ and, for each $\zeta \in K$, we have $\mathrm{{rank}}\,d_{\zeta } h=N$.

Theorem 1.8

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a holomorphic map $(N, N'\in {\mathbb {N}})$. Set

$$\begin{aligned} I_*:=\big \{ \iota \in I:\, h|_{U_{\iota }} \text { is nondegenerate}\big \}\,, \qquad U_{*}:=\bigcup _{\iota \in I_* } U_{\iota }, \end{aligned}$$

where $\{U_{\iota }\}_{\iota \in I}$ is the family of all connected components of U. Assume that a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$ has the HCP property and ${\hat{K}}\subset U$. Then the following three statements are equivalent:

• (i) h(K) has the HCP property;

• (ii) h(K) is L-regular (that is, $V_{h(K)}$ is continuous);

• (iii) $h(K)\subset h(K\cap U_{*})^{\widehat{\,\,}}$.

In particular, condition (iii) is the answer to Problem 1.5. Obviously, this condition is automatically satisfied if h nondegenerate. Hence, we get the following result.

Theorem 1.9

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a nondegenerate holomorphic map $(N, N'\in {\mathbb {N}})$. Assume that a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$ has the HCP property and ${\hat{K}}\subset U$. Then h(K) has the HCP property as well.

One of the most important applications of the HCP property concerns multivariate polynomial inequalities. More precisely, the HCP property is a sufficient condition for Markov’s inequality.

Definition 1.10

We say that a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$ satisfies Markov’s inequality (or: is a Markov set) if there exist $\varepsilon , C>0$ such that, for each polynomial $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$ and each $\alpha =(\alpha _1,\ldots ,\alpha _N)\in {\mathbb {N}}_0^N$,

$$\begin{aligned} {\Vert D^{\alpha }Q\Vert }_K\le \big ( C{(\mathrm{deg}\,Q)}^{\varepsilon }\big )^{|\alpha |}{\Vert Q\Vert }_{K}, \end{aligned}$$

(1.5)

where ${\mathbb {N}}_0:={\mathbb {N}}\cup \{0\}$, $D^{\alpha }Q:=\displaystyle \frac{\partial ^{|\alpha |}Q}{\partial z_1^{\alpha _1}\ldots \partial z_N^{\alpha _N}}$ and $|\alpha |:=\alpha _1+\cdots +\alpha _N$.

This is a generalization of the classical inequality due to Markov: If Q is a polynomial of one variable, then

$$\begin{aligned} {\Vert Q '\Vert }_{[-1,\,1]}\le (\deg Q)^2{\Vert Q\Vert }_{[-1,\,1]}\,. \end{aligned}$$

It is perhaps worth remarking that, for a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$, Markov’s inequality (Definition 1.10) is equivalent to the following condition: there exist $\varepsilon , D, M>0$ such that, for each polynomial $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$ with $\deg Q\le n$ ($n\in {\mathbb {N}}$),

$$\begin{aligned} {\Vert Q\Vert }_{K_{(Dn^{-\varepsilon })}}\le M{\Vert Q\Vert }_{K}. \end{aligned}$$

This follows easily from Cauchy’s inequalities and Taylor’s formula.

Markov type inequalities and related topics have been studied by many authors; see for instance [1,2,3, 6,7,8,9,10,11,12,13,14,15, 19,20,21, 24, 29, 30, 32, 34, 35, 38, 40, 47]. In this paper, we are interested in the following problem.

Problem 1.11

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a nondegenerate holomorphic map $(N, N'\in {\mathbb {N}})$. Assume that a compact set $\emptyset \ne K\subset {\mathbb {C}}^N$ satisfies Markov’s inequality and ${\hat{K}}\subset U$. Under what conditions does it happen that h(K) satisfies Markov’s inequality?

This problem has attracted considerable interest over the past decades, and certain partial results have been produced:

• Baran and Pleśniak [3, Theorem 2.5]. If additionally $N=N'$, K is polynomially convex, $h:U \rightarrow {\mathbb {C}}^N$ is nonsingular on K, and h(K) is nonpluripolar, then h(K) satisfies Markov’s inequality.

• Baran, Białas-Cież and Milówka [2, Theorem 4.2]. If additionally $N=N'=1$ and K is polynomially convex, then h(K) satisfies Markov’s inequality.

• Pierzchała [35, Theorem 1.4]. If additionally $U={\mathbb {C}}^N$ and h is a polynomial map, then h(K) satisfies Markov’s inequality. However, the proof of this result essentially relies on the assumption that h is a polynomial map and cannot be adapted to holomorphic maps.

In the present article, we give the following answer to the question raised in Problem 1.11.

Theorem 1.12

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a nondegenerate holomorphic map $(N, N'\in {\mathbb {N}})$. Assume that a compact set $K\subset {\mathbb {C}}^N$ satisfies Markov’s inequality, ${\hat{K}}\subset U$, and h(K) is a nonpluripolar subset of ${\mathbb {C}}^{N'}$. Then h(K) satisfies Markov’s inequality as well.

Theorem 1.12 “almost” solves Problem 1.11. The only issue here is the nonpluripolarity assumption on the set h(K). However, this assumption is really weak (in particular, pluripolar sets have Lebesgue measure zero; see [23, Corollary 2.9.10]). Furthermore, it is conjectured that all (nonempty) Markov sets are nonpluripolar. If this conjecture is true, then the set K of Theorem 1.12 is nonpluripolar and an elementary argument shows (cf. the proof of [36, Lemma 2.5]) that h(K) is nonpluripolar as well.

The proofs of Theorems 1.8 and 1.12 involve several ingredients. One of them is common to both proofs. It is the following result describing the geometry of nondegenerate holomorphic mappings.

Theorem 1.13

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a nondegenerate holomorphic map $(N, N'\in {\mathbb {N}})$. Assume that $K\subset U$ is a compact set. Then there exist $\varkappa , \theta , t_{*}>0$ and $q\in {\mathbb {N}}$ such that, for each $a\in K$, ${\mathbb {D}}(a,t_{*})\subset U$ and we can choose a polynomial map $Q_a:{\mathbb {C}}\rightarrow {\mathbb {C}}^{N'}$ with $\deg Q_a\le q$ such that

• $Q_a(0)=h(a)$,

• $\displaystyle \mathrm{dist}\Big (Q_a(t),\,{\mathbb {C}}^{N'}{\setminus } h\big ({\mathbb {D}}(a,t)\big )\Big )\ge \theta t^{\varkappa }$ for all $t\in (0,\,t_{*}]$.

Recall that in our notation ${\mathbb {D}}(a,t_{*}): =\displaystyle \big \{ z\in {\mathbb {C}}^N:\, |z-a|\le t_{*}\big \}\,.$

2 Proof of Theorem 1.13

2.1 Notation

For any $N,N'\in {\mathbb {N}}$, put

$$\begin{aligned} {\mathcal {A}}(N,N'):=\big \{ \sigma : \{1,\ldots ,N'\}\rightarrow \{1,\ldots , N\}:\, 1\le \sigma (1)<\cdots <\sigma (N')\le N\big \}\,. \end{aligned}$$

If $N'<N$ and $\sigma \in {\mathcal {A}}(N,N')$, then ${\bar{\sigma }}$ denotes the unique element of ${\mathcal {A}}(N,N-N')$ such that

$$\begin{aligned} \big \{ \sigma (1),\ldots , \sigma (N')\big \}\cup \big \{ {\bar{\sigma }}(1),\ldots , {\bar{\sigma }}(N-N')\big \}= \{1,\ldots , N\}\,. \end{aligned}$$

2.2 Key lemmas

We have divided the proof into a sequence of lemmas. The last lemma is the desired conclusion.

Take an open and bounded set $\Omega \subset {\mathbb {C}}^N$ such that $K\subset \Omega$, ${\overline{\Omega }}\subset U$ and $\Omega$ is a semianalytic subset of ${\mathbb {R}}^{2N}$. Moreover, set

$$\begin{aligned} E:=\big \{\zeta \in \Omega :\,\mathrm{{rank}}\,d_{\zeta } h=N'\big \}\,. \end{aligned}$$

Lemma 2.1

There exist $\theta _1,\theta _2,\upsilon >0$ and $d\in {\mathbb {N}}$ such that, for each $a\in {\overline{E}}$, we can choose a polynomial map $R_a:{\mathbb {C}}\rightarrow {\mathbb {C}}^N$ with $\deg R_a\le d$, satisfying the following conditions:

• (S1) $\mathrm{dist}\big (R_a(t),\,{\mathbb {C}}^N{\setminus } E\big ) \ge \theta _1t^{\upsilon }$ for all $t\in [0,\,1]$,

• (S2) $|R_a(t)-a|\le \theta _2t$ for all $t\in [0,\,1]$. In particular, $R_a(0)=a$.

Lemma 2.2

There exist $\theta _3,\theta _4, \omega >0$ such that, for each $a\in {\overline{E}}$ and each $t\in [0,\,1]$,

$$\begin{aligned} \theta _3\ge \sum _{\sigma \in {\mathcal {A}}(N,N')}\left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} \big (R_a(t)\big ) \right| \ge \theta _4t^{\omega }\,. \end{aligned}$$

Lemma 2.3

There exist $\theta _5,\theta _6, \varkappa >0$ such that, for each $a\in {\overline{E}}$ and each $t\in (0,\,1]$, ${\mathbb {D}}\big (R_a(t), \theta _5\big )\subset U$ and

$$\begin{aligned} {\mathbb {D}}\left( h\big ( R_a(t)\big ), \theta _6 t^{\varkappa } \right) \subset h\left( {\mathbb {D}}\big (R_a(t), \theta _5t\big ) \right) \,. \end{aligned}$$

Lemma 2.4

There exist $\theta , t_{*}>0$ and $q\in {\mathbb {N}}$ such that, for each $a\in K$, ${\mathbb {D}}(a,t_{*})\subset U$ and we can choose a polynomial map $Q_a:{\mathbb {C}}\rightarrow {\mathbb {C}}^{N'}$ with $\deg Q_a\le q$ such that

• $Q_a(0)=h(a)$,

• $\displaystyle \mathrm{dist}\Big (Q_a(t),\,{\mathbb {C}}^{N'}{\setminus } h\big ({\mathbb {D}}(a,t)\big )\Big )\ge \theta t^{\varkappa }$ for all $t\in (0,\,t_{*}]$.

2.3 Proofs of key lemmas

Proof of Lemma 2.1

Set

$$\begin{aligned} Y:=\big \{\zeta \in U:\,\mathrm{{rank}}\,d_{\zeta } h<N'\big \}\,. \end{aligned}$$

Note that

$$\begin{aligned} Y= \bigcap _{\sigma \in {\mathcal {A}}(N,N')} \left\{ \zeta \in U: \, \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (\zeta )=0\right\} \,. \end{aligned}$$

It follows that Y is a closed and semianalytic subset of $U\subset {\mathbb {R}}^{2N}$, which implies that $E= \Omega {\setminus } Y$ is an open, bounded and semianalytic subset of ${\mathbb {R}}^{2N}$. By [29, Theorem 6.4], there exist $\theta _1, \upsilon >0$ and $d\in {\mathbb {N}}$ such that, for each $x\in {\overline{E}}$, we can choose a polynomial map $P_x:{\mathbb {R}}\rightarrow {\mathbb {R}}^{2N}$ with $\deg P_x\le d$, satisfying the following conditions:

• $P_x(0)=x$,

• $\mathrm{dist}\big (P_x(t),\,{\mathbb {R}}^{2N}{\setminus } E\big ) \ge \theta _1t^{\upsilon }$ for all $t\in [0,\,1]$.

Take $\varphi _1,\ldots , \varphi _d: {\overline{E}}\rightarrow {\mathbb {R}}^{2N}$ such that, for each $x\in {\overline{E}}$ and each $t\in {\mathbb {R}}$,

$$\begin{aligned} P_x(t)=x+\varphi _1(x)t+\cdots +\varphi _d(x)t^d\,. \end{aligned}$$

By [30, Lemma 3.1], the maps $\varphi _1,\ldots , \varphi _d$ are bounded, and hence

$$\begin{aligned} C:={\Vert \varphi _1\Vert }_{ {\overline{E}}}+\cdots + {\Vert \varphi _d\Vert }_{ {\overline{E}}}<+\infty \,. \end{aligned}$$

For each $a\in {\overline{E}}$ and $t\in {\mathbb {R}}$, set $R_a(t):=\chi _N\big ( P_{a}(t) \big )$, where

$$\begin{aligned} \chi _N: {\mathbb {R}}^{2N}\ni (u_1,v_1,\ldots ,u_N,v_N)\mapsto (u_1+iv_1, \ldots , u_N+iv_N)\in {\mathbb {C}}^N\,. \end{aligned}$$

It is straightforward to see that conditions (S1) and (S2) hold with $\theta _2:=\sqrt{2}C$. $\square$

Proof of Lemma 2.2

Note that, for each $a\in {\overline{E}}$ and each $t\in [0,\,1]$, $R_a(t)\in {\overline{E}}$, and hence the first required estimate holds with the constant

$$\begin{aligned} \theta _3:=\max _{\zeta \in {\overline{E}}} \left( \sum _{\sigma \in {\mathcal {A}}(N,N')} \left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (\zeta ) \right| \right) \,. \end{aligned}$$

Let Y be as in the proof of Lemma 2.1. There are two cases to consider.

Case 1: $Y=\emptyset$. Then the second required estimate holds with $\omega :=1$ and

$$\begin{aligned} \theta _4:=\min _{\zeta \in {\overline{E}}} \left( \sum _{\sigma \in {\mathcal {A}}(N,N')} \left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (\zeta ) \right| \right) \,. \end{aligned}$$

Case 2: $Y\ne \emptyset$. By [28, p. 243], there exist $C_1, \omega _1>0$ such that, for each $\zeta \in {\overline{E}}$,

$$\begin{aligned} \sum _{\sigma \in {\mathcal {A}}(N,N')} \left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (\zeta ) \right| \ge C_1\mathrm{dist}(\zeta , Y)^{\omega _1}\,. \end{aligned}$$

(2.1)

Set $\theta _4:=C_1\theta _1^{\omega _1}$ and $\omega :=\upsilon \omega _1$, where $\theta _1, \upsilon >0$ are of Lemma 2.1. Fix $a\in {\overline{E}}$ and $t\in [0,\,1]$. By (2.1) and Lemma 2.1 (condition (S1)), we get

$$\begin{aligned} \sum _{\sigma \in {\mathcal {A}}(N,N')}\left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} \big (R_a(t)\big ) \right|&\ge C_1\Big (\mathrm{dist}\big (R_a(t),\,Y\big ) \Big )^{\omega _1}\\&\ge C_1\Big ( \mathrm{dist}\big (R_a(t),\,{\mathbb {C}}^N{\setminus } E\big )\Big )^{\omega _1} \ge \theta _4t^{\omega }\,, \end{aligned}$$

which yields the second required estimate. $\square$

Proof of Lemma 2.3

Let $\theta _3,\theta _4,\omega >0$ be of Lemma 2.2. Take $\epsilon \in (0,\,1)$, and also take $r_0>0$ such that $E_{(r_0)}\subset U$. Set

$$\begin{aligned}&\eta _1:= \max _{1\le j\le N'}\left( \sum _{1\le k\le N}{\left\| \frac{\partial h_j}{\partial z_k} \right\| }_{E}\right) \,,\\&\eta _2:=\max _{1\le j\le N'}\left( \sum _{\begin{array}{c} 1\le k\le N\\ 1\le \nu \le N \end{array}}{\left\| \frac{\partial ^2 h_j}{\partial z_k\partial z_{\nu }} \right\| }_{E_{( r_0)}}\right) \,. \end{aligned}$$

Furthermore, fix $a\in {\overline{E}}$, $t\in (0,\,1]$, and put $b=b(a,t):=R_a(t)$. Note that $b\in E$. By Lemma 2.2, there exists $\sigma \in {\mathcal {A}}(N,N')$ such that

$$\begin{aligned} \theta _3\ge \left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (b) \right| \ge \frac{\theta _4t^{\omega }}{\left( {\begin{array}{c}N\\ N'\end{array}}\right) }\,. \end{aligned}$$

(2.2)

Define $H_{\sigma }: U\rightarrow {\mathbb {C}}^N$ by the formula

$$\begin{aligned} H_{\sigma }(z):=\left. {\left\{ \begin{array}{ll} \big ( h(z), z_{{\bar{\sigma }}(1)},\ldots , z_{{\bar{\sigma }}(N-N')}\big ) &{} \mathrm{if}\,\,N'<N\\ h(z) &{} \mathrm{if}\,\, N'=N. \end{array}\right. } \right. \end{aligned}$$

Note that $d_b H_{\sigma }: {\mathbb {C}}^N\rightarrow {\mathbb {C}}^N$ is an isomorphism, because

$$\begin{aligned} \left| \mathrm{Jac}\,H_{\sigma }(b) \right| = \left| \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (b) \right| \ne 0\,. \end{aligned}$$

Set $\varrho _b:=1/\left\| (d_b H_{\sigma })^{-1} \right\|$. Here, and throughout this proof, $\Vert \,\,\,\Vert$ denotes the operator norm.

Note that, for each $\zeta \in E$,

$$\begin{aligned} \max _{\begin{array}{c} 1\le j\le N\\ 1\le k\le N \end{array}}\left| \frac{\partial H_{\sigma ,j}}{\partial z_k} (\zeta )\right| \le \max \{\eta _1,1\}\,, \end{aligned}$$

(2.3)

where

$$\begin{aligned} H_{\sigma ,j}:=\left. {\left\{ \begin{array}{ll} h_j &{} \mathrm{if}\,\,1\le j \le N'\\ z_{{\bar{\sigma }}(j-N')}&{} \mathrm{if}\,\, N'+1\le j\le N. \end{array}\right. } \right. \end{aligned}$$

For $1\le j\le N, 1\le k\le N$, let $\Delta _{jk}(\zeta )\in {\mathbb {C}}$ denote the entries of the classical adjoint of the jacobian matrix of the map $H_{\sigma }$ at the point $\zeta$. It follows from (2.3) that, for each $\zeta \in E$, $|\Delta _{jk}(\zeta )|\le M$, where $M=M(\eta _1, N)$ is a positive constant depending only on $\eta _1$ and N. For example, we can take

$$\begin{aligned} M:= \big ( \max \{\eta _1,1\}\big )^{N-1}(N-1)^{(N-1)/2} \,. \end{aligned}$$

Thus, we have in particular the following estimate:

$$\begin{aligned} \left\| (d_b H_{\sigma })^{-1} \right\| \le \frac{MN}{\left| \mathrm{Jac}\,H_{\sigma }(b) \right| }=\frac{MN}{\left| \displaystyle \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (b) \right| }\,, \end{aligned}$$

and hence

$$\begin{aligned} \varrho _b\ge \frac{\left| \displaystyle \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (b) \right| }{MN}\,. \end{aligned}$$

(2.4)

Set

$$\begin{aligned}&\eta := \max \left\{ \frac{\epsilon \eta _1}{r_0}, \frac{\epsilon }{r_0}, \eta _2, \frac{\epsilon \theta _3}{r_0MN} \right\} \,,\\&\theta _5 := \frac{\epsilon \theta _3}{MN\eta }\,,\\&\theta _6:= \frac{\epsilon (1-\epsilon )}{\eta }\left( \frac{\theta _4}{\left( {\begin{array}{c}N\\ N'\end{array}}\right) MN} \right) ^2\,,\\&\varkappa := 2\omega +1\,,\\&r=r(a,t):=\frac{\epsilon t}{MN\eta }\left| \displaystyle \frac{\partial (h_1,\ldots ,h_{N'})}{\partial (z_{\sigma (1)}, \ldots ,z_{\sigma (N')} )} (b) \right| \,. \end{aligned}$$

Note that $\theta _5$ and $\theta _6$ depend neither on a nor t. Moreover, (2.4) gives

$$\begin{aligned} r \le \frac{\epsilon \varrho _b}{\eta }\,. \end{aligned}$$

(2.5)

Clearly, $\Vert d_b H_{\sigma }\Vert \le \max \{\eta _1,1\}$, and hence, by (2.5),

$$\begin{aligned} r&\le \frac{\epsilon \varrho _b}{\eta }\le \frac{ r_0\varrho _b}{\max \{\eta _1,1\}}= \frac{ r_0}{\max \{\eta _1,1\}\left\| (d_b H_{\sigma })^{-1} \right\| }\le \frac{ r_0 \Vert d_b H_{\sigma }\Vert }{\max \{\eta _1,1\}}\le r_0\,. \end{aligned}$$

Consequently, $r\le r_0$ and ${\mathbb {D}}(b,r)\subset E_{(r_0)}\subset U$.

Put $g_b:= d_bH_{\sigma }-H_{\sigma }:U\rightarrow {\mathbb {C}}^N$. Observe that, for each $\zeta \in {\mathbb {D}}(b,r)$,

$$\begin{aligned} \Vert d_{\zeta }g_b\Vert \le \eta _2r\,. \end{aligned}$$

(2.6)

Indeed, $d_{\zeta }g_b=d_b H_{\sigma }- d_{\zeta } H_{\sigma }$ and therefore

$$\begin{aligned} \Vert d_{\zeta }g_b\Vert =\Vert d_b H_{\sigma }- d_{\zeta } H_{\sigma } \Vert \le \eta _2|\zeta -b|\le \eta _2r\,, \end{aligned}$$

which gives (2.6).

Consider the map $\psi _b:=(d_b H_{\sigma })^{-1}\circ g_b$. For $z,z'\in {\mathbb {D}}(b,r)$, we have

$$\begin{aligned} |\psi _b(z)-\psi _b(z')|&\le \left\| (d_b H_{\sigma })^{-1} \right\| \cdot |g_b(z)-g_b(z')|&\\&\le \eta _2 r\left\| (d_b H_{\sigma })^{-1} \right\| \cdot |z-z'|&\text {(by }(2.6)\text {)}\\&\le \frac{\eta r}{\varrho _b}|z-z'|\le \epsilon |z-z'|&\text {(by }(2.5)\text {)\,,} \end{aligned}$$

and hence

$$\begin{aligned} |\psi _b(z)-\psi _b(z')|\le \epsilon |z-z'|\,. \end{aligned}$$

(2.7)

[17, Theorem 4.4.1], together with (2.7), yields

$$\begin{aligned} {\mathbb {D}}\Big ( (d_b H_{\sigma })^{-1}\big ( H_{\sigma }(b) \big ) ,(1-\epsilon )r\Big ) \subset (d_b H_{\sigma })^{-1}\Big ( H_{\sigma }\big ( {\mathbb {D}}(b,r) \big ) \Big )\,. \end{aligned}$$

(2.8)

But, for each $z\in {\mathbb {C}}^N$ and $\tau >0$,

$$\begin{aligned} {\mathbb {D}}\big ( d_bH_{\sigma }(z), \varrho _b\tau \big )\subset d_bH_{\sigma }\big ({\mathbb {D}}(z,\tau )\big )\,. \end{aligned}$$

Combining this with (2.8), we get

$$\begin{aligned} {\mathbb {D}}\big ( H_{\sigma }(b) ,\varrho _b(1-\epsilon )r\big )\subset H_{\sigma }\big ( {\mathbb {D}}(b,r)\big )\,. \end{aligned}$$

Let

$$\begin{aligned} \pi : {\mathbb {C}}^N\ni (u_1,\ldots ,u_N)\mapsto (u_1, \ldots , u_{N'})\in {\mathbb {C}}^{N'}\,. \end{aligned}$$

The above inclusion implies that

$$\begin{aligned} \pi \Big ({\mathbb {D}}\big ( H_{\sigma }(b) ,\varrho _b(1-\epsilon )r\big )\Big )\subset ( \pi \circ H_{\sigma })\big ( {\mathbb {D}}(b,r)\big )\,, \end{aligned}$$

and hence

$$\begin{aligned} {\mathbb {D}}\big ( h(b) ,\varrho _b(1-\epsilon )r\big )\subset h\big ( {\mathbb {D}}(b,r)\big )\,. \end{aligned}$$

Since $\varrho _b(1-\epsilon )r\ge \theta _6t^{\varkappa }$ (see (2.2) and (2.4)) and $r\le \theta _5t$ (see (2.2)), we get

$$\begin{aligned} {\mathbb {D}}\big ( h(b), \theta _6 t^{\varkappa } \big ) \subset h\big ( {\mathbb {D}}(b, \theta _5t) \big ) \,, \end{aligned}$$

which is the desired conclusion. $\square$

Proof of Lemma 2.4

Let $\theta _2>0$, $d\in {\mathbb {N}}$ be of Lemma 2.1 and let $\theta _5,\theta _6,\varkappa >0$ be of Lemma 2.3. Take $q_0\in {\mathbb {N}}$ such that $q_0>\varkappa -1$ and put

$$\begin{aligned} \theta _7:=\frac{1}{\theta _2+\theta _5}\,,\qquad \theta :=\frac{\theta _6\theta _7^{\varkappa }}{2}\,, \qquad q:=dq_0\,. \end{aligned}$$

For each $u\in (-1,\,1)$, set

$$\begin{aligned} \varphi (u):=\sum _{\nu =q_0+1}^{+\infty }\left( {\begin{array}{c}\nu +N-1\\ \nu \end{array}}\right) u^{\nu -q_0-1}\,. \end{aligned}$$

Furthermore, take $r_0>0$ such that $E_{(r_0)}\subset U$.

Clearly, there exists $t_{*}>0$ such that $t_{*}\le \min \big \{r_0, 1/\theta _7 \big \}$ and

$$\begin{aligned} t_{*}^{q_0+1-\varkappa }\varphi \left( \frac{\theta _2\theta _7t_{*}}{r_0} \right) \le \frac{\theta }{{\Vert h\Vert }_{E_{(r_0)}}}\left( \frac{r_0}{\theta _2\theta _7} \right) ^{q_0+1}\,. \end{aligned}$$

(2.9)

Note that $t_{*}\le r_0<r_0/(\theta _2\theta _7)$.

Fix $a\in {\overline{E}}$ (observe that $K\subset \Omega \subset {\overline{E}}$). Define $Q_a=\big (Q_{a,1},\ldots , Q_{a,N'}\big ): {\mathbb {C}}\rightarrow {\mathbb {C}}^{N'}$ by the formula

$$\begin{aligned} Q_{a,j}(t):= \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^N\\ |\alpha |\le q_0 \end{array}}\frac{D^{\alpha }h_j(a)}{\alpha !}\big ( R_a\left( \theta _7t \right) -a \big )^{\alpha }\,, \end{aligned}$$

where $R_a:{\mathbb {C}}\rightarrow {\mathbb {C}}^N$ is the polynomial map of Lemma 2.1. Clearly, $\deg Q_a\le q$ and $Q_a(0)=h(a)$; see Lemma 2.1. Note also that, for each $t\in [0,\,t_{*}]$,

$$\begin{aligned} \left| h\big ( R_a\left( \theta _7t \right) \big ) -Q_a(t)\right| \le \theta t^{\varkappa }\,. \end{aligned}$$

(2.10)

Indeed, the estimate (S2) of Lemma 2.1 gives

$$\begin{aligned} \left| R_a\left( \theta _7 t\right) -a\right| \le \theta _2\theta _7t<r_0\,. \end{aligned}$$

(2.11)

In particular,

$$\begin{aligned} R_a\left( \theta _7 t\right) \in {\mathbb {D}}(a,r_0)\subset E_{(r_0)}\subset U\,. \end{aligned}$$

(2.12)

Hence, for each $j\le N'$,

$$\begin{aligned} h_j\big ( R_a\left( \theta _7t \right) \big )=\sum _{\alpha \in {\mathbb {N}}_0^N}\frac{D^{\alpha }h_j(a)}{\alpha !}\big ( R_a\left( \theta _7t \right) -a \big )^{\alpha }\,. \end{aligned}$$

(2.13)

Finally, for each $j\le N'$,

$$\begin{aligned}&\left| h_j\big ( R_a\left( \theta _7t \right) \big ) -Q_{a,j}(t)\right| \\&\le \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^N\\ |\alpha |\ge q_0+1 \end{array}}\frac{\left| D^{\alpha }h_j(a)\right| }{\alpha !}\left| R_a\left( \theta _7t \right) -a \right| ^{|\alpha |}&\text {(by } (2.13)\text {)}\\ {}&\le \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^N\\ |\alpha |\ge q_0+1 \end{array}} \frac{\left| D^{\alpha }h_j(a)\right| }{\alpha !} \left( \theta _2\theta _7t\right) ^{|\alpha |}&\text {(by } (3.11)\text {)} \\&\le \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^N\\ |\alpha |\ge q_0+1 \end{array}} {\Vert h_j\Vert }_{E_{(r_0)}} \left( \frac{\theta _2\theta _7t}{r_0} \right) ^{|\alpha |}&\text {(by }(2.12)\text { and Cauchy's inequalities)} \\&= \sum _{\nu =q_0+1}^{+\infty }{\Vert h_j\Vert }_{E_{(r_0)}}\left( {\begin{array}{c}\nu +N-1\\ \nu \end{array}}\right) \left( \frac{\theta _2\theta _7t}{r_0} \right) ^{\nu } \\&\le {\Vert h_j\Vert }_{E_{(r_0)}}\left( \frac{\theta _2\theta _7}{r_0} \right) ^{q_0+1}t^{\varkappa }t_{*}^{q_0+1-\varkappa }\varphi \left( \frac{\theta _2\theta _7t_{*}}{r_0} \right) \\&\le \theta t^{\varkappa }&\text {(by }(2.9)\text {)} \,, \end{aligned}$$

which yields (2.10).

It follows from (2.11) that, for each $t\in (0,\,t_{*}]$,

$$\begin{aligned} {\mathbb {D}}\big ( R_a\left( \theta _7t \right) , \theta _5\theta _7t \big )\subset {\mathbb {D}}(a,t)\subset {\mathbb {D}}(a,t_{*}) \subset U\,. \end{aligned}$$

(2.14)

Therefore, for each $t\in (0,\,t_{*}]$,

$$\begin{aligned}&\ \,\mathrm{dist}\Big (Q_a(t),\,{\mathbb {C}}^{N'}{\setminus } h\big ({\mathbb {D}}(a,t)\big )\Big )\\&\quad \ge \mathrm{dist}\Big ( h\big ( R_a\left( \theta _7t\right) \big ) ,\,{\mathbb {C}}^{N'}{\setminus } h\big ({\mathbb {D}}(a,t)\big )\Big ) -\theta t^{\varkappa }&\text {(by }(2.10)\text {)}\\&\quad \ge \mathrm{dist}\bigg ( h\big ( R_a\left( \theta _7t\right) \big ) ,\,{\mathbb {C}}^{N'}{\setminus } h \Big ( {\mathbb {D}}\big ( R_a\left( \theta _7t \right) , \theta _5\theta _7t \big ) \Big ) \bigg ) -\theta t^{\varkappa }&\text {(by }(2.14)\text {)}\\&\quad \ge \theta _6\left( \theta _7t\right) ^{\varkappa }-\theta t^{\varkappa } =\theta t^{\varkappa }&\text {(by Lemma }2.3\text {)}\,, \end{aligned}$$

which completes the proof of the lemma (and hence proves Theorem 1.13). $\square$

3 Proof of Theorem 1.9

Proof of Theorem 1.9

Take an open, bounded set $\Omega \subset {\mathbb {C}}^N$ such that ${\hat{K}}\subset \Omega$ and ${\overline{\Omega }}\subset U$. Furthermore, take a compact and polynomially convex set $E\subset {\mathbb {C}}^N$ such that $K\subset \mathrm{Int}E$ and $E\subset \Omega$; cf. [22, Proof of Lemma 2.7.4]. By the uniform version of the Bernstein-Walsh-Siciak theorem (see [36, 41] and see also [33]), there exist $C_1>0$ and $\rho \in (0,\,1)$ with the following property: for each holomorphic and bounded function $f: \Omega \rightarrow {\mathbb {C}}$ and each $\nu \in {\mathbb {N}}$, there exists a polynomial $W_{\nu }\in {\mathbb {C}}[z_1,\ldots ,z_N]$ with $\deg W_{\nu }\le \nu$ and such that

$$\begin{aligned} {\Vert f-W_{\nu }\Vert }_E\le C_1{\Vert f\Vert }_{\Omega }\,\rho ^{\nu }\,. \end{aligned}$$

(3.1)

By Theorem 1.13, there exist $\varkappa , \theta , t_{*}>0$ and $q\in {\mathbb {N}}$ such that, for each $a\in K$, ${\mathbb {D}}(a,t_{*})\subset U$ and we can choose a polynomial map $Q_a:{\mathbb {C}}\rightarrow {\mathbb {C}}^{N'}$ with $\deg Q_a\le q$ such that

• (i) $Q_a(0)=h(a)$,

• (ii) $\displaystyle \mathrm{dist}\Big (Q_a(t),\,{\mathbb {C}}^{N'}{\setminus } h\big ({\mathbb {D}}(a,t)\big )\Big )\ge \theta t^{\varkappa }$ for all $t\in (0,\,t_{*}]$.

We can clearly assume that $\varkappa \in {\mathbb {N}}$ and $K_{(t_{*})}\subset E$.

Note that the set h(K) is nonpluripolar; cf. [36, Proof of Lemma 2.5]. Therefore, by (1.1) and (1.2), $\Phi _{h(K)}$ is locally bounded in ${\mathbb {C}}^{N'}$. In particular, there exists $k\in {\mathbb {N}}$ such that

$$\begin{aligned} \rho ^k \sup _{h(\Omega )} \Phi _{h(K)}\le 1\,. \end{aligned}$$

(3.2)

Since the set K has the HCP property, there exist $C_2, \mu >0$ such that, for each $z\in K_{(t_{*})}$,

$$\begin{aligned} \Phi _{K}(z)\le 1+C_2\left( \mathrm{dist}(z,\,K)\right) ^{\mu }. \end{aligned}$$

(3.3)

Set

$$\begin{aligned}&\gamma := \frac{\mu }{\varkappa (2\mu +1)}\,,\\&\tau _{*}:= t_{*}^{\varkappa (2\mu +1)}\,,\\&C_3:=2\sqrt{\frac{2N'-1}{\theta ^{1/\varkappa }}} \left( \sqrt{\tau _{*}^{2\gamma }\frac{2N'-1}{\theta ^{1/\varkappa }}} +\sqrt{\tau _{*}^{2\gamma }\frac{2N'-1}{\theta ^{1/\varkappa }}+1 }\right) \,,\\&C_4:=\sup _{\tau \in (0,\,\tau _{*}]} \frac{ \big ( 1+\max \{C_2,C_3\}\tau ^{\gamma }\big )^{\max \{q,\varkappa \}+k} -1}{ \tau ^{\gamma } }\\ {}&\,\,\,\,\,\,\,\,= \frac{ \big ( 1+\max \{C_2,C_3\}\tau _{*}^{\gamma }\big )^{\max \{q,\varkappa \}+k} -1}{ \tau _{*}^{\gamma } }\,. \end{aligned}$$

We claim that, for each $y\in h(K)_{(\tau _{*})}$,

$$\begin{aligned} \Phi _{h(K)}(y)\le 1+C_4\left( \mathrm{dist}\big (y,\,h(K)\big )\right) ^{\gamma }\,, \end{aligned}$$

(3.4)

which then proves the theorem.

To see the claim, fix a polynomial $P\in {\mathbb {C}}[y_1,\ldots ,y_{N'}]$ with $\deg P\le n$ ($n\in {\mathbb {N}}$), and also fix $y\in h(K)_{(\tau _{*})}$. Take $a\in K$ such that $\mathrm{dist}\big (y,\,h(K)\big )=|y-h(a)|$ and set $\tau :=|y-h(a)|$. Note that $\tau \le \tau _{*}$. It suffices to show that

$$\begin{aligned} |P(y)|\le \big (1+C_4\tau ^{\gamma }\big )^n{\Vert P\Vert }_{h(K)}\,. \end{aligned}$$

(3.5)

We may assume that $\tau >0$, because otherwise (3.5) is trivial. Put $r:=\tau ^{\gamma /\mu }$ and let $H_r: {\mathbb {C}}^{N'+1}\rightarrow {\mathbb {C}}^{N'}$ be the map

$$\begin{aligned} H_r(s,\chi _1,\ldots ,\chi _{N'})&=Q_a\left( \frac{r}{N'+1}\big (s+\chi _1+\cdots +\chi _{N'}\big )\right) \\&\quad + \theta \left( \frac{r}{N'+1}\right) ^{\varkappa }\big ( (s-\chi _1)^{\varkappa },\ldots , (s-\chi _{N'})^{\varkappa } \big )\,. \end{aligned}$$

Clearly, $H_r$ is a polynomial map with $\deg H_r\le \max \{q,\varkappa \}$. Using (i) and (ii) we easily verify that

$$\begin{aligned} H_r\big ( [0,\,1]^{N'+1}\big )\subset h\big ({\mathbb {D}}(a,r)\big )\,. \end{aligned}$$

(3.6)

Let $\zeta (y)=\big ( \zeta _1(y),\ldots , \zeta _{N'}(y)\big )\in {\mathbb {C}}^{N'}$ be such that

$$\begin{aligned} \theta \left( \frac{r}{N'+1}\right) ^{\varkappa } \big ( \zeta _1^{\varkappa }(y),\ldots , \zeta _{N'}^{\varkappa }(y)\big )=y-h(a)\,. \end{aligned}$$

(3.7)

Note that

$$\begin{aligned} |\zeta (y)|\le \left( \frac{\tau }{\theta } \right) ^{1/\varkappa }\frac{N'+1}{r}\,. \end{aligned}$$

(3.8)

Set

$$\begin{aligned}&s(y):=\frac{\zeta _1(y)+\cdots +\zeta _{N'}(y)}{N'+1}\,,\\&\chi _1(y):=\frac{\zeta _1(y)+\cdots +\zeta _{N'}(y)}{N'+1}-\zeta _{1}(y)\,,\\&\quad \vdots \\&{\chi }_{N'}(y):=\frac{\zeta _1(y)+\cdots +\zeta _{N'}(y)}{N'+1}-\zeta _{N'}(y)\,. \end{aligned}$$

By (i) and (3.7),

$$\begin{aligned} H_r\big (s(y),\chi _1(y),\ldots ,\chi _{N'}(y)\big )=y\,. \end{aligned}$$

(3.9)

On account of (3.8), we have moreover

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} |s(y)|\le \displaystyle \left( \frac{\tau }{\theta } \right) ^{1/\varkappa }\frac{N'}{r}\,,\\ &{} |\chi _1(y)|\le \displaystyle \left( \frac{\tau }{\theta } \right) ^{1/\varkappa }\frac{2N'-1}{r}\,,\\ &{}\quad \vdots \\ &{} |\chi _{N'}(y)|\le \displaystyle \left( \frac{\tau }{\theta } \right) ^{1/\varkappa }\frac{2N'-1}{r}\,. \end{array}\right. } \end{aligned}$$

(3.10)

Recall that, for each $w\in {\mathbb {C}}$,

$$\begin{aligned} \Phi _{[-1,\,1]}(w)=\big | w+\sqrt{w^2-1} \big |\,, \end{aligned}$$

where the square root is so chosen that $\big | w+\sqrt{w^2-1} \big |\ge 1$; see the last formula in Example 1.3. Consequently,

$$\begin{aligned} \Phi _{[0,\,1]}(w)=\big | 2w-1+2\sqrt{w^2-w} \big |\,, \end{aligned}$$

and hence

$$\begin{aligned} \Phi _{[0,\,1]}(w)\le 1 +2\sqrt{|w|}\big ( \sqrt{|w|}+\sqrt{|w|+1} \big ) \end{aligned}$$

(3.11)

for all $w\in {\mathbb {C}}$.

By [44, Proposition 5.9],

$$\begin{aligned}&\Phi _{[0,\,1]^{N'+1}} \big ( s(y),\chi _1(y),\ldots ,\chi _{N'}(y)\big )\\&\quad = \max \Big \{ \Phi _{[0,\,1]}\big (s(y)\big ), \Phi _{[0,\,1]}\big (\chi _1(y)\big ),\ldots , \Phi _{[0,\,1]}\big (\chi _{N'}(y)\big ) \Big \} \\&\quad \le \overbrace{1+ 2\sqrt{\left( \frac{\tau }{\theta }\right) ^{1/\varkappa }\frac{2N'-1}{r}} \left( \sqrt{\left( \frac{\tau }{\theta }\right) ^{1/\varkappa }\frac{2N'-1}{r}}+ \sqrt{\left( \frac{\tau }{\theta }\right) ^{1/\varkappa }\frac{2N'-1}{r}+1} \right) }^{ \text {by }(3.10)\text { and }(3.11)} \\&\quad \le 1+C_3\tau ^{\gamma }\,. \end{aligned}$$

Hence

$$\begin{aligned} \Phi _{[0,\,1]^{N'+1}} \big ( s(y),\chi _1(y),\ldots ,\chi _{N'}(y)\big )\le 1+C_3\tau ^{\gamma }\,. \end{aligned}$$

(3.12)

Note that

$$\begin{aligned} |P(y)|\le \big ( 1+C_3\tau ^{\gamma }\big )^{n\max \{q,\varkappa \}}{\Vert P\Vert }_{ h\left( {\mathbb {D}}(a,r)\right) }\,. \end{aligned}$$

(3.13)

Indeed,

$$\begin{aligned} |P(y)|&= \left| (P\circ H_r)\big ( s(y),\chi _1(y),\ldots ,\chi _{N'}(y)\big )\right|&\text {(by } (3.9)\text {)}\\&\le \left( \Phi _{[0,\,1]^{N'+1}} \big ( s(y),\chi _1(y),\ldots ,\chi _{N'}(y)\big ) \right) ^{n\max \{q,\varkappa \}}{\Vert P\circ H_r\Vert }_{[0,\,1]^{N'+1}}&\text {(by }(1.3)\text {)}\\&\le \big ( 1+C_3\tau ^{\gamma }\big )^{n\max \{q,\varkappa \}}{\Vert P\circ H_r\Vert }_{[0,\,1]^{N'+1}}&\text {(by }(3.12)\text {)}\\&\le \big ( 1+C_3\tau ^{\gamma }\big )^{n\max \{q,\varkappa \}} {\Vert P\Vert }_{ h\left( {\mathbb {D}}(a,r)\right) }&\text {(by } (3.6)\text {)}\,, \end{aligned}$$

which yields (3.13).

By (3.1), for each $j\in {\mathbb {N}}$, there exists a polynomial $R_j\in {\mathbb {C}}[z_1,\ldots , z_N]$ with $\deg R_j\le jkn$ such that

$$\begin{aligned} {\Vert P^j\circ h-R_j\Vert }_{E}\le C_1{\Vert P^j\circ h\Vert }_{\Omega }\,\rho ^{jkn}\,. \end{aligned}$$

Then

$$\begin{aligned} {\Vert P^j\circ h-R_j\Vert }_{E}&\le C_1{\Vert P^j\circ h\Vert }_{\Omega }\,\rho ^{jkn}&\\&\le C_1\Big (\sup _{h(\Omega )} \Phi _{h(K)}\Big )^{jn} {\Vert P\Vert }_{h(K)}^j\,\rho ^{jkn }&\text {(by }(1.3)\text {)}\\&\le C_1{\Vert P\Vert }_{h(K)}^j&\text {(by }(3.2)\text {)}\,. \end{aligned}$$

Consequently,

$$\begin{aligned} {\Vert P^j\circ h-R_j\Vert }_{E}\le C_1{\Vert P\Vert }_{h(K)}^j\,. \end{aligned}$$

(3.14)

Since $r\le t_{*}$, it follows that ${\mathbb {D}}(a,r)\subset {K}_{(t_{*})}\subset E$. Therefore,

$$\begin{aligned} \Vert P\Vert _{h\left( {\mathbb {D}}(a,r)\right) }^j&\le {\Vert P^j\circ h-R_j\Vert }_{{\mathbb {D}}(a,r)} +{\Vert R_j\Vert }_{{\mathbb {D}}(a,r)}&\\&\le C_1{\Vert P\Vert }_{h(K)}^j +{\Vert R_j\Vert }_{{\mathbb {D}}(a,r)}&\text {(by }(3.14)\text {)}\\&\le C_1{\Vert P\Vert }_{h(K)}^j +\Big (\sup _{{\mathbb {D}}(a,r)}\Phi _K\Big )^{jkn}{\Vert R_j\Vert }_K&\text {(by }(1.3)\text {)} \\&\le C_1{\Vert P\Vert }_{h(K)}^j +(1+C_2r^{\mu })^{jkn}{\Vert R_j\Vert }_K&\text {(by }(3.3)\text {)}\,. \end{aligned}$$

But

$$\begin{aligned} {\Vert R_j\Vert }_K&\le {\Vert P^j\circ h-R_j\Vert }_{K}+ {\Vert P^j\circ h\Vert }_{K}&\\&\le (C_1+1){\Vert P\Vert }_{h(K)}^j&\text {(by }(3.14)\text {)}\,. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert P\Vert _{h\left( {\mathbb {D}}(a,r)\right) }^j \le (2C_1+1)(1+C_2r^{\mu })^{jkn}{\Vert P\Vert }_{h(K)}^j\,. \end{aligned}$$

Since $j\in {\mathbb {N}}$ was arbitrary, we get

$$\begin{aligned} \Vert P\Vert _{h\left( {\mathbb {D}}(a,r)\right) }\le (1+C_2r^{\mu })^{kn}{\Vert P\Vert }_{h(K)}\,. \end{aligned}$$

Together with (3.13), this implies that

$$\begin{aligned} |P(y)|&\le \big ( 1+C_3\tau ^{\gamma }\big )^{n\max \{q,\varkappa \}} {\Vert P\Vert }_{ h\left( {\mathbb {D}}(a,r)\right) }\\&\le (1+C_2r^{\mu })^{kn}\big ( 1+C_3\tau ^{\gamma }\big )^{n\max \{q,\varkappa \}}{\Vert P\Vert }_{ h(K) }\\&= (1+C_2\tau ^{\gamma })^{kn}\big ( 1+C_3\tau ^{\gamma }\big )^{n\max \{q,\varkappa \}}{\Vert P\Vert }_{ h(K) }\\&\le \big ( 1+\max \{C_2,C_3\}\tau ^{\gamma }\big )^{n(\max \{q,\varkappa \}+k)}{\Vert P\Vert }_{ h(K) }\\&\le (1+C_4\tau ^{\gamma })^{n}{\Vert P\Vert }_{ h(K) }\,. \end{aligned}$$

The above estimates yield (3.5) and hence (3.4). The proof of the theorem is complete, because (3.4) means that h(K) has the HCP property, which is our assertion. $\square$

4 Proof of Theorem 1.8

Proof of Theorem 1.8

For each $\zeta \in U{\setminus } U_{*}=\bigcup _{\iota \in I{\setminus } I_* } U_{\iota }$, we have $\mathrm{{rank}}\,d_{{\zeta }} h\le N'-1$. It follows from [28, p. 254] that $h(U{\setminus } U_{*})$ is a countable union of submanifolds of dimension $\le N'-1$. In particular, the set $h(U{\setminus } U_{*})$ is pluripolar (recall that countable unions of pluripolar sets are pluripolar; see [23, Corollary 4.7.7]) and hence $h(K{\setminus } U_{*})$ is pluripolar as well.

$\boxed {\hbox {(i)}\implies \hbox {(ii)}.}$ Trivial (see Introduction).

$\boxed {\hbox {(ii)}\implies \hbox {(iii)}.}$ Assume that h(K) is L-regular. To obtain a contradiction, suppose that $h(K)\not \subset h(K\cap U_{*})^{\widehat{\,\,}}$, and take $b\in h(K){\setminus } h(K\cap U_{*})^{\widehat{\,\,}}$. Note that $h(K\cap U_{*})\ne \emptyset$ (otherwise, h(K) would be pluripolar, in contradiction with $V_{h(K)}$ being continuous; see (1.1)) and that $V_{h(K\cap U_{*}) }(b)>0$ (see [44, Corollary 4.14]). Since $h(K{\setminus } U_{*})$ is pluripolar, [23, Theorem 5.2.4 and Corollary 5.2.5] imply that

$$\begin{aligned} V_{h(K)}^*=V_{h(K{\setminus } U_{*})\cup h(K\cap U_{*}) }^*=V_{h(K\cap U_{*}) }^*\,, \end{aligned}$$

and hence

$$\begin{aligned} \begin{aligned} 0&=V_{h(K)}(b)=V_{h(K)}^*(b)=V_{h(K\cap U_{*}) }^*(b)\ge V_{h(K\cap U_{*}) }(b)>0\,, \end{aligned} \end{aligned}$$

a contradiction. Therefore, $h(K)\subset h(K\cap U_{*})^{\widehat{\,\,}}$, which is our claim.

$\boxed {\hbox {(iii)} \implies \hbox {(i)}.}$ Suppose that (iii) holds, that is, $h(K)\subset h(E)^{\widehat{\,\,}}$, where $E:= K\cap U_{*}$. We can, clearly, assume that $U_{*}\ne U$, because otherwise Theorem 1.9 immediately completes the proof. In brief outline, the idea of the proof is the following:

• we use the hypothesis that K has the HCP property (4.1) to show that E satisfies Markov’s inequality (4.8);

• then we show, with the aid of (4.1) and (4.8), that E has the HCP property (4.9);

• finally, we apply Theorem 1.9 to the set E and the map $h|_{U_{*}}$ to deduce that h(E) has the HCP property (4.11);

• the proof of (i) concludes by observing, via the hypothesis (iii), that $V_{h(K)}=V_{h(E)}$.

The HCP property of K means that there exist $\theta _1, \mu >0$ such that, for each $z\in K_{(1)}$,

$$\begin{aligned} \Phi _{K}(z)\le 1+\theta _1\left( \mathrm{dist}(z,\,K)\right) ^{\mu }\,. \end{aligned}$$

(4.1)

By [46, Remark 3.7], $\mu \le 1$.

Consider $f: U\rightarrow {\mathbb {C}}$ given by

$$\begin{aligned} f(z):=\left. {\left\{ \begin{array}{ll} 1 &{} \mathrm{if}\,\,z\in U_{*}\\ 0 &{} \mathrm{if}\,\, z\in U{\setminus } U_{*}. \end{array}\right. } \right. \end{aligned}$$

Also, take a compact and polynomially convex set $\Delta \subset {\mathbb {C}}^N$ such that ${\hat{K}}\subset \mathrm{Int}\Delta$ and $\Delta \subset U$; cf. [22, Proof of Lemma 2.7.4]. By [44, Theorem 8.5(1)], there exist $\theta _2>0$, $\rho \in (0,\,1)$, and a sequence of polynomials $W_{\nu }\in {\mathbb {C}}[z_1,\ldots ,z_N]$ ($\nu \in {\mathbb {N}}$) with $\deg W_{\nu }\le \nu$ such that, for each $\nu \in {\mathbb {N}}$,

$$\begin{aligned} {\Vert f-W_{\nu }\Vert }_{\Delta }\le \theta _2{\rho ^{\nu }}\,. \end{aligned}$$

(4.2)

We can, clearly, assume that $\theta _2\ge 1$.

Take $\nu _0\in {\mathbb {N}}$ such that $2\theta _2\rho ^{\nu _0}<1$. We now show that

$$\begin{aligned} {\hat{E}}\subset U_{*}\,. \end{aligned}$$

(4.3)

To this end, take $c\in {\hat{E}}$, and suppose, towards a contradiction, that $c\notin U_{*}$. Since $c\in {\hat{K}}{\setminus } U_{*}\subset U{\setminus } U_{*}$, we have from (4.2)

$$\begin{aligned} {\Vert 1-W_{\nu _0}\Vert }_E<\frac{1}{ 2} \quad \text {and}\quad |W_{\nu _0}(c)|<\frac{1}{ 2}\,. \end{aligned}$$

Hence,

$$\begin{aligned} {\Vert 1-W_{\nu _0}\Vert }_E< |1-W_{\nu _0}(c)|\,, \end{aligned}$$

which implies that $c\notin {\hat{E}}$, a contradiction.

Similarly, we show that E is nonpluripolar. Indeed, striving for a contradiction, assume that E is pluripolar. Note that $\emptyset \ne h(K)\subset h(E)^{\widehat{\,\,}}$, which implies that $E\ne \emptyset$. Take ${\tilde{c}}\in E$. Then, from (4.2),

$$\begin{aligned} {\Vert W_{\nu _0}\Vert }_{K{\setminus } E}<\frac{1}{ 2} \quad \text {and}\quad |W_{\nu _0}({\tilde{c}})|>\frac{1}{ 2}\,. \end{aligned}$$

Hence, ${\tilde{c}}\notin (K{\setminus } E)^{\widehat{\,\,}}$, and by [23, Theorem 5.2.4 and Corollary 5.2.5],

$$\begin{aligned} \begin{aligned} 0&=V_K({\tilde{c}})=V_K^*({\tilde{c}}) =V_{E\cup (K{\setminus } E)}^*({\tilde{c}})= V_{K{\setminus } E}^*({\tilde{c}}) \ge V_{K{\setminus } E}({\tilde{c}})>0 \,, \end{aligned} \end{aligned}$$

which is a contradiction.

Nonpluripolarity of E implies that $\Phi _E$ is locally bounded in ${\mathbb {C}}^N$; see (1.1) and (1.2). In particular, there exists $m\in {\mathbb {N}}$ such that $\theta _2\rho ^m<1$ and

$$\begin{aligned} \sup _K \Phi _E\le \frac{1}{\theta _2\rho ^m}\,. \end{aligned}$$

(4.4)

For each nonconstant polynomial $T\in {\mathbb {C}}[z_1,\ldots ,z_N]$ set $P^T:=W_{m\deg T}T$. Note that, for each $z\in \Delta \cap U_{*}$,

$$\begin{aligned} |P^T(z)|\ge \big (1-\theta _2\rho ^{m\deg T}\big ) |T(z)|\ge (1-\theta _2\rho ^m)|T(z)|\,. \end{aligned}$$

(4.5)

Indeed, by (4.2),

$$\begin{aligned} \begin{aligned} |P^T(z)|&= |W_{m\deg T}(z)| \cdot |T(z)|\ge \big ( 1-|W_{m\deg T}(z)-1| \big )|T(z)|\\&\ge \big (1-\theta _2\rho ^{m\deg T}\big )|T(z)|\,. \end{aligned} \end{aligned}$$

We now show that, for each nonconstant polynomial $T\in {\mathbb {C}}[z_1,\ldots ,z_N]$,

$$\begin{aligned} {\Vert P^T\Vert }_K\le \big (1+\theta _2\rho ^{m\deg T}\big ) {\Vert T\Vert }_E\le (1+\theta _2\rho ^m){\Vert T\Vert }_E\,. \end{aligned}$$

(4.6)

To this end, fix $z\in K$. By (4.2),

$$\begin{aligned} \begin{aligned} |P^T(z)|&= |W_{m\deg T}(z)| \cdot |T(z)|\\&\le \big ( |f(z)|+|f(z)- W_{m\deg T}(z)|\big )|T(z)|\\&\le \big (|f(z)|+ \theta _2\rho ^{m\deg T}\big )|T(z)|\,. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} |P^T(z)|\le \left. {\left\{ \begin{array}{ll} \big (1+\theta _2\rho ^{m\deg T}\big ) {\Vert T\Vert }_E&{} \mathrm{if}\,\,z\in E\\ \theta _2\rho ^{m\deg T}|T(z)| &{} \mathrm{if}\,\, z\in K{\setminus } E. \end{array}\right. } \right. \end{aligned}$$

(4.7)

Moreover,

$$\begin{aligned} \theta _2\rho ^{m\deg T}|T(z)|&\le (\theta _2\rho ^{m})^{\deg T}|T(z)|&\\&\le \left( \theta _2\rho ^{m}\sup _K\Phi _E \right) ^{\deg T}{\Vert T\Vert }_E&\text {(by }(1.3)\text {)} \\&\le {\Vert T\Vert }_E&\text {(by }(4.4)\text {).} \end{aligned}$$

Combining the above estimates with (4.7) yields (4.6).

Take $\theta _3\in (0,\,1]$ such that $E_{(\theta _3)}\subset \Delta \cap U_{*}$, and set

$$\begin{aligned} \begin{aligned}&\theta _4:=\frac{ \exp \big ( \theta _1\theta _3^{\mu }(m+1)\big ) }{ \theta _3 } \cdot \frac{ 1+\theta _2\rho ^m }{ 1-\theta _2\rho ^m }\,,\\&\theta _5:=\sup _{ t\in (0,\,N\theta _3\theta _4] } \frac{ \exp (t)-1 }{ t }=\frac{ \exp (N\theta _3\theta _4)-1 }{ N\theta _3\theta _4 }\,,\\&\theta _6:=\max \left\{ \theta _1 ,\, \frac{ 2\theta _2\rho ^m }{ (1-\theta _2\rho ^m)\theta _3^{\mu } } \right\} \,,\\&\theta _7:=\sup _{ t\in (0,\,\theta _3^{\mu }\theta _6] }\frac{ (1+t)^{m+2}-1 }{ t }=\frac{ (1+\theta _3^{\mu }\theta _6)^{m+2}-1 }{ \theta _3^{\mu }\theta _6 }\,,\\&\theta _8:=\max \left\{ N\theta _3^{1-\mu }\theta _4\theta _5, \,\theta _6\theta _7 \right\} \,. \end{aligned} \end{aligned}$$

We now show that, for each polynomial $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$ and each $\alpha =(\alpha _1,\ldots ,\alpha _N)\in {\mathbb {N}}_0^N$,

$$\begin{aligned} {\Vert D^{\alpha }Q\Vert }_E\le \big ( \theta _4{(\mathrm{deg}\,Q)}^{1/\mu }\big )^{|\alpha |}{\Vert Q\Vert }_{E}\,. \end{aligned}$$

(4.8)

(If Q is constant and $\alpha =0\in {\mathbb {N}}_0^N$ , then $\big ( \theta _4{(\mathrm{deg}\,Q)}^{1/\mu }\big )^{|\alpha |}:= 1$.) Obviously, it suffices to check (4.8) for $\alpha \in {\mathbb {N}}_0^N$ such that $|\alpha |=1$. So, fix such an $\alpha$, and also fix $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$. Put $d:=\deg Q$. We may assume that $d\ge 1$, because (4.8) is trivial for Q being constant.

Then, for each $z\in E$,

$$\begin{aligned} |D^{\alpha }Q(z)|&\le \frac{d^{1/\mu }}{\theta _3}{\Vert Q\Vert }_{ {\mathbb {D}}(z,\theta _3d^{-1/\mu })}&\text {(by Cauchy's inequalities)}\\&\le \frac{d^{1/\mu }}{\theta _3(1-\theta _2\rho ^m)}{\Vert P^Q \Vert }_{{\mathbb {D}}(z,\theta _3d^{-1/\mu })}&\text {(by }(4.5)\text {)}\\&\le \frac{d^{1/\mu }}{\theta _3(1-\theta _2\rho ^m)}\left( 1+\frac{\theta _1\theta _3^{\mu }}{d} \right) ^{d(m+1)}{\Vert P^Q \Vert }_K&\text {(by }(1.3)\text { and }(4.1)\text {)}\\&\le \frac{d^{1/\mu }\exp \big (\theta _1\theta _3^{\mu }(m+1)\big )}{\theta _3(1-\theta _2\rho ^m)} {\Vert P^Q \Vert }_K\\&\le \frac{d^{1/\mu } \exp \big ( \theta _1\theta _3^{\mu }(m+1)\big ) }{ \theta _3 } \cdot \frac{ 1+\theta _2\rho ^m }{ 1-\theta _2\rho ^m }{\Vert Q\Vert }_E&\text {(by }(4.6)\text {)}\\&=\theta _4 d^{1/\mu }{\Vert Q\Vert }_E\,, \end{aligned}$$

which implies (4.8).

We are now in a position to show that, for each $a\in E_{(\theta _3)}$,

$$\begin{aligned} \Phi _{E}(a)\le 1+\theta _8\left( \mathrm{dist}(a,\,E)\right) ^{\mu }\,. \end{aligned}$$

(4.9)

To prove this, fix $a\in E_{(\theta _3)}$ and $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$ such that $d:=\deg Q>0.$ We need to check that

$$\begin{aligned} |Q(a)|\le \big (1+\theta _8\delta ^{\mu }\big )^d{\Vert Q\Vert }_E\,, \end{aligned}$$

(4.10)

where $\delta := \mathrm{dist}(a,\,E)$.

Case 1: $\delta \le \theta _3d^{-1/\mu }$. Take $b\in E$ such that $|a-b|=\delta$. Then

$$\begin{aligned} |Q(a)|&= \left| \sum _{\alpha \in {\mathbb {N}}_0^{N}} \frac{D^{\alpha }Q(b)}{\alpha !}(a-b)^{\alpha } \right|&\\&\le \sum _{\alpha \in {\mathbb {N}}_0^{N}} \frac{{\Vert D^{\alpha }Q\Vert }_E}{\alpha !}\delta ^{|\alpha |}&\\&\le \sum _{\alpha \in {\mathbb {N}}_0^{N}} \frac{\big ( \theta _4d^{1/\mu }\delta \big )^{|\alpha |} }{\alpha !} {\Vert Q\Vert }_E&\text {(by }(4.8)\text {)}\\&=\exp \big (N\theta _4d^{1/\mu }\delta \big ){\Vert Q\Vert }_E&\\&=\exp \big (d N\theta _4(d^{1/\mu }\delta )^{1-\mu }\delta ^{\mu } \big ){\Vert Q\Vert }_E&\\&\le \exp \big (d N\theta _3^{1-\mu }\theta _4\delta ^{\mu } \big ){\Vert Q\Vert }_E&\text {(since }\mu \le 1\text { and }\delta \le \theta _3d^{-1/\mu })\\&\le \big ( 1+N\theta _3^{1-\mu }\theta _4\theta _5\delta ^{\mu } \big )^d{\Vert Q\Vert }_E&\text {(since }\delta \le \theta _3\text {)}\\&\le \big (1+\theta _8\delta ^{\mu } \big )^{d} {\Vert Q\Vert }_E\,,&\end{aligned}$$

which yields (4.10) in case 1.

Case 2: $\delta >\theta _3d^{-1/\mu }$. Then

$$\begin{aligned} |Q(a)|&\le \frac{1}{1-\theta _2\rho ^m}|P^Q(a)|&\text {(by } (4.5)\text {)}\\&\le \frac{1}{1-\theta _2\rho ^m}\big ( \Phi _{K}(a)\big )^{d(m+1)}{\Vert P^Q\Vert }_K&\text {(by }(1.3)\text {)} \\&\le \frac{1}{1-\theta _2\rho ^m} \big ( 1+\theta _1\delta ^{\mu } \big )^{d(m+1)}{\Vert P^Q\Vert }_K&\text {(by }(4.1)\text {)}\\&\le \frac{1+\theta _2\rho ^m}{1-\theta _2\rho ^m} \big ( 1+\theta _1\delta ^{\mu } \big )^{d(m+1)}{\Vert Q\Vert }_E&\text {(by } (4.6)\text {)}\\&= \left( 1+\frac{2\theta _2\rho ^m}{1-\theta _2\rho ^m} \right) \big ( 1+\theta _1\delta ^{\mu } \big )^{d(m+1)}{\Vert Q\Vert }_E&\\&\le \left( 1+\frac{2\theta _2\rho ^m }{1-\theta _2\rho ^m}\cdot \frac{d\delta ^{\mu }}{\theta _3^{\mu }} \right) \big ( 1+\theta _1\delta ^{\mu } \big )^{d(m+1)}{\Vert Q\Vert }_E&\text {(since }\delta >\theta _3d^{-1/\mu }\text {)}\\&\le \big (1+d\theta _6\delta ^{\mu } \big ) \big ( 1+\theta _1\delta ^{\mu } \big )^{d(m+1)}{\Vert Q\Vert }_E&\\&\le \big (1+\theta _6\delta ^{\mu } \big )^d \big ( 1+\theta _1\delta ^{\mu } \big )^{d(m+1)}{\Vert Q\Vert }_E&\\&\le \big (1+\theta _6\delta ^{\mu } \big )^{d(m+2)} {\Vert Q\Vert }_E&\\&\le \big (1+\theta _6\theta _7\delta ^{\mu } \big )^{d} {\Vert Q\Vert }_E&\text {(since }\delta \le \theta _3\text {)}\\&\le \big (1+\theta _8\delta ^{\mu } \big )^{d} {\Vert Q\Vert }_E\,,&\\ \end{aligned}$$

which establishes the estimate (4.10).

To complete the proof, note that h(E) has the HCP property. Indeed, since ${\hat{E}}\subset U_{*}$ (see (4.3)) and E has the HCP property (see (4.9)), we can apply Theorem 1.9 to the set E and the map $h|_{U_{*}}$. Therefore, there exist $\theta , \gamma >0$ such that, for all $y,y'\in {\mathbb {C}}^{N'}$,

$$\begin{aligned} |V_{h(E)}(y)-V_{h(E)}(y')|\le \theta |y-y'|^{\gamma } \,. \end{aligned}$$

(4.11)

Recall also that we are now working under the assumption that (iii) holds, and so

$$\begin{aligned} h(E)\subset h(K)\subset h(E){^{\widehat{\,\,}}}\,. \end{aligned}$$

Consequently,

$$\begin{aligned} V_{h(E)}\ge V_{h(K)} \ge V_{h(E)^{\widehat{\,\,}}}\,. \end{aligned}$$

On the other hand, [23, Theorem 5.1.7] tell us that $V_{h(E)^{\widehat{\,\,}}}=V_{h(E)}$. Therefore, $V_{h(K)}=V_{h(E)}$. Together with (4.11), this completes the proof of (i) under the assumption that (iii) holds. $\square$

5 Proof of Theorem 1.12

Proof of Theorem 1.12

Take an open, bounded set $\Omega \subset {\mathbb {C}}^N$ such that ${\hat{K}}\subset \Omega$ and ${\overline{\Omega }}\subset U$. Since h(K) is a nonpluripolar subset of ${\mathbb {C}}^{N'}$ and $h(\Omega )$ is bounded,

$$\begin{aligned} C_1:=\sup _{h(\Omega )} \Phi _{h(K)} <+\infty \,; \end{aligned}$$

see (1.1) and (1.2).

As in the proof of Theorem 1.9, we show that there exist a set $E\subset {\mathbb {C}}^N$ and constants $C_2>0$, $\rho \in (0,\,1)$ such that

• $K\subset \mathrm{Int}E$ and $E\subset \Omega$,

• for each holomorphic and bounded function $f: \Omega \rightarrow {\mathbb {C}}$ and each $\nu \in {\mathbb {N}}$, there exists a polynomial $R_{\nu }\in {\mathbb {C}}[z_1,\ldots ,z_N]$ with $\deg R_{\nu }\le \nu$ and such that

  $$\begin{aligned} {\Vert f-R_{\nu }\Vert }_E\le C_2{\Vert f\Vert }_{\Omega }\,\rho ^{\nu }\,. \end{aligned}$$

  (5.1)

Furthermore, take $k\in {\mathbb {N}}$ and $t_1>0$ such that

$$\begin{aligned} C_1\rho ^k\le 1 \end{aligned}$$

(5.2)

and

$$\begin{aligned} K_{(t_1)}\subset E\,. \end{aligned}$$

(5.3)

Let $\varepsilon , C>0$ be of the definition of Markov’s inequality for the set K. That is, for each polynomial $R\in {\mathbb {C}}[z_1,\ldots ,z_N]$ and each $\beta \in {\mathbb {N}}_0^N$,

$$\begin{aligned} {\Vert D^{\beta }R\Vert }_K\le \big ( C{(\mathrm{deg}\,R)}^{\varepsilon }\big )^{|\beta |}{\Vert R\Vert }_{K}\,. \end{aligned}$$

(5.4)

Moreover, let $\varkappa , \theta , t_{*}>0$ and $q\in {\mathbb {N}}$ be of Theorem 1.13. Set

$$\begin{aligned}&\kappa := \varkappa (2+\varepsilon )\,,\\&C_3:=\min \left\{ t_{*}, t_1, \frac{1}{Ck^{\varepsilon }} \right\} \,,\\&C_4:= \sup _{ j\in {\mathbb {N}} } \left( \frac{ 1+\displaystyle \frac{1}{\sqrt{2}j} }{ 1-\displaystyle \frac{1}{\sqrt{2}j}}\right) ^{jq}\,,\\&C_5:=\frac{\theta C_3^{\varkappa }}{2^{\varkappa +1}}\,,\\&C_6:=C_2C_42^{N'}+(C_2+1) C_4e^N2^{N'}\,. \end{aligned}$$

Fix a polynomial $P\in {\mathbb {C}}[y_1,\ldots ,y_{N'}]$ with $\deg P\le n$ ($n\in {\mathbb {N}}$), and also fix $y\in h(K)_{(C_5n^{-\kappa })}$. We now show that

$$\begin{aligned} |P(y)|\le C_6 {\Vert P\Vert }_{h(K)}\,. \end{aligned}$$

(5.5)

Set $l=l(n):=kn$. By (5.1), there exists a polynomial $R_{l}\in {\mathbb {C}}[z_1,\ldots ,z_N]$ with $\deg R_l\le l$ and such that

$$\begin{aligned} {\Vert P\circ h-R_l\Vert }_E\le C_2{\Vert P\circ h\Vert }_{\Omega }\,\rho ^l\,. \end{aligned}$$

(5.6)

Note that

$$\begin{aligned} {\Vert P\circ h-R_l\Vert }_E\le C_2{\Vert P\Vert }_{h(K)}\,. \end{aligned}$$

(5.7)

Indeed,

$$\begin{aligned} {\Vert P\circ h-R_l\Vert }_{E}&\le C_2{\Vert P\Vert }_{h(\Omega )}\,\rho ^{l}&\text {(by }(5.6)\text {)}\\&\le C_2\rho ^{l}\Big (\sup _{h(\Omega )} \Phi _{h(K)}\Big )^{n} {\Vert P\Vert }_{h(K)}&\text {(by }(1.3)\text {)}\\&=C_2(C_1\rho ^k)^n {\Vert P\Vert }_{h(K)}&\\&\le C_2{\Vert P\Vert }_{h(K)}&\text {(by }(5.2)\text {)}\,, \end{aligned}$$

which proves (5.7).

Put $v_n:=C_3/n^{\varepsilon }$ and $\tau _n:=v_n/2n^2$. Take $a\in K$ such that $\mathrm{dist}\big (y,\,h(K)\big )=|y-h(a)|$. Note that $v_n\le t_1$, and hence

$$\begin{aligned} {\mathbb {D}}(a,v_n)\subset E\,; \end{aligned}$$

(5.8)

see (5.3).

Let $Q_a:{\mathbb {C}}\rightarrow {\mathbb {C}}^{N'}$ be the polynomial map of Theorem 1.13. In particular, $Q_a(0)=h(a)$. Then

$$\begin{aligned} |P(y)|&= \left| \sum _{\alpha \in {\mathbb {N}}_0^{N'}} \frac{D^{\alpha }P\big (h(a)\big )}{\alpha !}\big (y-h(a)\big )^{\alpha } \right|&\\&\le \sum _{\alpha \in {\mathbb {N}}_0^{N'}} \frac{\left| D^{\alpha }P\big (Q_a(0)\big )\right| }{\alpha !}\left( \frac{C_5}{n^{\kappa }} \right) ^{|\alpha |}&\text {(since }y\in h(K)_{(C_5n^{-\kappa })}\text {)}\\&\le \sum _{\alpha \in {\mathbb {N}}_0^{N'}} \frac{{\Vert D^{\alpha }P\circ Q_a\Vert }_{[\tau _n,\,v_n]}}{\alpha !}\left( \Phi _{[\tau _n,\,v_n]}(0)\right) ^{nq}\left( \frac{C_5}{n^{\kappa }} \right) ^{|\alpha |}&\text {(by }(1.3)\text {)}\\&= \sum _{\alpha \in {\mathbb {N}}_0^{N'}} \frac{{\Vert D^{\alpha }P\circ Q_a\Vert }_{[\tau _n,\,v_n]}}{\alpha !} \left( \frac{ 1+\displaystyle \frac{1}{\sqrt{2}n} }{ 1-\displaystyle \frac{1}{\sqrt{2}n}}\right) ^{nq} \left( \frac{C_5}{n^{\kappa }} \right) ^{|\alpha |}\,. \end{aligned}$$

Here we have used the following formula:

$$\begin{aligned} \Phi _{[\tau ,\,v]}(u)=\frac{ 1+\sqrt{\displaystyle \frac{\tau -u}{v-u}}}{ \left| 1-\sqrt{\displaystyle \frac{\tau -u}{v-u}}\right| }\,, \end{aligned}$$

which is valid for all $\tau ,v\in {\mathbb {R}}$ with $\tau <v$ and all $u\in (-\infty , \,\tau ]\cup (v,\,+\infty )$; see for example [31, Lemma 2.1]. Consequently,

$$\begin{aligned} |P(y)|\le \sum _{\alpha \in {\mathbb {N}}_0^{N'}} \frac{C_4}{\alpha !} \left| D^{\alpha }P\circ Q_a\big (s_n(\alpha ) \big ) \right| \left( \frac{C_5}{n^{\kappa }} \right) ^{|\alpha |} \end{aligned}$$

(5.9)

for some $s_n(\alpha )\in [\tau _n,\,v_n]$.

Theorem 1.13 and the estimates $0<s_n(\alpha )\le v_n\le t_{*}$ imply that

$$\begin{aligned} {\mathbb {D}}\Big ( Q_a\big (s_n(\alpha )\big ), \theta s_n(\alpha )^{\varkappa }\Big )\subset h\big ( {\mathbb {D}}(a,v_n)\big )\,. \end{aligned}$$

(5.10)

Therefore,

$$\begin{aligned} |P(y)|&\le \sum _{\alpha \in {\mathbb {N}}_0^{N'}}C_4\frac{ {\Vert P\Vert }_{h({\mathbb {D}}(a,v_n))} }{\big (\theta s_n(\alpha )^{\varkappa }\big )^{|\alpha |}} \left( \frac{C_5}{n^{\kappa }} \right) ^{|\alpha |}&\text {(by } (5.9), (5.10),\text { and Cauchy's ineq.)}\\&\le \sum _{\alpha \in {\mathbb {N}}_0^{N'}}C_4\left( \frac{C_5}{\theta \tau _n^{\varkappa }n^{\kappa }} \right) ^{|\alpha |}{\Vert P\Vert }_{h({\mathbb {D}}(a,v_n))}&\\&=\sum _{\alpha \in {\mathbb {N}}_0^{N'}}\frac{C_4}{2^{|\alpha |}}{\Vert P\Vert }_{h({\mathbb {D}}(a,v_n))}&\\&= C_42^{N'}{\Vert P\circ h \Vert }_{{\mathbb {D}}(a,v_n) }&\\&\le C_42^{N'} \left( { \Vert P\circ h-R_l\Vert }_E+ {\Vert R_l\Vert }_{{\mathbb {D}}(a,v_n)}\right)&\text {(by }(5.8)\text {).} \end{aligned}$$

Hence, by (5.7),

$$\begin{aligned} |P(y)|\le C_42^{N'}\left( C_2{\Vert P\Vert }_{h(K)}+{\Vert R_l\Vert }_{{\mathbb {D}}(a,v_n)}\right) \,. \end{aligned}$$

(5.11)

Take $z_0\in {\mathbb {D}}(a,v_n)$ such that $|R_l(z_0)|={\Vert R_l\Vert }_{{\mathbb {D}}(a,v_n)}$. Then

$$\begin{aligned} {\Vert R_l\Vert }_{ {\mathbb {D}}(a,v_n) }&= \left| \sum _{\beta \in {\mathbb {N}}_0^{N} } \frac{D^{\beta }R_l(a)}{\beta !} \big (z_0-a\big )^{\beta } \right|&\\&\le \sum _{ \beta \in {\mathbb {N}}_0^{N} }\frac{ \left| D^{\beta }R_l(a)\right| }{ \beta ! }v_n^{|\beta |}&\\&\le \sum _{ \beta \in {\mathbb {N}}_0^{N} } \frac{(Cl^{\varepsilon })^{|\beta |} {\Vert R_l\Vert }_K}{\beta !}v_n^{|\beta |}&\text {(by }(5.4)\text {)}\\&= \sum _{ \beta \in {\mathbb {N}}_0^{N} } \frac{(CC_3k^{\varepsilon })^{|\beta |} }{\beta !}{\Vert R_l\Vert }_K\\&\le \sum _{ \beta \in {\mathbb {N}}_0^{N} }\frac{{\Vert R_l\Vert }_K}{\beta !}&\\&= e^N{\Vert R_l\Vert }_K&\\&\le e^N \big ( {\Vert P\circ h-R_l\Vert }_K+{\Vert P\circ h\Vert }_K \big )&\\&\le (C_2+1)e^N {\Vert P\Vert }_{h(K)}&\text {(by }(5.7)\text {)} \,. \end{aligned}$$

Combining this with (5.11), we get

$$\begin{aligned} |P(y)|&\le C_42^{N'}\left( C_2{\Vert P\Vert }_{h(K)}+(C_2+1)e^N {\Vert P\Vert }_{h(K)}\right) = C_6{\Vert P\Vert }_{h(K)}\,, \end{aligned}$$

which establishes the estimate (5.5).

By the remark following Definition 1.10 and by (5.5), h(K) satisfies Markov’s inequality, which is the desired conclusion. $\square$

6 A refinement of Theorems 1.9 and 1.12

We conclude this paper with one more result, which is a slight strengthening of Theorems 1.9 and 1.12.

Theorem 6.1

Let $h:U\rightarrow {\mathbb {C}}^{N'}$, where $U\subset {\mathbb {C}}^N$ is an open set, be a nondegenerate holomorphic map $(N, N'\in {\mathbb {N}})$. Assume that ${\mathcal {Z}}\subset {\mathbb {C}}^N$ is a compact set with $\hat{{\mathcal {Z}}}\subset U$. Then there exist $\eta _1,\eta _2>0$ such that:

• (1) For each compact set $\emptyset \ne K\subset {\mathcal {Z}}$ having the HCP property with the exponent $\mu >0$ (that is, (1.4) holds with some $\varpi >0$), the set h(K) has the HCP property with the exponent $\eta _1\mu$.

• (2) For each compact set $K\subset {\mathcal {Z}}$ satisfying Markov’s inequality with the exponent $\varepsilon >0$ (that is, (1.5) holds with some $C>0$) and such that h(K) is a nonpluripolar subset of ${\mathbb {C}}^{N'}$, the set h(K) satisfies Markov’s inequality with the exponent $\eta _2\varepsilon$.

Proof

Let $\varkappa >0$ be of Theorem 1.13 applied to ${\mathcal {Z}}$ (instead of K) and take ${\tilde{\varkappa }}\in {\mathbb {N}}$ such that ${\tilde{\varkappa }}\ge \varkappa$. Set $\eta _1:=1/(3{\tilde{\varkappa }})$ and $\eta _2:=3\varkappa$. Fix a compact set $\emptyset \ne K\subset {\mathcal {Z}}$. Note that ${\hat{K}}\subset \hat{{\mathcal {Z}}}\subset U$. Put

$$\begin{aligned} r:=\sup \big \{ |z_1|:\,(z_1,\ldots ,z_N)\in K\big \}\,, \end{aligned}$$

and take $a\in K$ such that $|a_1|=r$. If $r>0$, then for each $n\in {\mathbb {N}}$, let $Q_n\in {\mathbb {C}}[z_1,\ldots ,z_N]$ be defined by $Q_n(z):=\left( z_1/r\right) ^n$. Clearly, ${\Vert Q_n\Vert }_K=1$. Note that K being a Markov set implies $r>0$. This is immediately seen by considering the polynomial $Q(z):=z_1$.

Assume first that K has the HCP property with the exponent $\mu >0$, that is, there exists $\varpi >0$ such that, for each $z\in K_{(1)}$,

$$\begin{aligned} V_{K}(z)\le \varpi \left( \mathrm{dist}(z,\,K)\right) ^{\mu }\,. \end{aligned}$$

(6.1)

Then $r>0$ and, for each $n\in {\mathbb {N}}$, we have from (1.2), (1.3) and (6.1):

$$\begin{aligned} \frac{n}{r}&< n^{1/\mu }\left( 1+\frac{1}{n^{1/\mu }r} \right) ^n= n^{1/\mu }{\Vert Q_n\Vert }_{{\mathbb {D}}(a,n^{-1/\mu })}\\&\le n^{1/\mu }\left( \sup _{{\mathbb {D}}(a,n^{-1/\mu })} \Phi _K\right) ^n{\Vert Q_n\Vert }_K\le n^{1/\mu } e^{\varpi }\,. \end{aligned}$$

Thus, $n/r\le n^{1/\mu } e^{\varpi }$, which implies $\mu \le 1$; see also [46, Remark 3.7]. Analysis of the proof of Theorem 1.9 shows that the set h(K) has the HCP property with the exponent $\mu /\big ({\tilde{\varkappa }}(2\mu +1)\big )$. However, $\mu /\big ({\tilde{\varkappa }}(2\mu +1)\big )\ge \mu /(3{\tilde{\varkappa }})= \eta _1\mu$. Therefore, h(K) has the HCP property with the exponent $\eta _1\mu$, and (1) is proved.

In order to prove (2), assume that h(K) is a nonpluripolar subset of ${\mathbb {C}}^{N'}$ and K satisfies Markov’s inequality with the exponent $\varepsilon >0$, that is, there exists $C>0$ such that, for each polynomial $Q\in {\mathbb {C}}[z_1,\ldots ,z_N]$ and each $\alpha \in {\mathbb {N}}_0^N$,

$$\begin{aligned} {\Vert D^{\alpha }Q\Vert }_K\le \big ( C{(\mathrm{deg}\,Q)}^{\varepsilon }\big )^{|\alpha |}{\Vert Q\Vert }_{K}\,. \end{aligned}$$

(6.2)

Then $r>0$ and, for each $n\in {\mathbb {N}}$, we have from (6.2):

$$\begin{aligned} \frac{n}{r} = \left| \frac{\partial Q_n}{\partial z_1}(a) \right| \le Cn^{\varepsilon }{\Vert Q_n\Vert }_K= Cn^{\varepsilon }\,. \end{aligned}$$

It follows that $n/r\le Cn^{\varepsilon }$, which implies $\varepsilon \ge 1$. A careful look at the proof of Theorem 1.12 reveals that the set h(K) satisfies Markov’s inequality with the exponent $\varkappa (2+\varepsilon )$. But $\varkappa (2+\varepsilon )\le 3\varkappa \varepsilon =\eta _2\varepsilon$. Consequently, h(K) satisfies Markov’s inequality with the exponent $\eta _2\varepsilon$, which proves (2). $\square$