Separation Theorems for Group Invariant Polynomials

Abstract

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of $\textit{GL}(n,{\mathbb {C}})$, K is a set in ${\mathbb {C}}^{n}$ that is invariant under the action of G and z is a point in ${\mathbb {C}}^{n}\setminus K$ that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in ${\mathbb {C}}^{n}$ and $z\notin K$ then there exists a symmetric polynomial that separates z and K.

1 Introduction

The complex Hahn–Banach Separation Theorem establishes that if $K\subset {\mathbb {C}}^{n}$ is a closed convex set and z is a point in ${\mathbb {C}}^{n}\setminus K$, then there exists a complex linear form f with $\sup _{w\in K}\{\textit{Re}(f(w))\}<\textit{Re}(f(z))$, see [3, Theorem 2.12] and [5, Theorem 3.4]. Thus, if we also assume that K is balanced we can replace the real part of f by the modulus of f in the inequality, obtaining

$$\begin{aligned} \sup _{w\in K}\vert f(w)\vert <\vert f(z)\vert . \end{aligned}$$

(1.1)

If K is neither convex nor balanced, the above inequality does not hold in general. It seems natural to wonder if assuming a different geometric condition on the set K, it is possible to find a “good,” perhaps non-linear, function f that still satisfies the inequality. For instance, if we assume that the set K is symmetric (i.e., if $w\in K$ then $-w\in K$) and $z\notin K$, is it possible to find a “good” function f satisfying equation (1.1) and such that $f(w)=f(-w)$ for all $w\in {\mathbb {C}}^{n}$?

In this work, we answer this question in the affirmative under the assumption that the set K is invariant under the action of a finite group of linear transformations. The function f will then turn out to be a polynomial.

Definition 1.1

We say that a polynomial P separates a set K and a point z if

$$\begin{aligned} \sup _{w\in K}\vert P(w)\vert < \vert P(z)\vert . \end{aligned}$$

As usual, we denote by ${\mathcal {P}}({\mathbb {C}}^{n})$ the set of all complex-valued polynomials defined on ${\mathbb {C}}^{n}$ and by $\textit{GL}(n,{\mathbb {C}})$ the general linear group of degree n, consisting of the set of $n\times n$ complex invertible matrices. We consider the natural group action of $GL(n,{\mathbb {C}})$ on ${\mathbb {C}}^{n}$. For a subgroup $G\le GL(n,{\mathbb {C}})$ and a set $K\subset {\mathbb {C}}^{n}$ we denote by

$$\begin{aligned} \langle G,K\rangle =\{g w:g\in G, w\in K\} \end{aligned}$$

the action of the group G on the set K and we say that K is invariant under the action of G if $\langle G,K\rangle =K$. Given a finite subgroup $G\le GL(n,{\mathbb {C}})$, and a polynomial $P\in {\mathcal {P}}({\mathbb {C}}^{n})$, we say that P is an invariant polynomial under the group G or G-invariant if $P(z)=P(g z)$ for all $g\in G$ and all $z\in {\mathbb {C}}^{n}$. For more details about the theory of invariant polynomials under the action of finite groups, we recommend [2, Chap. 7].

The main result of this paper will be presented in Sect. 2. We show that if K is a set in ${\mathbb {C}}^{n}$ which is invariant under the action of G and z is a point in ${\mathbb {C}}^{n}\setminus K$ that can be separated from K by a polynomial, then z can be separated from K by a G-invariant polynomial. Furthermore, if K and z can be separated by a homogeneous polynomial then K and z can be separated by a G-invariant homogenous polynomial. In Sect. 3, we study separation theorems for polynomially convex sets. Section 4 is devoted to some examples of sets K that are polynomially convex and invariant under the action of a group G and also to some specific examples of G-invariant polynomials that separate points of ${\mathbb {C}}^{n}\setminus K$ and K.

It is worth mentioning that Theorem 2.3 and Corollary 2.4 presented in Sect. 2 and the examples provided in Sect. 4 also hold in the real case with similar proofs.

2 G-Invariant Separating Polynomials and Finite Groups

First we show that given a finite subgroup G of $\textit{GL}(n,{\mathbb {C}})$ we can find arbitrarily small convex balanced compact sets with non-empty interior that are invariant under the action of G. We denote by $B_{\ell _{2}^{n}}(R)$ the closed euclidean ball of center zero and radius $R>0$ in ${\mathbb {C}}^{n}$. When $R=1$ we write $B_{\ell _{2}^{n}}$ instead of $B_{\ell _{2}^{n}}(1)$.

Lemma 2.1

Given a finite subgroup G of $\textit{GL}(n,{\mathbb {C}})$ and a positive number R, there exists a convex balanced compact set $K\subset {\mathbb {C}}^{n}$ with non-empty interior that is invariant under the action of G and $K\subset B_{\ell _{2}^{n}}(R)$.

Proof

Consider $K'=\langle G,B_{\ell _{2}^{n}}\rangle$. Then, $\langle G,K'\rangle =K'$ and since G is finite, $K'$ is compact and hence $K'\subset B_{\ell _{2}^{n}}(M)$ for some positive constant M. Then, the absolutely convex hull of $K'$, $\textit{absconv}(K')$, is convex, balanced, and satisfies that $\textit{absconv}(K')\subset B_{\ell _{2}^{n}}(M)$. Now, for every $w\in \textit{absconv}(K')$ there exists m vectors $w_{1},\ldots ,w_{m}\in K'$ and m complex numbers $\lambda _{1},\ldots ,\lambda _{m}\in {\mathbb {C}}$ with $\sum _{j=1}^{m}\vert \lambda _{j}\vert \le 1$ such that $w=\sum _{j=1}^{m}\lambda _{j}w_{j}$. Therefore, for every $g\in G$,

$$\begin{aligned} g w=g \left( \sum _{j=1}^{m}\lambda _{j}w_{j}\right) =\sum _{j=1}^{m}\lambda _{j}(g w_{j})\in \textit{absconv}(K'). \end{aligned}$$

Thus $\textit{absconv}(K')$ is invariant under the action of G.

Notice that $\textit{absconv}(K')$ contains $B_{\ell _{2}^{n}}$. Hence $\textit{absconv}(K')$ has non-empty interior. Since $K'$ is a compact set in a finite-dimensional space, $\textit{absconv}(K')$ is compact. The set $K=\frac{R}{M}{} \textit{absconv}(K')$ satisfies the desired conditions. $\square$

The following is probably folkloric.

Lemma 2.2

Given r complex numbers of modulus one, $\{z_{1}\ldots ,z_{r}\}$, there exists a strictly increasing sequence of natural numbers $(m_{k})_{k=1}^{\infty }$ such that the sequence $(z_{j}^{m_{k}})$ converges to 1 for $j=1,\ldots ,r$.

Proof

Denote by $S^{r}$ the cartesian product of r copies of the unit sphere $S=\{w\in {\mathbb {C}}:\vert w\vert =1\}$. $S^{r}$ is a compact subset of ${\mathbb {C}}^{r}$. Therefore given the sequence $\big ((z_{1}^{t},\ldots ,z_{r}^{t})\big )_{t=0}^{\infty }$ there exists a subsequence $\{(z_{1}^{t_{k}},\ldots ,z_{r}^{t_{k}})\}_{k=0}^{\infty }$ convergent to a point $(w_{1},\ldots ,w_{r})\in S^{r}$. Thus, for every natural number n, there exists k(n) such that $\Vert (z_{1}^{t_{k}},\ldots ,z_{r}^{t_{k}})- (w_{1},\ldots ,w_{r})\Vert < \frac{1}{2n}$ for every $k\ge k(n)$. Consider $k'(1)> k(1)$ and, by induction, choose $k'(n)> k(n)$ such that $m_{n}=t_{k'(n)}-t_{k(n)}$ satisfies $m_{n+1}>m_{n}$ for every n. Now, we have that

$$\begin{aligned} \vert z_{j}^{m_{n}}-1\vert&= \vert z_{j}^{t_{k'(n)}}-z_{j}^{t_{k(n)}}\vert \le \vert z_{j}^{t_{k'(n)}}-w_{j}\vert +\vert w_{j}-z_{j}^{t_{k(n)}}\vert < \frac{1}{n}, \end{aligned}$$

for $j=1,\ldots , r$ and every $n\in {\mathbb {N}}$. $\square$

Now we present the main result of this note.

Theorem 2.3

Let G be a finite subgroup of $\textit{GL}(n,{\mathbb {C}})$ and K a set in ${\mathbb {C}}^{n}$ that is invariant under the action of G. If z is an element in ${\mathbb {C}}^{n}\setminus K$ that can be separated from K by a polynomial Q, then there exists a G-invariant polynomial P that separates z and K. Furthermore, if Q is homogeneous, then P can be chosen to be homogeneous.

Proof

Without loss of generality we can assume that

$$\begin{aligned} \sup _{w\in K} \vert Q(w)\vert \le \delta < 1\text { and }\vert Q(z)\vert >1 \end{aligned}$$

for some real number $\delta$.

Let us consider the sequence of polynomials

$$\begin{aligned} P_{m}(w)=\sum _{g\in G}\big (Q(g w)\big )^{m}. \end{aligned}$$

(2.1)

Then $P_{m}$ is a G-invariant polynomial. Furthermore, if Q is a homogeneous polynomial, then P is also homogeneous. We will show that $P_{m}$ separates z and K for some big non-negative entire number m.

On the one hand, since K is invariant under the action of G, for every w in K and every $g\in G$, we have that gw is in K. Therefore $\vert Q(g w)\vert \le \delta <1$ and $\vert \big (Q(g w)\big )^{m}\vert \le \delta ^{m}$ for every non-negative entire number m. Thus $\sup _{w\in K}\vert P_{m}(w)\vert \le \vert G\vert \delta ^{m}$ which converges to zero as m tends to infinity.

For each g in G, let $\theta _g$ be a real number such that $Q( g z)=|Q( g z)|\text {e}^{i\theta _g}$. Consider $0<\epsilon <1$. By Lemma 2.2, there exists an increasing sequence of natural numbers $\{m_{k}\}_{k=1}^{\infty }$ such that $\vert (\text {e}^{i\theta _{g}})^{m_{k}}-1\vert <\epsilon$ for every g in the finite set G and every $k\in {\mathbb {N}}$. Thus, for every natural number k,

$$\begin{aligned} \vert P_{m_{k}}(z)\vert&=\Bigg \vert \sum _{g\in G}\big (Q(g z)\big )^{m_k}\Bigg \vert =\Bigg \vert \sum _{g\in G}\big \vert (Q(g z))\vert ^{m_k} \text {e}^{i\theta _{g}m_{k}}\Bigg \vert \\&\ge \sum _{g\in G}\big \vert (Q(g z))\vert ^{m_k} - \sum _{g\in G}\big \vert (Q(g z))\vert ^{m_k}\big \vert \text {e}^{i\theta _{g}m_{k}}-1\big \vert \\&\ge (1-\epsilon ) \sum _{g\in G}\big \vert (Q(g z))\vert ^{m_k}\\&\ge (1-\epsilon )\vert Q(z)\vert ^{m_k}.\\ \end{aligned}$$

Since $\vert Q(z)\vert >1$ and $\sup _{w\in K}\vert P_{m_{k}}(w)\vert <\big \vert G\big \vert \delta ^{m_k}$, we have, for k big enough,

$$\begin{aligned} \vert P_{m_{k}}(z)\vert >\sup _{w\in K}\vert P_{m_{k}}(w)\vert . \end{aligned}$$

$\square$

Corollary 2.4

Let G be a finite subgroup of $\textit{GL}(n,{\mathbb {C}})$ and K a closed convex balanced subset of ${\mathbb {C}}^{n}$. If K is invariant under the action of G and z is an element in ${\mathbb {C}}^{n}\setminus K$, then there exists a homogeneous G-invariant polynomial P that separates z and K.

Proof

Since K is a closed convex set, there exists a complex linear form Q with

$$\begin{aligned} \sup _{w\in K}\{\textit{Re} (Q(w))\}<\textit{Re}( Q(z)). \end{aligned}$$

(2.2)

Moreover, since K is balanced

$$\begin{aligned} \sup _{w\in K}\vert Q(w)\vert =\sup _{w\in K}\{\textit{Re} (Q(w))\}<\textit{Re}( Q(z))\le \vert Q(z)\vert . \end{aligned}$$

By Theorem 2.3, we can find a homogeneous polynomial $P_{m}$ invariant under the action of G that separates the point z and the set K. $\square$

The condition of K being balanced in Corollary 2.4 cannot be removed in general. Indeed, if K is a convex compact set invariant under the action of G and z is a point not in K, as a consequence of Theorem 3.1 below, there exists a G-invariant polynomial that separates z and K. But, if K is not balanced, then there exists a point $z_0\in K$ and a complex number $\lambda$ of modulus less than or equal to one with $\lambda z_0\notin K$. However, for any m-homogeneous polynomial $\vert P(\lambda z_0)\vert =\vert \lambda ^m P( z_0)\vert \le \vert P( z_0)\vert \le \sup _{z \in K}|P(z)|$. Hence, any G-invariant separating polynomial of $\lambda z_0$ and K cannot be homogeneous.

3 G-Invariant Separating Polynomials and Polynomially Convex Sets

The definition of separating polynomial is closely related to that of polynomially convex set. Recall that the polynomially convex hull of a compact subset K of ${\mathbb {C}}^{n}$ is the set

$$\begin{aligned} \widehat{K} = \left\{ z \in {\mathbb {C}}^{n} : \vert P (z)\vert \le \sup _{w \in K}|P(w)|\; \text {for every polynomial } P \right\} . \end{aligned}$$

A compact set K of ${\mathbb {C}}^{n}$ is polynomially convex if $\hat{K}=K$. By definition we have that for a compact set $K\subset {\mathbb {C}}^{n}$, every point in ${\mathbb {C}}^{n}\setminus K$ can be separated from K by a polynomial if and only if the set K is polynomially convex. Some classical examples of polynomially convex sets are compact convex sets and the closure of polynomial polyhedrons. Here, for a set of polynomials $p_{1},\ldots ,p_{m}$ on ${\mathbb {C}}^{n}$ and a positive number r, the polynomial polyhedron defined by $p_{1},\ldots ,p_{m}$ of radius r is the set

$$\begin{aligned} K=\{(w_{1},\ldots ,w_{n})\in {\mathbb {C}}^{n}:\vert w_{j}\vert< r, \vert p_{j}(w)\vert < r, 1\le j\le n\}. \end{aligned}$$

See [9] for a comprehensive reference about polynomial convexity.

Since we are working with polynomials of several complex variables, one can ask if it is possible to obtain a proof of Theorem 2.3 by using only techniques of complex analysis. The following result is a weaker version of Theorem 2.3 proved in this way. Moreover, the proof presented here has some limitations. This proof only works for compact sets, and if we assume that the compact set K is convex and balanced, as in Corollary 2.4, we cannot ensure that the separating polynomials are homogeneous. Nevertheless, we feel that the argument is worth describing.

Theorem 3.1

Let G be a finite subgroup of $\textit{GL}(n,{\mathbb {C}})$. If K is a compact polynomially convex set that is invariant under the action of G and z is an element in ${\mathbb {C}}^{n}\setminus K$, then there exists a G-invariant polynomial P separating z and K.

Proof

Since G is a finite group, the orbit of the point z under the action of G, $Orb(z)=\{g z:g\in G\}$, has at most $\vert G\vert$ elements. In particular, Orb(z) has zero length in ${\mathbb {C}}^{n}$ and $Orb(z)\cup K$ is compact. Then, by [9, Corollary 1.6.3], $Orb(z)\cup K$ is polynomially convex.

Since K is invariant under the action of the group G, the sets K and Orb(z) are disjoint. Thus, as a consequence of the Oka–Weil theorem [9, Theorem 1.5.1], there exists a polynomial Q with $\vert Q(w)-3/2\vert \le \frac{1}{2}$ for all $w\in Orb(z)$ and $\sup _{w\in K}\vert Q(w)\vert \le \delta <1$. Since G is a subgroup of $\textit{GL}(n,{\mathbb {C}})$, every element g of G is a linear transformation in ${\mathbb {C}}^{n}$. Therefore $Q\circ g$ is again a polynomial and since K is invariant under the action of G, $\sup _{w\in K}\vert (Q\circ g)(w)\vert <\delta$, for all $g\in G$. If we consider the polynomial

$$\begin{aligned} P(w)=\sum _{g\in G}(Q\circ g)(w) \end{aligned}$$

we have that for every $w\in K$, $\vert P(w)\vert \le \delta \vert G\vert$, and

$$\begin{aligned} \vert P(z)\vert \ge \textit{Re}(P(z))= \textit{Re}\left( \sum _{g\in G}Q(gz)\right) \ge \sum _{g\in G} 1=\vert G\vert . \end{aligned}$$

Therefore P separates z and K. $\square$

Remark 3.2

Let G be a finite subgroup of $\textit{GL}(n,{\mathbb {C}})$ and K a compact set invariant under the action of G. The existence, for each $z\in {\mathbb {C}}^{n}\setminus K$, of a G-invariant polynomial that separates z and K is actually a characterization of K being polynomially convex. That is, $\widehat{K}=K$ if and only if every element of ${\mathbb {C}}^{n}\setminus K$ can be separated from K by a G-invariant polynomial. One implication is given by Theorem 3.1. It is easy to check that the converse implication also holds.

We finish this section by presenting a way to find polynomially convex sets that are invariant under the action of finite unitary reflection groups.

We recall that a finite unitary reflection group G is a finite subgroup of $\textit{GL}(n,{\mathbb {C}})$ of unitary transformations that is generated by the reflections that it contains. By reflection we understand a linear transformation $T:{\mathbb {C}}^{n}\rightarrow {\mathbb {C}}^{n}$ that fixes pointwise only a hyperplane of dimension $n-1$. The finite unitary groups were studied and classified by Shephard and Todd [8] and Flatto [4]. Even more, Hilbert proved that for any finite unitary reflection group G there exists a finite family of polynomials $p_{1},\ldots ,p_{r}$ that are G-invariant and form a basis for the set of G-invariant polynomials. That is, given a G-invariant polynomial P, there exists a unique polynomial Q such that

$$\begin{aligned} P(w)=Q(p_{1}(w),\ldots ,p_{r}(w)). \end{aligned}$$

Chevalley showed that in fact for finite unitary reflection groups this basis can be chosen to be exactly n homogeneous G-invariant polynomials. See [2, §2.5 Theorem 4] and [1, Theorem (A)] for the details.

A classic example of finite unitary reflection group is the symmetric group of order n, consisting of the group of permutations of the set $\{1,\ldots ,n\}$ and denoted by $S_{n}$. $S_{n}$ can be naturally considered as a subgroup of $\textit{GL}(n,{\mathbb {C}}^{n})$, the action of a permutation $\sigma \in S_{n}$ on a point $(w_{1},\ldots ,w_{n})\in {\mathbb {C}}^{n}$ being given by $\sigma (w_{1},\ldots ,w_{n})=(w_{\sigma (1)},\ldots ,w_{\sigma (n)})$. For this group, a basis of $S_{n}$-invariant polynomials is given by the set of polynomials

$$\begin{aligned} p_{m}(w_{1},\ldots ,w_{n})=\sum _{1\le j_{1}<\ldots < j_{m}\le n}w_{j_{1}} \cdots w_{j_{m}}, \quad j=1,\ldots ,n; \end{aligned}$$

see [2, Theorem 3].

To prove the following proposition, we will use the definition of proper maps between open connected sets $\Omega$ and $\Omega '$ in ${\mathbb {C}}^{n}$. Recall that a map $F:\Omega \rightarrow \Omega '$ is said to be proper if for every compact set $C\subset \Omega '$, $F^{-1}(C)$ is compact in $\Omega$; see, e.g., [6, Chap. 15].

Proposition 3.3

Let G be a finite unitary reflection subgroup of $\textit{GL}(n,{\mathbb {C}})$ with invariant polynomials $p_{1},\ldots ,p_{n}$. Consider the mapping F from ${\mathbb {C}}^{n}$ to ${\mathbb {C}}^{n}$ defined by $F(w)=(p_{1}(w),\ldots ,p_{n}(w))$. Then, for every polynomially convex compact set X in ${\mathbb {C}}^{n}$, the compact set $K=F^{-1}(X)$ is polynomially convex and invariant under the action of G.

Proof

Fix a polynomially convex compact set X in ${\mathbb {C}}^{n}$. The same proof used in [7, Proposition 2.1] can be used to show that the map $F:{\mathbb {C}}^n \mapsto {\mathbb {C}}^n$ is proper. Since X is polynomially convex, by [9, Theorem 1.6.24], K is a polynomially convex compact set. To finish we only need to show that $\langle G,K\rangle =K$. For this, fix $w\in K$. Then, for any $g\in G$ and any polynomial $p_{i}$, since $p_{i}$ is G-invariant, we have that $p_{i}(g w)=p_{i}(w)$. Therefore, $g w\in F^{-1}(F(w))\subseteq K$. $\square$

4 Examples

In Theorem 2.3, we proved that if K is invariant under the action of a group G and if a point z can be separated from K by a polynomial Q, then there exists a G-invariant polynomial that separates z and K. Naturally the first idea to find this G-invariant polynomial is to consider the polynomial

$$\begin{aligned} P(w)=\sum _{g\in G}Q(gw). \end{aligned}$$

Our first example shows that this approach does not work in general and that taking powers in equation (2.1) is necessary.

Example 1

Let us consider the set

$$\begin{aligned} K=\{(w,w)\in {\mathbb {C}}^{2}:w\in {\mathbb {C}}\}. \end{aligned}$$

Then K is invariant under the action of the group $S_{2}$. Furthermore, every point that does not belong to K can be separated from K by the polynomial $Q(w_{1},w_{2})=w_{1}-w_{2}$. However the polynomial

$$\begin{aligned} P(w_{1},w_{2})=Q(w_{1},w_{2})+Q(w_{2},w_{1}) \end{aligned}$$

is identically zero. Hence P is $S_{2}$-invariant but does not separate any point in ${\mathbb {C}}^{n}\setminus K$ from K.

To continue, we present examples of sets that are invariant under the action of a group.

Example 2

Given a finite group $G\le GL(n,{\mathbb {C}}^{n})$, a set $\{p_{1},\ldots ,p_{r}\}$ of G-invariant polynomials and r non-negative numbers $R_{1},\ldots ,R_{r}$, the set

$$\begin{aligned} K=\cap _{j=1}^{r}p_{j}^{-1}(R_{j}{\mathbb {D}}) \end{aligned}$$

is invariant under the action of G. Here $R{\mathbb {D}}$ denotes the closed disk of center zero and radius $R>0$ in ${\mathbb {C}}$. Furthermore, every point not in K can be separated from K by one of the G-invariant polynomials $p_{j}$, $j=1,\ldots ,r$. Notice that K is not necessarily bounded.

For the rest of the examples that we consider, the sets K are polynomially convex.

Example 3

(Finite unitary reflection groups)

• (a) For every $p\in [1,\infty )$, the set

  $$\begin{aligned} B_{\ell _{p}^{n}}=\left\{ (w_{1},\ldots ,w_{n})\in {\mathbb {C}}^{n}:\Vert (w_{1},\ldots ,w_{n})\Vert _{p}=\left( \sum _{j=1}^{n}\vert w_{i}\vert ^{p}\right) ^{\frac{1}{p}}\le 1\right\} \end{aligned}$$

  is convex, hence polynomially convex and invariant under action of the symmetric group $S_{n}$.

• (b) For every finite unitary reflection group G with basis of invariant polynomials $\{p_{1},\ldots ,p_{n}\}$, the closure of the polynomial polyhedron defined by $p_{1},\ldots ,p_{n}$ and radius r,

  $$\begin{aligned} K=\{w=(w_{1},\ldots ,w_{n})\in {\mathbb {C}}^{n}: \vert p_{j}(w)\vert \le r, 1\le j\le n\}, \end{aligned}$$

  is polynomially convex and invariant under the action of G.

• (c) Now we give an easy example of a set in ${\mathbb {C}}^{2}$ that is polynomially convex and balanced but not convex. As we mentioned before, the set of $S_{2}$-invariant polynomials is generated by the polynomials $p_{1}(w_{1},w_{2})=w_{1}+w_{2}$ and $p_{2}(w_{1},w_{2})=w_{1}w_{2}$. Then, for each $R>0$, the set

  $$\begin{aligned} K=(R{\mathbb {D}}\times \{0\})\cup (\{0\}\times R{\mathbb {D}}), \end{aligned}$$

  is the inverse image of $X=R{\mathbb {D}}\times \{0\}$ by the proper mapping $F(w_1,w_2)=(w_1+w_2,w_1w_2)$. X is a convex set and hence polynomially convex in ${\mathbb {C}}^{2}$. Therefore, by Proposition 3.3, any point in ${\mathbb {C}}^{2}\setminus K$ can be separated from K by a polynomial which is invariant under permutations.

Next, we provide an example where the group G is not a finite unitary reflection group.

Example 4

(Symmetric sets) If we consider any balanced, bounded, and convex set K in ${\mathbb {C}}^{n}$, then K is polynomially convex and invariant under the action of the group $G=\{Id, -Id\}\le GL(n,{\mathbb {C}}^{n})$, where Id denotes the identity permutation. Notice that since G only fixes the origin in ${\mathbb {C}}^{n}$, G is not a finite unitary reflection group for $n\ge 2$.

To continue, we present some specific examples of separating polynomials for common groups and polynomially convex sets in ${\mathbb {C}}^{n}$. The following example can be seen as a particular case of Proposition 3.3. However, here we present a direct proof of the result. The idea presented in this construction was the motivation for the proof of Theorem 2.3.

Example 5

(The polydisk center zero and radius r) Let K be the closed polydisk in ${\mathbb {C}}^{n}$ of radius $r>0$ and center zero, $K=\{(w_{1},\ldots ,w_{n})\in {\mathbb {C}}^{n}:\vert w_{j}\vert \le r\}$, and G the group $S_{n}$. As we mentioned before, a polynomial P is invariant under the group $S_{n}$ if for any permutation $\sigma \in S_{n}$ and any point $(w_{1},\ldots , w_{n})$,

$$\begin{aligned} P(w_{1},\ldots , w_{n})=P(w_{\sigma (1)},\ldots , w_{\sigma (n)}). \end{aligned}$$

For $j=1,\ldots ,n$ let us denote by $Q_{j}$ the linear functional on ${\mathbb {C}}^{n}$ defined by $Q_{j}(w_{1},\ldots ,w_{n})=\frac{w_{j}}{r}$. If $z=(z_{1},\ldots ,z_{n})$ is a point in ${\mathbb {C}}^{n}$ not contained in K, then at least one of the coordinates $z_{j_{0}}$ has modulus bigger than r. Then the linear functional $Q_{j_{0}}$ satisfies that $\vert Q_{j_{0}}(z)\vert >1=\sup _{w\in K} \vert Q_{j_{0}}(w)\vert$.

Then, for every natural number m the polynomial

$$\begin{aligned} P_{m}(w)=\sum _{j=1}^{n} (Q_{j}(w))^{m} \end{aligned}$$

is an m-homogeneous $S_{n}$-invariant polynomial. As a consequence of Lemma 2.2, and employing the same ideas used in the proof of Theorem 2.3, the $\limsup$ of the sequence $\{\vert P_{m}(z)\vert \}_{m=1}^{\infty }$ goes to infinity as m goes to infinity. Therefore there exists $m_0$ such that for for $m\ge m_0$ we have $|P_m(z)|>n$. However,

$$\begin{aligned} \sup _{w\in K}\vert P_{m}(w)\vert \le n \end{aligned}$$

for every natural number m. Hence for $m\ge m_0$ the m-homogeneous $S_{n}$-invariant polynomial $P_{m}$ separates the point z and the set K.

In the following example, we explicitly give the separating polynomials for some of the sets presented in Example 3. It is worth mentioning that in the above example, all the separating polynomials depend on the function $Q_{1}$ and only the number m depends on the point z that we want to separate from K. In the following example, the separating polynomials cannot be obtained in this way and it is necessary to consider different linear forms Q and different numbers m, both of them depending on the point z that we want to separate from K.

Example 6

(Finite unitary reflection groups)

• (a) If we consider $K=B_{\ell _{2}^{n}}\subset {\mathbb {C}}^{n}$ and the group $S_{n}$, then for every $(z_{1},\ldots ,z_{n})\notin K$ we have that the linear form

  $$\begin{aligned} Q(w_{1},\ldots ,w_{n})=\frac{1}{\Vert z\Vert _{2}^{2}}\sum _{j=1}^{n}\overline{z_{j}}w_{j} \end{aligned}$$

  satisfies $Q(z)=1=\Vert Q\Vert$ and $\sup _{w\in K}\vert Q(w)\vert <1$. Therefore a similar argument to the one given above shows that for a natural number m the m-homogeneous polynomial

  $$\begin{aligned} P_{m}(w)=\sum _{g\in S_{n}}(Q(g w))^{m} \end{aligned}$$

  is $S_{n}$-invariant and separates the point z and the set K.

• (b) Let us consider $G=S_{3}\times S_{2}$. Then G is a finite unitary reflection group of order 12. Let us consider the set $K=B_{\ell _{2}^{3}}\times 2{\mathbb {D}}^{2}$, i.e.,

  $$\begin{aligned} K:=\{(w_{1},\ldots ,w_{5}):\vert w_{1}\vert ^{2}+\vert w_{2}\vert ^{2}+ \vert w_{3}\vert ^{2}\le 1\text { and }\vert w_{4}\vert ,\vert w_{5}\vert \le 2\}. \end{aligned}$$

  Clearly $\langle G,K\rangle =K$ and K is polynomially convex. However, it is easy to see that K is not invariant under the group $S_{5}$.

  Let z be a point not in K. Since z is not in K then $\vert z_{1}\vert ^{2}+\vert z_{2}\vert ^{2}+ \vert z_{3}\vert ^{2}> 1$ or one of the coordinates $z_{4},z_{5}$ has modulus bigger than two.

  If $\vert z_{1}\vert ^{2}+\vert z_{2}\vert ^{2}+ \vert z_{3}\vert ^{2}> 1$, then we consider the linear functional Q and the m-homogeneous polynomials $P_{m}$ defined by

  $$\begin{aligned} Q(w_{1},\ldots ,w_{5})= & {} \frac{1}{(\vert z_{1}\vert ^{2}+\vert z_{2}\vert ^{2}+ \vert z_{3}\vert ^{2})^{\frac{1}{2}}}\sum _{j=1}^{3}\overline{z_{j}}w_{j}\ \text { and } \\ \quad P_{m}(w)= & {} \sum _{g\in S_{3}}(Q(g w))^{m}. \end{aligned}$$

  If $z_{4}$ or $z_{5}$ has modulus bigger than two, we consider the m-homogeneous polynomials $P_{m}$ defined by

  $$\begin{aligned} P_{m}(w_{1},\ldots ,w_{5})=\Big (\frac{w_{4}}{2}\Big )^{m}+\Big (\frac{w_{5}}{2}\Big )^{m}. \end{aligned}$$

  The same argument that was given in the above examples shows that for some natural number m the polynomial $P_{m}$ separates the point z and the compact K.

Example 6(b) can be generalized in the following way. Let us consider the sets $K_{1}\subset {\mathbb {C}}^{n_{1}},\ldots , K_{r}\subset {\mathbb {C}}^{n_{r}}$ invariant under the action of the groups $G_{1},\ldots ,G_{r}$, respectively. Then, if $(z_{1},\ldots ,z_{r})$ is a point not in $K_{1}\times \cdots \times K_{r}$, for some $j\in \{1,\ldots ,r\}$, $z_{j}\notin K_{j}$. If $P_{j}$ is a polynomial in ${\mathbb {C}}^{n_{j}}$ that separates $z_{j}$ from $K_{j}$, then, for some natural number m the polynomial

$$\begin{aligned} P(w)=\sum _{g\in G_{j}}(P_{j}(gw))^{m} \end{aligned}$$

is a $G_{j}$-invariant polynomial that separates $z_{j}$ and $K_{j}$.

By considering the natural embedding of the set $K_{j}$ in $K_{1}\times \cdots \times K_{r}$ and the natural embedding of the group $G_{j}$ in the group $G_{1}\times \cdots \times G_{r}$ we have that the polynomial associated to P is $G_{1}\times \cdots \times G_{r}$-invariant and separates the point $(z_{1},\ldots ,z_{r})$ and the set $K_{1}\times \cdots \times K_{r}$.