Homogeneous Plurisubharmonic Polynomials in Higher Dimensions

Abstract

We prove several results on homogeneous plurisubharmonic polynomials on ${\mathbb {C}}^n$, $n\in {\mathbb {Z}}_{\ge 2}$. Said results are relevant to the problem of constructing local bumpings at boundary points of pseudoconvex domains of finite D’Angelo 1-type in ${\mathbb {C}}^{n+1}$.

1 Introduction

Local bumpings at boundary points of certain bounded, smoothly bounded pseudoconvex domains of finite D’Angelo 1-type in ${\mathbb {C}}^{n+1}$, $n\in {\mathbb {Z}}_{\ge 1}$, have been used both in the construction of peak functions (e.g. [1, 5, 8]) and in the construction of integral kernels for solving the ${\overline{\partial }}$-equation (e.g. [4, 9]).

As explained in [2, 3], the problem of constructing such local bumpings naturally leads to the study of homogeneous plurisubharmonic polynomials on ${\mathbb {C}}^n$. Furthermore, in [7], results on homogeneous plurisubharmonic polynomials on ${\mathbb {C}}^2$ by Bharali, Stensønes [3], applied in combination with results from [6], played an important role in establishing sup-norm estimates for solutions to the ${\overline{\partial }}$-equation for a large class of pseudoconvex domains in ${\mathbb {C}}^3$. Specifically, the crucial results on homogeneous plurisubharmonic polynomials on ${\mathbb {C}}^2$ are the following:

Result

([3, Proposition 1]) Let P be a homogeneous, plurisubharmonic, non-pluriharmonic polynomial on ${\mathbb {C}}^2$. Then there are at most finitely many complex lines through the origin in ${\mathbb {C}}^2$ along which P is harmonic.

Result

([3, Theorem 2]) Let $P:{\mathbb {C}}^2\rightarrow {\mathbb {R}}$ be a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms. Assume that there exists a holomorphic function $g:{\mathbb {C}}^2\rightarrow {\mathbb {C}}$, nonsingular on a non-empty open set $U\subseteq {\mathbb {C}}^2$, such that P is harmonic along every level set of $g|_U$.

Then there exist a homogeneous, subharmonic polynomial $s:{\mathbb {C}}\rightarrow {\mathbb {R}}$ and a homogeneous holomorphic polynomial $h:{\mathbb {C}}^2\rightarrow {\mathbb {C}}$, such that $P=s\circ h$ on ${\mathbb {C}}^2$.

Result

([3, Theorem 3]) Let $P:{\mathbb {C}}^2\rightarrow {\mathbb {R}}$ be a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms and assume that P is homogeneous of degree $2d_1$ in $z_1, \overline{z_1}$ and homogeneous of degree $2d_2$ in $z_2, \overline{z_2}$, where $d_1 , d_2 \in {\mathbb {Z}}_{\ge 1}$. Then, there exist a homogeneous, subharmonic polynomial $s:{\mathbb {C}}\rightarrow {\mathbb {R}}$ without harmonic terms and integers $j,l\in {\mathbb {Z}}_{\ge 1}$, such that

$$\begin{aligned} P(z_1 , z_2 ) = s({z_1}^j {z_2}^l )\text { for all }(z_1 , z_2 )\in {\mathbb {C}}^2\text {.} \end{aligned}$$

When attempting to adapt the methods from [7] to higher dimensions, it is natural to ask for generalizations of the above-mentioned results on homogeneous plurisubharmonic polynomials on ${\mathbb {C}}^2$ to higher dimensions. Specifically, it is natural to ask the following questions:

Question A

Given a homogeneous, plurisubharmonic, non-pluriharmonic polynomial P on ${\mathbb {C}}^n$, $n\ge 2$, is it true that there are at most finitely many complex hyperplanes through the origin in ${\mathbb {C}}^n$ along which P is pluriharmonic?

Question B

Let $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$, $n\in {\mathbb {Z}}_{\ge 2}$, be a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms. Assume that there exists a holomorphic map $G:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$, $1\le m\le n-1$, nonsingular on a non-empty open set $U\subseteq {\mathbb {C}}^n$, such that P is pluriharmonic along every level set of $G|_U$.

Are there necessarily a homogeneous, plurisubharmonic polynomial $Q:{\mathbb {C}}^{m}\rightarrow {\mathbb {R}}$ and holomorphic polynomials $F_1, \dots ,F_m:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$, all homogeneous of the same degree, such that $P=Q\circ (F_1 ,\dots ,F_m )$ on ${\mathbb {C}}^n$?

Question C

Let $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$, $n\in {\mathbb {Z}}_{\ge 2}$, be a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms and assume P is homogeneous in l variables separately, $1\le l\le n-1$.

Are there necessarily a homogeneous, plurisubharmonic polynomial $Q:{\mathbb {C}}^{n-l}\rightarrow {\mathbb {R}}$ and holomorphic polynomials $F_1, \dots ,F_{n-l}:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$, all homogeneous of the same degree, such that $P=Q\circ (F_1 ,\dots ,F_{n-l} )$ on ${\mathbb {C}}^n$?

The purpose of this paper is to provide a detailed answer to Questions A, B and C. A formal statement of the results can be found in Sect. 2.

The answer to Question A is “yes” (Proposition 2.1).

The answer to Question B is “no” in general, even if we additionally assume that the component functions of G are holomorphic polynomials which are all homogeneous of the same degree (Proposition 2.2). However, the answer is “yes” in the special case $m=1$, i.e., when the polynomial is pluriharmonic along the level sets of a single holomorphic function (Theorem 2.3).

The answer to Question C is “no” in general (Proposition 2.2). However, in the special case where the polynomial is homogeneous in all n variables separately, the answer is “yes” (Corollary 2.5). Furthermore, we get that the answer is “almost yes”, or “yes, up to certain singular holomorphic coordinate changes” in the general setting (Theorem 2.4). In many cases, the latter theorem can be used to get the desired bumping results. This is important, since Proposition 2.2 shows that the result by Bharali, Stensønes [3, Theorem 3] does not generalize in this setting.

2 Statement of Results

In this section, we state the results of this paper. All the proofs can be found in the later sections.

The answer to Question A is “yes”. We have:

Proposition 2.1

Let $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$, $n\in {\mathbb {Z}}_{\ge 2}$, be a homogeneous, plurisubharmonic, non-pluriharmonic polynomial. Then there are at most finitely many complex hyperplanes through the origin in ${\mathbb {C}}^n$ along which P is pluriharmonic.

The answer to Question B is “no”, even if we additionally assume that the component functions of G are holomorphic polynomials which are all homogeneous of the same degree. The answer to Question C is “no” as well. All of this is implied by the following:

Proposition 2.2

Let $P:{\mathbb {C}}^3\rightarrow {\mathbb {R}}$,

$$\begin{aligned} P(z,w_1,w_2)=|z|^2\cdot (|w_1 |^4+|w_1^2-w_1 w_2|^2+|w_2|^4)\text {.} \end{aligned}$$

Then P is a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms and P is homogeneous in one variable separately (see Definition 3.3). Furthermore, away from the coordinate hyperplanes, P is pluriharmonic along the level sets of $G:{\mathbb {C}}^3\rightarrow {\mathbb {C}}^2$, $G(z,w_1 ,w_2 )=(zw_1^2,zw_2^2)$.

However, there do not exist a homogeneous, plurisubharmonic polynomial $Q:{\mathbb {C}}^2\rightarrow {\mathbb {R}}$ and holomorphic polynomials $F_1 ,F_2:{\mathbb {C}}^3\rightarrow {\mathbb {C}}$, homogeneous of the same degree, such that $P=Q\circ (F_1 , F_2 )$ on ${\mathbb {C}}^3$.

Nevertheless, the answer to Question B in the special case where $m=1$ is “yes”:

Theorem 2.3

Let $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$, $n\in {\mathbb {Z}}_{\ge 2}$, be a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms. Assume that there exists a holomorphic function $G:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$, nonsingular on a non-empty open set $U\subseteq {\mathbb {C}}^n$, such that P is pluriharmonic along every level set of $G|_U$.

Then, there exist a homogeneous, subharmonic polynomial $s:{\mathbb {C}}\rightarrow {\mathbb {R}}$ without harmonic terms and a homogeneous holomorphic polynomial $h:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$, such that $P=s\circ h$ on ${\mathbb {C}}^n$.

Note that Theorem 2.3 generalizes the upper mentioned result by Bharali, Stensønes [3, Theorem 2] to higher dimension.

Although the answer to Question C is “no”, we get the following result, which says that the answer is “yes, up to singular holomorphic coordinate changes”:

Theorem 2.4

Let $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$, $n\in {\mathbb {Z}}_{\ge 2}$, be a non-constant plurisubharmonic polynomial without pluriharmonic terms and assume that P is homogeneous of degree 2k, $k\in {\mathbb {Z}}_{\ge 1}$. Let $1\le l\le n-1$ and assume that P is homogeneous of degree $2d_j$, $d_j\in {\mathbb {Z}}_{>0}$, in $z_j$, $\overline{z_j}$ for $j=1,\dots ,l$ (see Definition 3.3). Assume furthermore that $k-D>0$, where $D=d_1 +\dots +d_l$. Write $d:=\gcd (d_1 ,\dots ,d_l ,k)\in {\mathbb {Z}}_{\ge 1}$.

Then, there exists a plurisubharmonic polynomial $Q:{\mathbb {C}}^{n-l}\rightarrow {\mathbb {R}}$ without pluriharmonic terms, homogeneous of degree $2k-2D$, with the property that both the holomorphic and the anti-holomorphic degree (see Sect. 3) of every term appearing in Q are divisible by the integer $(k-D)/d$, such that we have for all $(z_1 ,\dots ,z_n)\in {\mathbb {C}}^n$:

$$\begin{aligned} P(z_1 ,\dots ,z_n )=Q(\tau z_{l+1},\dots ,\tau z_n )\text {,} \end{aligned}$$

for every solution $\tau \in {\mathbb {C}}$ of $\tau ^{(k-D)/d}=z_1^{d_1/d}\cdots z_l^{d_l/d}$.

Alternatively, we can carry out a singular holomorphic coordinate change $\Phi :{\mathbb {C}}^n\rightarrow {\mathbb {C}}^n$, $(z_1 ,\dots ,z_n )\mapsto ({z_1}^{(k-D)/d},\dots ,{z_l}^{(k-D)/d},z_{l+1},\dots ,z_n )$ and write

$$\begin{aligned} (P\circ \Phi )(z_1 ,\dots ,z_n )=Q(z_1^{d_1/d}\cdots z_l^{d_l/d}z_{l+1},\dots ,z_1^{d_1/d}\cdots z_l^{d_l/d}z_n )\text {.} \end{aligned}$$

Note that, without the assumptions $d_1 ,\dots ,d_l ,k-D>0$ in Theorem 2.4, P is effectively a polynomial in fewer than n variables, hence we can ignore that case.

As a corollary (of the proof) of Theorem 2.4 we get that the answer to Question C in the special case where P is homogeneous in all n variables separately is “yes”:

Corollary 2.5

Let $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$, $n\in {\mathbb {Z}}_{\ge 2}$, be a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms and assume that P is homogeneous of degree $2d_j$, $d_j\in {\mathbb {Z}}_{>0}$, in $z_j$, $\overline{z_j}$ for $j=1,\dots ,n$ (see Definition 3.3). Write $d:=\gcd (d_1 ,\dots ,d_n )\in {\mathbb {Z}}_{\ge 1}$.

Then, there exists a homogeneous, subharmonic polynomial $s:{\mathbb {C}}\rightarrow {\mathbb {R}}$ without harmonic terms, such that

$$\begin{aligned} P(z_1 ,\dots ,z_n )=s({z_1}^{d_1 /d}\cdots {z_n}^{d_n /d}) \end{aligned}$$

for all $(z_1 ,\dots ,z_n )\in {\mathbb {C}}^n$.

Note that Corollary 2.5 generalizes the upper mentioned result by Bharali, Stensønes [3, Theorem 3] to higher dimension.

3 Preliminaries

For the remainder of this section we fix an integer $n\ge 2$ and a non-constant polynomial $P:{\mathbb {C}}^n\rightarrow {\mathbb {R}}$ with the following properties:

• P is ${\mathbb {R}}$-homogeneous of degree 2k, for some positive integer k,

• P is plurisubharmonic,

• P does not have any pluriharmonic terms (i.e., purely holomorphic or purely anti-holomorphic terms).

In particular, there exists a collection $(a_{{\alpha },{\beta }})_{({\alpha },{\beta })\in {\mathcal {J}}}$ of complex numbers where

• ${\mathcal {J}}$ is the set of all pairs $({\alpha },{\beta })\in ({\mathbb {Z}}_{\ge 0})^n\times ({\mathbb {Z}}_{\ge 0})^n$ satisfying $|\alpha |>0$, $|\beta |>0$ and $|\alpha |+|\beta |=2k$,

• $\overline{a_{{\alpha },{\beta }}}=a_{{\beta },{\alpha }}$ for all $({\alpha },{\beta })\in {\mathcal {J}}$,

such that

$$\begin{aligned} P(z)=\sum _{({\alpha },{\beta })\in {\mathcal {J}}} a_{\alpha ,\beta }z^\alpha {\overline{z}}^\beta \end{aligned}$$

for all $z=(z_1 , \dots , z_n )\in {\mathbb {C}}^n$. Here we are making use of the usual multi-index notation: $|\alpha |=\alpha _1 +\dots +\alpha _n$ and $z^\alpha =z_1^{\alpha _1}\cdots z_n^{\alpha _n}$ (and analogously for $\beta$ and ${\overline{z}}^\beta$). If $a_{\alpha ,\beta }\ne 0$, then we say that $|\alpha |$ (resp. $|\beta |$) is the holomorphic (resp. anti-holomorphic) degree of the term $a_{\alpha ,\beta }z^\alpha {\overline{z}}^\beta$.

Furthermore, let ${\mathcal {L}}(P;p,V)$ denote the Levi form of P at the point $p\in {\mathbb {C}}^n$ in direction $V=(V_1 ,\dots ,V_n )^t\in {\mathbb {C}}^n$, i.e.,

$$\begin{aligned} {\mathcal {L}}(P;p,V)= (V_1,\dots ,V_n ) \begin{pmatrix} \frac{\partial ^2 P}{\partial {z_1}\partial \overline{z_1}}(p) &{} \dots &{} \frac{\partial ^2 P}{\partial {z_1}\partial \overline{z_n}}(p) \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial ^2 P}{\partial {z_n}\partial \overline{z_1}}(p) &{} \dots &{} \frac{\partial ^2 P}{\partial {z_n}\partial \overline{z_n}}(p) \end{pmatrix} \begin{pmatrix} \overline{V_1} \\ \vdots \\ \overline{V_n} \end{pmatrix}\text {.} \end{aligned}$$

Lemma 3.1

Let ${\mathcal {A}}=\{\alpha \in ({\mathbb {Z}}_{\ge 0})^n:|\alpha |=k\text { and }a_{\alpha ,\alpha }\ne 0\}$ and let

$$\begin{aligned} {\mathcal {C}}=\left\{ (c_1 ,\dots ,c_n )\in {\mathbb {C}}^n:\sum _{j=1}^{n}\alpha _j c_j =0\text { for all }\alpha \in {\mathcal {A}}\right\} \text {.} \end{aligned}$$

Then we have for all $(c_1 ,\dots ,c_n )\in {\mathcal {C}}$ and for all $z=(z_1 ,\dots ,z_n )\in {\mathbb {C}}^n$:

$$\begin{aligned} {\mathcal {L}}(P;z,(c_1 z_1 ,\dots ,c_n z_n )^t)=0\text {.} \end{aligned}$$

Proof

For all $z=(z_1 ,\dots ,z_n )\in {\mathbb {C}}^n$, $(c_1 ,\dots ,c_n )\in {\mathbb {C}}^n$ a straightforward calculation shows that

$$\begin{aligned} {\mathcal {L}}(P;z,(c_1 z_1 ,\dots ,c_n z_n )^t) =\sum _{(\alpha ,\beta )\in {\mathcal {J}}}a_{\alpha ,\beta }\cdot \Bigg ({\sum _{j=1}^{n} \alpha _j c_j}\Bigg )\cdot \overline{\Bigg ({\sum _{j=1}^{n}\beta _j c_j}\Bigg )}\cdot z^\alpha {\overline{z}}^\beta \text {.} \end{aligned}$$

Assume for the sake of a contradiction that the claim is wrong. We then find some $(c_1 ,\dots ,c_n )\in {\mathcal {C}}$ and some $r_1 ,\dots ,r_n\in {\mathbb {R}}_{\ge 0}$, $\phi _1 ,\dots ,\phi _n\in [0,2\pi )$, such that:

$$\begin{aligned} {\mathcal {L}}(P;(r_1 e^{i\phi _1},\dots ,r_n e^{i\phi _n}),(c_1 r_1 e^{i\phi _1},\dots ,c_n r_n e^{i\phi _n})^t)\ne 0\text {.} \end{aligned}$$

By continuity, and since P is plurisubharmonic, we then get:

$$\begin{aligned} 0&<\int _{0}^{2\pi }\dots \int _{0}^{2\pi }{\mathcal {L}}(P;(r_1 e^{i\theta _1},\dots ,r_n e^{i\theta _n}),(c_1 r_1 e^{i\theta _1},\dots ,c_n r_n e^{i\theta _n})^t) {d}\theta _1\dots {d}\theta _n\\&=\sum _{(\alpha ,\beta )\in {\mathcal {J}}}a_{\alpha ,\beta }\cdot \Bigg ({\sum _{j=1}^{n} \alpha _j c_j}\Bigg )\cdot \overline{\Bigg ({\sum _{j=1}^{n}\beta _j c_j}\Bigg )}\cdot {r_1}^{\alpha _1 +\beta _1}\cdots {r_n}^{\alpha _n +\beta _n}\\&\quad \cdot \left( \int _{0}^{2\pi } e^{i(\alpha _1 -\beta _1 )\theta _1} {d}\theta _1\right) \cdots \left( \int _{0}^{2\pi } e^{i(\alpha _n -\beta _n )\theta _n} {d}\theta _n\right) \\&=(2\pi )^n\sum _{\alpha \in {\mathcal {A}}}a_{\alpha ,\alpha }\cdot \Bigg |{\sum _{j=1}^{n} \alpha _j c_j}\Bigg |^2 \cdot {r_1}^{\alpha _1 +\alpha _1}\cdots {r_n}^{\alpha _n +\alpha _n}\\&=0\text {,} \end{aligned}$$

where the last equality is due to the fact that $(c_1 ,\dots ,c_n )\in {\mathcal {C}}$. We have arrived at the desired contradiction; the claim follows. $\square$

For all $\beta \in ({\mathbb {Z}}_{\ge 0})^n$ with $1\le |\beta |\le 2k-1$ we define a homogeneous holomorphic polynomial $P_{\beta }:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$,

$$\begin{aligned} P_{\beta }(z)=\sum _{\alpha :|\alpha |=2k-|\beta |}a_{\alpha ,\beta }z^{\alpha }\text {.} \end{aligned}$$

In particular, we can write

$$\begin{aligned} P(z)=\sum _{\beta :1\le |\beta |\le 2k-1}{{\overline{z}}^{\beta }P_{\beta }(z)}\text {.} \end{aligned}$$

Lemma 3.2

Assume that there exists a holomorphic map $G:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$, $1\le m\le n-1$, nonsingular on a non-empty open set $U\subseteq {\mathbb {C}}^n$, such that P is pluriharmonic along every level set of $G|_U$.

Then, for all $i_1 ,\dots ,i_m,L\in \{1,\dots ,n\}$ (not necessarily pairwise distinct) and for all $\beta \in ({\mathbb {Z}}_{\ge 0})^n$ with $1\le |\beta |\le 2k-1$, the following equality holds on ${\mathbb {C}}^n$:

$$\begin{aligned} \det \begin{pmatrix} \frac{\partial G_1}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_1}{\partial z_{i_m}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_m}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_m}{\partial z_{i_m}} \end{pmatrix}\cdot \frac{\partial P_{\beta }}{\partial z_L} =\sum _{j=1}^{m}\det \begin{pmatrix} \frac{\partial G_1}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_1}{\partial z_{i_m}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_{j-1}}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_{j-1}}{\partial z_{i_m}} \\ \frac{\partial P_{\beta }}{\partial z_{i_1}} &{} \dots &{} \frac{\partial P_{\beta }}{\partial z_{i_m}} \\ \frac{\partial G_{j+1}}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_{j+1}}{\partial z_{i_m}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_m}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_m}{\partial z_{i_m}} \end{pmatrix}\cdot \frac{\partial G_j}{\partial z_L}\text {.} \end{aligned}$$

Proof

Fix a point $p\in U$. Then there exist an open neighborhood $W\subseteq U$ of p and holomorphic maps $K_1 ,\dots ,K_{n-m}:W\rightarrow {\mathbb {C}}^n\setminus \{0\}$, such that $\{K_1 (z),\dots ,K_{n-m}(z)\}$ is a basis for the null space of $G'(z)\in {\mathbb {C}}^{m\times n}$ for all $z\in W$. For $l=1,\dots ,n-m$, denote the component functions of $K_l$ as ${K_l}^{(1)},\dots ,{K_l}^{(n)}$. Since P is pluriharmonic along every level set of $G|_U$, we get

$$\begin{aligned} 0=({K_l}^{(1)},\dots ,{K_l}^{(n)}) \begin{pmatrix} \frac{\partial ^2 P}{\partial {z_1}\partial \overline{z_1}} &{} \dots &{} \frac{\partial ^2 P}{\partial {z_1}\partial \overline{z_n}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial ^2 P}{\partial {z_n}\partial \overline{z_1}} &{} \dots &{} \frac{\partial ^2 P}{\partial {z_n}\partial \overline{z_n}} \end{pmatrix} \begin{pmatrix} \overline{{K_l}^{(1)}} \\ \vdots \\ \overline{{K_l}^{(n)}} \end{pmatrix} \end{aligned}$$

on W for $l=1,\dots ,n-m$; but since the Complex Hessian matrix of P is positive semidefinite, we even get

$$\begin{aligned} (0,\dots ,0)=({K_l}^{(1)},\dots ,{K_l}^{(n)}) \begin{pmatrix} \frac{\partial ^2 P}{\partial {z_1}\partial \overline{z_1}} &{} \dots &{} \frac{\partial ^2 P}{\partial {z_1}\partial \overline{z_n}} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial ^2 P}{\partial {z_n}\partial \overline{z_1}} &{} \dots &{} \frac{\partial ^2 P}{\partial {z_n}\partial \overline{z_n}} \end{pmatrix}\text {.} \end{aligned}$$

Writing

$$\begin{aligned} P(z)=\sum _{\beta :1\le |\beta |\le 2k-1}{{\overline{z}}^{\beta }P_{\beta }(z)}\text {,} \end{aligned}$$

as above, we get for $I=1,\dots ,n$:

$$\begin{aligned} 0&= \sum _{J=1}^{n}{K_l}^{(J)}(z)\cdot \frac{{\partial }^2 P}{\partial z_J\partial \overline{z_I}}(z)\\&=\sum _{\beta :1\le |\beta |\le 2k-1}{\left( \beta _I\overline{z_1}^{\beta _1}\cdots \overline{z_I}^{\beta _I -1}\cdots \overline{z_n}^{\beta _n}\cdot \sum _{J=1}^{n}{K_l}^{(J)}(z)\frac{\partial P_{\beta }}{\partial z_J}(z)\right) } \end{aligned}$$

for all $z\in W$, $l=1,\dots ,n-m$; hence

$$\begin{aligned} 0&=\sum _{I=1}^{n}\overline{z_I}\cdot 0\\&=\sum _{\beta :1\le |\beta |\le 2k-1}{\left( |\beta |\cdot {\overline{z}}^{\beta }\cdot \sum _{J=1}^{n}{K_l}^{(J)}(z)\frac{\partial P_{\beta }}{\partial z_J}(z)\right) }\text {.} \end{aligned}$$

Owing to the fact that the ${K_l}^{(J)}$ and the $P_\beta$ are holomorphic, we then get for all $\beta \in ({{\mathbb {Z}}_{\ge 0}})^n$ with $1\le |\beta |\le 2k-1$:

$$\begin{aligned} 0=\sum _{J=1}^{n}{K_l}^{(J)}(z)\frac{\partial P_{\beta }}{\partial z_J}(z)\text { for all }z\in W, l\in \{1,\dots ,n-m\}\text {,} \end{aligned}$$

i.e., $K_l(z)=({K_l}^{(1)}(z),\dots ,{K_l}^{(n)}(z))^t$ is in the null space of the matrix $P_{\beta }'(z)\in {\mathbb {C}}^{1\times n}$. But this implies that the null space of the matrix

$$\begin{aligned} \begin{pmatrix} \frac{\partial G_1}{\partial z_1}(z) &{} \dots &{} \frac{\partial G_1}{\partial z_n}(z) \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_m}{\partial z_1}(z) &{} \dots &{} \frac{\partial G_m}{\partial z_n}(z) \\ \frac{\partial P_\beta }{\partial z_1}(z) &{} \dots &{} \frac{\partial P_\beta }{\partial z_n}(z) \end{pmatrix}\in {\mathbb {C}}^{(m+1)\times n} ,z\in W\text {,} \end{aligned}$$

is $(n-m)$-dimensional for all $\beta$ with $1\le |\beta |\le 2k-1$, so the rank of said matrix is m. Hence, given $i_1 ,\dots ,i_m,L\in \{1,\dots ,n\}$, we have

$$\begin{aligned} 0=\det \begin{pmatrix} \frac{\partial G_1}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_1}{\partial z_{i_m}} &{} \frac{\partial G_1}{\partial z_L} \\ \vdots &{} \ddots &{} \vdots &{} \vdots \\ \frac{\partial G_m}{\partial z_{i_1}} &{} \dots &{} \frac{\partial G_m}{\partial z_{i_m}} &{} \frac{\partial G_m}{\partial z_L} \\ \frac{\partial P_\beta }{\partial z_{i_1}} &{} \dots &{} \frac{\partial P_\beta }{\partial z_{i_m}} &{} \frac{\partial P_\beta }{\partial z_L} \end{pmatrix} \end{aligned}$$

on W for all $\beta$ with $1\le |\beta |\le 2k-1$; noting that all the entries of the latter matrix are holomorphic on ${\mathbb {C}}^n$, the identity theorem gives that the determinant vanishes on all of ${\mathbb {C}}^n$. The claim follows by Laplace expanding by the last column and calculating. $\square$

Definition 3.3

Given $l\in \{1,\dots ,n\}$, we say that P is homogeneous in l variables separately, provided there exist integers $1\le i_1<\dots <i_l\le n$ and integers $d_{i_1},\dots ,d_{i_l}\ge 1$, such that for every $({\alpha },{\beta })\in {\mathcal {J}}$ with ${a_{{\alpha },{\beta }}}\ne 0$ we have $\alpha _{i_1}+\beta _{i_1}=2d_{i_1},\dots , \alpha _{i_l}+\beta _{i_l}=2d_{i_l}$. In this case we say that P is homogeneous of degree $2d_{i_j}$ in $z_{i_j},\overline{z_{i_j}}$ for all $j\in \{1,\dots ,l\}$.

Note 3.4

We restrict attention to even degrees in Definition 3.3 due to the plurisubharmonicity requirement. Note furthermore that, in the case $l=n-1$, the polynomial P is necessarily homogeneous in all n variables separately, since P is homogeneous.

Lemma 3.5

Let $1\le l\le n-1$ and assume that P is homogeneous of degree $2d_j$, $d_j\in {\mathbb {Z}}_{>0}$, in $z_j$, $\overline{z_j}$ for $j=1,\dots ,l$. Assume furthermore that $k-D>0$, where $D=d_1 +\dots +d_l$.

Then, away from the coordinate hyperplanes, P is pluriharmonic along the level sets of $G:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^{n-l}$,

$$\begin{aligned} G(z_1 ,\dots ,z_n )={z_1}^{d_1}\cdots {z_l}^{d_l}\cdot \left( {z_{l+1}}^{k-D},{z_{l+2}}^{k-D},\dots ,{z_n}^{k-D}\right) \text {.} \end{aligned}$$

Proof

G is nonsingular on $U:=\{(z_1 ,\dots ,z_n )\in {\mathbb {C}}^n:z_1\ne 0,\dots ,z_n\ne 0 \}$. Given $z\in U$, we have to show that ${\mathcal {L}}(P;z,V)=0$ for all V in the null space of $G'(z)\in {\mathbb {C}}^{(n-l)\times n}$. But since the Complex Hessian matrix of P is positive semidefinite, it suffices to verify this for a basis of said null space. If $z\in U$, then the collection of vectors

forms a basis for the null space of $G'(z)\in {\mathbb {C}}^{(n-l)\times n}$. Hence, with ${\mathcal {A}}$ and ${\mathcal {C}}$ as in Lemma 3.1, it suffices to show that $C_j\in {\mathcal {C}}$ for $j\in \{1,\dots ,l\}$, where

To this end, we consider some $j\in \{1,\dots ,l\}$ and some $\alpha \in {\mathcal {A}}$. Since $a_{\alpha ,\alpha }\ne 0$ and by assumption on P we then have $\alpha _1 =d_1 ,\dots ,\alpha _l =d_l$, and $\alpha _{l+1}+\dots +\alpha _n =k-D$. Writing $C_j=:({c_1}^{(j)} ,\dots ,{c_n}^{(j)} )$, we then have:

$$\begin{aligned} \sum _{s=1}^{n} \alpha _s {c_s}^{(j)} =(k-D)\alpha _j -d_j (\alpha _{l+1}+\dots +\alpha _n )=0\text {,} \end{aligned}$$

as desired. $\square$

4 Proof of Proposition 2.1

Proof of Proposition 2.1

We proceed by induction on the dimension n. The 2-dimensional case was handled by Bharali, Stensønes [3], so let $n\ge 3$ and assume the claim holds in dimensions $2,\dots ,n-1$. Let P be as in the statement of the theorem and assume for the sake of a contradiction that P is pluriharmonic along infinitely many complex hyperplanes through $0\in {\mathbb {C}}^n$. We then find a sequence $({\mathcal {H}}_j )_{j\in {\mathbb {Z}}_{\ge 1}}$ of pairwise distinct such hyperplanes. It is furthermore easy to see that there exists a complex hyperplane A through $0\in {\mathbb {C}}^n$, such that P is not pluriharmonic along A. Since P is pluriharmonic along each ${\mathcal {H}}_j$, $j\in {\mathbb {Z}}_{\ge 1}$, we get that P is pluriharmonic along $A\cap {\mathcal {H}}_j$ for all $j\in {\mathbb {Z}}_{\ge 1}$. Hence, by induction, the set $\{A\cap {\mathcal {H}}_j :j\in {\mathbb {Z}}_{\ge 1} \}$ is finite. So, there exists an $(n-2)$-dimensional complex vector subspace V of A, such that $A\cap {\mathcal {H}}_j = V$ for infinitely many $j\in {\mathbb {Z}}_{\ge 1}$. Thus, after deleting some members of the sequence if necessary, we can assume that

$$\begin{aligned} A\cap {\mathcal {H}}_j = V\text { for all }j\in {\mathbb {Z}}_{\ge 1}\text {.} \end{aligned}$$

It is easy to verify that there exists a complex hyperplane B through $0\in {\mathbb {C}}^n$, such that P is not pluriharmonic along B and B does not contain V. By repeating the same argument and again deleting some members of the sequence if necessary, we find an $(n-2)$-dimensional complex vector subspace W of B, such that

$$\begin{aligned} B\cap {\mathcal {H}}_j = W\text { for all }j\in {\mathbb {Z}}_{\ge 1}\text {.} \end{aligned}$$

Hence every ${\mathcal {H}}_j$ contains $V+W$. However, B contains W but does not contain V, so we get that $V+W$ is at least $(n-1)$-dimensional. We conclude that ${\mathcal {H}}_j = V+W$ for all $j\in {\mathbb {Z}}_{\ge 1}$. Since the members of the sequence $({\mathcal {H}}_j )_{j\in {\mathbb {Z}}_{\ge 1}}$ were chosen to be pairwise distinct, we have arrived at the desired contradiction. $\square$

5 Proof of Proposition 2.2

Let P and G be as in the statement of Proposition 2.2. It is obvious that P is indeed a non-constant, homogeneous, plurisubharmonic polynomial without pluriharmonic terms and that P is homogeneous of degree 2 in z, ${\overline{z}}$, so P is homogeneous in one variable separately. Furthermore, away from the coordinate hyperplanes, P is pluriharmonic along the level sets of G by Lemma 3.5.

Assume for the sake of a contradiction that there exist a homogeneous, plurisubharmonic polynomial $Q:{\mathbb {C}}^2\rightarrow {\mathbb {R}}$ and holomorphic polynomials $F_1 ,F_2:{\mathbb {C}}^3\rightarrow {\mathbb {C}}$, homogeneous of the same degree, such that $P=Q\circ (F_1 , F_2 )$ on ${\mathbb {C}}^3$. As in Sect. 3 we write

$$\begin{aligned} P(z,w_1 ,w_2 )=&{\overline{z}}\cdot {\overline{w_1}^2}\cdot \big (2z{w_1}^2 -zw_1 w_2\big )+{\overline{z}}\cdot \overline{w_1}\cdot \overline{w_2}\cdot \big (zw_1 w_2 -z{w_1}^2\big )\\&+{\overline{z}}\cdot \overline{w_2}^2\cdot \big (z{w_2}^2\big )\text {.} \end{aligned}$$

Even though the holomorphic map $(F_1 , F_2 ):{\mathbb {C}}^3\rightarrow {\mathbb {C}}^2$ is (a priori) not necessarily non-singular, an argument analogous to the proof of Lemma 3.2 gives that the matrix

$$\begin{aligned} \begin{pmatrix} \frac{\partial F_1}{\partial z} &{} \frac{\partial F_1}{\partial w_1} &{} \frac{\partial F_1}{\partial w_2} \\ \frac{\partial F_2}{\partial z} &{} \frac{\partial F_2}{\partial w_1} &{} \frac{\partial F_2}{\partial w_2} \\ 2{w_1}^2-w_1 w_2 &{} 4zw_1 -zw_2 &{} -zw_1 \\ w_1 w_2 -{w_1}^2 &{} -2zw_1 +zw_2 &{} zw_1 \\ {w_2}^2 &{} 0 &{} 2zw_2 \end{pmatrix} \end{aligned}$$

has the same rank as the matrix

$$\begin{aligned} \begin{pmatrix} \frac{\partial F_1}{\partial z} &{} \frac{\partial F_1}{\partial w_1} &{} \frac{\partial F_1}{\partial w_2} \\ \frac{\partial F_2}{\partial z} &{} \frac{\partial F_2}{\partial w_1} &{} \frac{\partial F_2}{\partial w_2} \end{pmatrix} \end{aligned}$$

at every point of some non-empty open subset U of ${\mathbb {C}}^3$, which does not meet the coordinate hyperplanes. In particular, said rank is 2 and $(F_1 ,F_2 )$ is non-singular on U; hence $F_1$ and $F_2$ are both non-constant and homogeneous of degree $d:=\deg F_1 =\deg F_2 \ge 1$. After adding the fourth row (of the former matrix) to the third row and applying the identity theorem, we get that the following holds on ${\mathbb {C}}^3$ for $j\in \{1,2\}$:

$$\begin{aligned} 0=\det \begin{pmatrix} \frac{\partial F_j}{\partial z} &{} \frac{\partial F_j}{\partial w_1} &{} \frac{\partial F_j}{\partial w_2} \\ {w_1}^2 &{} 2zw_1 &{} 0\\ {w_2}^2 &{} 0 &{} 2zw_2 \end{pmatrix}\text {,} \end{aligned}$$

and, using that $F_j$ is homogeneous, a calculation then gives

$$\begin{aligned} 3z\frac{\partial F_j}{\partial z}=z\frac{\partial F_j}{\partial z}+w_1\frac{\partial F_j}{\partial w_1}+w_2\frac{\partial F_j}{\partial w_2}=d\cdot F_j\text {.} \end{aligned}$$

From this we readily deduce that d/3 is a positive integer and that there exist holomorphic polynomials $0\not \equiv f_1 ,f_2:{\mathbb {C}}^2\rightarrow {\mathbb {C}}$, homogeneous of degree 2d/3, such that

$$\begin{aligned} F_j (z,w_1 ,w_2 )=z^{\frac{d}{3}}\cdot f_j (w_1 ,w_2 )\text { on }{\mathbb {C}}^3 \end{aligned}$$

for $j\in \{1,2\}$. Since $Q:{\mathbb {C}}^2\rightarrow {\mathbb {R}}$ is a homogeneous, plurisubharmonic (and clearly also non-pluriharmonic) polynomial, its degree $\deg Q$ is even. But d/3 is an integer and $6=\deg P=d\cdot \deg Q$, so we necessarily have $\deg Q=2$ and $d=3$. In particular, $f_j$ is homogeneous of degree 2 and $F_j(z, w_1 ,w_2 )=z\cdot f_j (w_1 ,w_2 )$ on ${\mathbb {C}}^3$ for $j\in \{1,2\}$. Since P does not have any pluriharmonic terms, we can assume that Q does not have any pluriharmonic terms either. So there exist $a,c\in {\mathbb {R}}$, $b\in {\mathbb {C}}$, such that we have for all $(x,y)\in {\mathbb {C}}^2$:

$$\begin{aligned} Q(x,y)=a\cdot |x|^2 +b\cdot x{\overline{y}}+{\overline{b}}\cdot {\overline{x}}y+c\cdot |y|^2\text {.} \end{aligned}$$

For $j\in \{1,2\}$ we furthermore find $\sigma _j ,\rho _j ,\mu _j\in {\mathbb {C}}$, such that we have for all $(w_1 ,w_2 )\in {\mathbb {C}}^2$:

$$\begin{aligned} f_j (w_1 ,w_2 )=\sigma _j {w_1}^2 +\rho _j w_1 w_2 +\mu _j {w_2}^2\text {.} \end{aligned}$$

A calculation then shows that

$$\begin{aligned}&Q(z\cdot f_1 (w_1 ,w_2 ),z\cdot f_2 (w_1 ,w_2 ))\\&\quad = |z|^2\cdot \Big ( \overline{w_1}^2\cdot g_1 (w_1 ,w_2 ) +\overline{w_1 w_2}\cdot h(w_1 ,w_2 ) +\overline{w_2}^2\cdot g_2 (w_1 ,w_2 ) \Big ) \end{aligned}$$

for some $g_1 ,h,g_2:{\mathbb {C}}^2\rightarrow {\mathbb {C}}$ contained in the ${\mathbb {C}}$-vector space ${\mathcal {V}}$ spanned by $f_1$ and $f_2$; we trivially have $\dim _{\mathbb {C}} {\mathcal {V}}\le 2$. Recalling that $P=Q\circ (F_1 ,F_2 )$, we necessarily have

$$\begin{aligned} g_1 (w_1 ,w_2 )=2{w_1}^2 -w_1 w_2\text {,}&h(w_1 ,w_2 )=w_1 w_2 -{w_1}^2\text {,}&g_2 (w_1 ,w_2 )={w_2}^2 \end{aligned}$$

which implies that $\dim _{\mathbb {C}} {\mathcal {V}}\ge 3$. We have arrived at the desired contradiction.

6 Proof of Theorem 2.3

In this section we will (without further comment) identify holomorphic polynomials ${\mathbb {C}}^n\rightarrow {\mathbb {C}}$ with elements of the polynomial ring ${\mathbb {C}}[z_1 ,\dots , z_n ]$ in the obvious way.

Notation 6.1

Let $0\ne g\in {\mathbb {C}}[z_1 ,\dots , z_n ]$ be a homogeneous polynomial of positive degree. Then the set

$$\begin{aligned} \{m\in {\mathbb {Z}}_{>0}:g={\widetilde{g}}^m\text { for some homogeneous }{\widetilde{g}}\in {\mathbb {C}}[z_1 ,\dots ,z_n ] \} \end{aligned}$$

is clearly non-empty and bounded from above. Hence it has a maximum, which we denote as $M_g\in {\mathbb {Z}}_{>0}$.

Lemma 6.2

Let $0\ne g\in {\mathbb {C}}[z_1 ,\dots , z_n ]$ be a homogeneous polynomial of positive degree and let $h\in {\mathbb {C}}[z_1 ,\dots , z_n ]$ be any homogeneous polynomial with $g=h^{M_g}$ (see Notation 6.1). If $0\ne f\in {\mathbb {C}}[z_1 ,\dots , z_n ]$ is a homogeneous polynomial of positive degree satisfying

$$\begin{aligned} (\deg g)\cdot g\cdot \frac{\partial f}{\partial z_l}=(\deg f)\cdot f\cdot \frac{\partial g}{\partial z_l}\text { for all }l\in \{1,\dots ,n\}\text {,} \end{aligned}$$

then $M_g\cdot (\deg f)/(\deg g)$ is a positive integer and there exists a $c\in {\mathbb {C}}\setminus \{0\}$, such that $f=c\cdot h^{M_g\cdot \frac{\deg f}{\deg g}}$.

Proof

Let $g,h,f\in {\mathbb {C}}[z_1 ,\dots ,z_n ]$ be as in the statement of the lemma. Write

$$\begin{aligned} g=u{p_1}^{\alpha _1}\cdots {p_m}^{\alpha _m}\text {,} \end{aligned}$$

where m is a positive integer (since $\deg g>0$), $u\in {\mathbb {C}}\setminus \{0\}$ is a unit, $p_1,\dots ,p_m\in {\mathbb {C}}[z_1 ,\dots ,z_n ]$ are pairwise non-associate primes, and $\alpha _1 ,\dots ,\alpha _m$ are positive integers. Since $\deg f>0$, we get that g divides $f\cdot (\partial g)/(\partial z_l )$ for $l=1,\dots ,n$. Hence, considering any $s\in \{1,\dots ,m\}$, we get that ${p_s}^{\alpha _s}$ divides

$$\begin{aligned} f\cdot \frac{\partial g}{\partial z_l}=f\cdot u\cdot \sum _{j=1}^{m}{p_1}^{\alpha _1}\cdots \widehat{{p_j}^{\alpha _j}}\cdots {p_m}^{\alpha _m}\cdot \alpha _j\cdot {p_j}^{\alpha _j -1}\cdot \frac{\partial p_j}{\partial z_l}\text {.} \end{aligned}$$

If $j\ne s$, then the corresponding summand is trivially divisible by ${p_s}^{\alpha _s}$. But this implies that ${p_s}^{\alpha _s}$ divides

$$\begin{aligned} f\cdot u\cdot {p_1}^{\alpha _1}\cdots \widehat{{p_s}^{\alpha _s}}\cdots {p_m}^{\alpha _m}\cdot \alpha _s\cdot {p_s}^{\alpha _s -1}\cdot \frac{\partial p_s}{\partial z_l}\text {.} \end{aligned}$$

Hence

$$\begin{aligned} p_s\biggm | f\cdot {p_1}^{\alpha _1}\cdots \widehat{{p_s}^{\alpha _s}}\cdots {p_m}^{\alpha _m}\cdot \frac{\partial p_s}{\partial z_l}\text {.} \end{aligned}$$

Since $p_s$ is prime, it divides one of the factors. Since the primes $p_1,\dots ,p_m$ are pairwise non-associate, we get for all $l\in \{1,\dots ,n\}$:

$$\begin{aligned} p_s\bigm | f\text { or }p_s\biggm |\frac{\partial p_s}{\partial z_l}\text {.} \end{aligned}$$

But since $p_s$ is prime and hence $(\partial p_s )/(\partial z_{l_s} )\ne 0$ for some $l_s\in \{1,\dots ,n\}$, we get $p_s\bigm | f$. We have shown that every prime factor of g divides f. But by reversing the roles of f and g and repeating the same argument, we also get that every prime factor of f divides g. We conclude that there exist a unit $v\in {\mathbb {C}}\setminus \{0\}$ and positive integers $\beta _1 ,\dots ,\beta _m$, such that

$$\begin{aligned} f=v{p_1}^{\beta _1}\cdots {p_m}^{\beta _m}\text {.} \end{aligned}$$

By assumption we then have for all $l\in \{1,\dots ,n\}$:

$$\begin{aligned}&(\deg g)\cdot u{p_1}^{\alpha _1}\cdots {p_m}^{\alpha _m}\cdot v\cdot \sum _{j=1}^{m}{p_1}^{\beta _1}\cdots \widehat{{p_j}^{\beta _j}}\cdots {p_m}^{\beta _m}\cdot \beta _j\cdot {p_j}^{\beta _j -1}\cdot \frac{\partial p_j}{\partial z_l}\\&\quad = (\deg f)\cdot v{p_1}^{\beta _1}\cdots {p_m}^{\beta _m}\cdot u\cdot \sum _{j=1}^{m}{p_1}^{\alpha _1}\cdots \widehat{{p_j}^{\alpha _j}}\cdots {p_m}^{\alpha _m}\cdot \alpha _j\cdot {p_j}^{\alpha _j -1}\cdot \frac{\partial p_j}{\partial z_l}\text {,} \end{aligned}$$

and hence

$$\begin{aligned} 0=\sum _{j=1}^{m}p_1\cdots \widehat{p_j}\cdots p_m\cdot \frac{\partial p_j}{\partial z_l}\cdot (\beta _j (\deg g)-\alpha _j (\deg f)) \text {.} \end{aligned}$$

Considering any $s\in \{1,\dots ,m\}$, we note that $p_s$ obviously divides the jth summand for $j\ne s$, and hence

$$\begin{aligned} p_s\biggm | p_1\cdots \widehat{p_s}\cdots p_m\cdot \frac{\partial p_s}{\partial z_l}\cdot (\beta _s (\deg g)-\alpha _s (\deg f))\text {.} \end{aligned}$$

Using again that the primes $p_1,\dots ,p_m$ are pairwise non-associate and considering some $l_s\in \{1,\dots ,n\}$ with $(\partial p_s )/(\partial z_{l_s})\ne 0$, we get

$$\begin{aligned} p_s\bigm | (\beta _s (\deg g)-\alpha _s (\deg f))\text {,} \end{aligned}$$

i.e., $\beta _s (\deg g)-\alpha _s (\deg f)=0$. So, since s was chosen arbitrarily, we have

$$\begin{aligned} \frac{\beta _j}{\alpha _j}=\frac{\deg f}{\deg g}\text { for all }j\in \{1,\dots ,m\}\text {.} \end{aligned}$$

Since $g=h^{M_g}$, we can write

$$\begin{aligned} h=w{p_1}^{\gamma _1}\cdots {p_m}^{\gamma _m}\text {,} \end{aligned}$$

where $w\in {\mathbb {C}}\setminus \{0\}$, $w^{M_g}=u$, and $\gamma _j=\alpha _j /M_g$ is a positive integer for $j=1,\dots ,m$. Due to the defining properties of $M_g$ (see Notation 6.1) we furthermore have $\gcd (\gamma _1 ,\dots ,\gamma _m )=1$, i.e., there exist $d_1 ,\dots ,d_m\in {\mathbb {Z}}$, such that

$$\begin{aligned} 1=\sum _{j=1}^{m}d_j\gamma _j\text {.} \end{aligned}$$

Hence

$$\begin{aligned} M_g\cdot \frac{\deg f}{\deg g}=\sum _{j=1}^{m}d_j\cdot \gamma _j\cdot M_g\cdot \frac{\deg f}{\deg g}=\sum _{j=1}^{m}d_j\cdot \frac{\alpha _j}{M_g}\cdot M_g\cdot \frac{\beta _j}{\alpha _j}=\sum _{j=1}^{m}d_j\beta _j \end{aligned}$$

is a (positive) integer, as desired. Finally, we compute

$$\begin{aligned} h^{M_g\cdot \frac{\deg f}{\deg g}}&=w^{M_g\cdot \frac{\deg f}{\deg g}}\cdot \prod _{j=1}^{m}\left( {p_j}^{\gamma _j}\right) ^{M_g\cdot \frac{\beta _j}{\alpha _j}}\\&=w^{M_g\cdot \frac{\deg f}{\deg g}}\cdot \prod _{j=1}^{m}\left( {p_j}^{\frac{\alpha _j}{M_g}}\right) ^{M_g\cdot \frac{\beta _j}{\alpha _j}}\\&=w^{M_g\cdot \frac{\deg f}{\deg g}}{p_1}^{\beta _1}\cdots {p_m}^{\beta _m}\\&=\frac{w^{M_g\cdot \frac{\deg f}{\deg g}}}{v}\cdot f\text {,} \end{aligned}$$

and the claim follows. $\square$

Armed with Lemma 6.2, we can provide a proof for Theorem 2.3.

Proof of Theorem 2.3

As in Sect. 3 we write

$$\begin{aligned} P(z)=\sum _{\beta :1\le |\beta |\le 2k-1}{{\overline{z}}^{\beta }P_{\beta }(z)}\text {.} \end{aligned}$$

We then apply Lemma 3.2 and get for all $\beta \in ({\mathbb {Z}}_{\ge 0})^n$ with $1\le |\beta |\le 2k-1$ and for all $j,l\in \{1,\dots ,n\}$:

$$\begin{aligned} \frac{\partial P_\beta }{\partial z_l}\frac{\partial G}{\partial z_j}=\frac{\partial P_\beta }{\partial z_j}\frac{\partial G}{\partial z_l}\text { on }{\mathbb {C}}^n\text {.} \end{aligned}$$

Writing $G=G(0)+q+R$, where $q:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$ is a non-constant, homogeneous, holomorphic polynomial and $R:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$ is a holomorphic function whose Taylor series at 0 does not involve any terms of degree $\le \deg q$, we get, owing to the fact that the $P_\beta$ are homogeneous:

$$\begin{aligned} \frac{\partial P_\beta }{\partial z_l}\frac{\partial q}{\partial z_j}=\frac{\partial P_\beta }{\partial z_j}\frac{\partial q}{\partial z_l}\text { on }{\mathbb {C}}^n \end{aligned}$$

for all $\beta \in ({\mathbb {Z}}_{\ge 0})^n$ with $1\le |\beta |\le 2k-1$ and for all $j,l\in \{1,\dots ,n\}$. By multiplying with $z_j$ and then summing over j, we get, using that both $P_\beta$ and q are homogeneous:

$$\begin{aligned} (\deg P_\beta )\cdot P_\beta \cdot \frac{\partial q}{\partial z_l}=(\deg q)\cdot q\cdot \frac{\partial P_\beta }{\partial z_l}\text { on }{\mathbb {C}}^n \end{aligned}$$

for all $\beta ,l$. Let $h:{\mathbb {C}}^n\rightarrow {\mathbb {C}}$ be any homogeneous, holomorphic polynomial with $q=h^{M_q}$ (see Notation 6.1). Lemma 6.2 then implies that, for a given $\beta$, the following is true:

• if $M_q\cdot \frac{\deg P_\beta }{\deg q}=M_q\cdot \frac{\deg P_\beta }{M_q\cdot \deg h}=\frac{2k-|\beta |}{\deg h}$ is not a positive integer, then $P_\beta \equiv 0$,

• if $\frac{2k-|\beta |}{\deg h}$ is a positive integer, then there exists a $c_\beta \in {\mathbb {C}}$, such that

  $$\begin{aligned} P_\beta =c_\beta \cdot h^{M_q\cdot \frac{\deg P_\beta }{\deg q}}=c_\beta \cdot h^{\frac{2k-|\beta |}{\deg h}} \end{aligned}$$

  (if $P_\beta \equiv 0$, take $c_\beta =0$, otherwise apply Lemma 6.2).

Since $P\not \equiv 0$, there exists a positive integer L, such that

$$\begin{aligned} \{1,\dots ,L\}=\left\{ \frac{2k-|\beta |}{\deg h}:\beta \in ({\mathbb {Z}}_{\ge 0})^n,1\le |\beta |\le 2k-1, \frac{2k-|\beta |}{\deg h}\in {\mathbb {Z}}_{>0} \right\} \text {.} \end{aligned}$$

Hence we can write on ${\mathbb {C}}^n$:

$$\begin{aligned} P(z)&=\sum _{l=1}^{L}\sum _{\beta :|\beta |=2k-l\cdot \deg h}{\overline{z}}^{\beta }\cdot c_\beta \cdot (h(z))^l\\&=\sum _{l=1}^{L} (h(z))^l\cdot \sum _{\beta :|\beta |=2k-l\cdot \deg h}c_\beta \cdot {\overline{z}}^{\beta }\\&=\sum _{l=1}^{L} \overline{h(z)}^l\cdot \sum _{\beta :|\beta |=2k-l\cdot \deg h}\overline{c_\beta }\cdot {z}^{\beta }\text {,} \end{aligned}$$

where the last equality is due to the fact that P is real-valued. Since $h^l\not \equiv 0$, we can find an $\alpha _l\in ({\mathbb {Z}}_{\ge 0})^n$ with $|\alpha _l |=l\cdot \deg h$, such that $z^{\alpha _l}$ appears with coefficient $\gamma _l\ne 0$ in the Taylor expansion of $h^l$ at 0 (note that $\alpha _l$ is not uniquely determined in general). Recalling that

$$\begin{aligned} P(z)=\sum _{\beta :1\le |\beta |\le 2k-1}{{\overline{z}}^{\beta }P_{\beta }(z)}\text {,} \end{aligned}$$

we see that we necessarily have (recall that h is homogeneous):

$$\begin{aligned} P_{\alpha _l}(z)=\overline{\gamma _l}\cdot \sum _{\beta :|\beta |=2k-l\cdot \deg h}\overline{c_\beta }\cdot {z}^{\beta }\text {,} \end{aligned}$$

and hence

$$\begin{aligned} P(z)=\sum _{l=1}^{L} \overline{h(z)}^l\cdot \frac{1}{\overline{\gamma _l}}\cdot P_{\alpha _l}(z)\text {.} \end{aligned}$$

Assume for the sake of a contradiction that $2k/(\deg h)$ is not an integer. Then we have for all $l=1,\dots ,L$ that

$$\begin{aligned} \frac{2k-|\alpha _l |}{\deg h}=\frac{2k}{\deg h}-l\not \in {\mathbb {Z}}, \end{aligned}$$

which implies that $P_{\alpha _l}\equiv 0$ (see above). But then $P\equiv 0$ and we arrive at the desired contradiction. Hence $2k/(\deg h)\in {\mathbb {Z}}$. But then we have that $\frac{2k-|\alpha _l |}{\deg h}$ is a positive integer for $l=1,\dots ,L$, which, using the above, implies that

$$\begin{aligned} P_{\alpha _l}=c_{\alpha _l}\cdot h^{\frac{2k-|\alpha _l |}{\deg h}}=c_{\alpha _l}\cdot h^{\frac{2k}{\deg h}-l}\text {.} \end{aligned}$$

We now define $s:{\mathbb {C}}\rightarrow {\mathbb {R}}$,

$$\begin{aligned} \tau \mapsto \sum _{l=1}^{L}\frac{c_{{\alpha _l}}}{\overline{\gamma _l}}\cdot {\overline{\tau }}^l\cdot \tau ^{\frac{2k}{\deg h}-l}\text {.} \end{aligned}$$

Note that s is indeed real-valued. It is now easy to see that s and h have all the desired properties. $\square$

7 Proof of Theorem 2.4 and Corollary 2.5

In this section we will provide a proof for Theorem 2.4. We will not provide a separate proof for Corollary 2.5, since it will be obvious from the proof of Theorem 2.4.

Proof of Theorem 2.4

We adapt the notation from Sect. 3. In particular we write

$$\begin{aligned} P(z)=\sum _{\beta :1\le |\beta |\le 2k-1}{{\overline{z}}^{\beta }P_{\beta }(z)}=\sum _{\beta \in B}{{\overline{z}}^{\beta }P_{\beta }(z)}\text {,} \end{aligned}$$

where $B=\{\beta \in ({{\mathbb {Z}}_{\ge 0}})^n:1\le |\beta |\le 2k-1\text { and }P_{\beta }\not \equiv 0 \}$. By assumption on P we find, for every $\beta \in B$, a holomorphic polynomial $q_{\beta }:{\mathbb {C}}^{n-l}\rightarrow {\mathbb {C}}$, homogeneous of degree $2k-(2d_1 +\dots +2d_l +\beta _{l+1}+\dots +\beta _n )$, such that

$$\begin{aligned} P_\beta (z_1,\dots ,z_n )={z_1}^{2d_1 -\beta _1}\cdots {z_l}^{2d_l -\beta _l}q_{\beta }(z_{l+1},\dots ,z_{n})\text {.} \end{aligned}$$

For ease of notation we write $m=n-l$ and $(w_1 ,\dots ,w_m )=(z_{l+1},\dots ,z_{n})$. We will switch back and forth between notations whenever convenient. By Lemma 3.5, away from the coordinate hyperplanes, P is pluriharmonic along the level sets of $G:{\mathbb {C}}^n\rightarrow {\mathbb {C}}^{m}$,

$$\begin{aligned} G(z_1 ,\dots ,z_l ,w_1 ,\dots ,w_m )={z_1}^{d_1}\cdots {z_l}^{d_l}\cdot \left( {w_1}^{k-D},\dots ,{w_m}^{k-D}\right) \text {.} \end{aligned}$$

Lemma 3.2 then gives that the following holds on ${\mathbb {C}}^n$ for all $\beta \in B$ and for all $\nu \in \{1,\dots ,n\}$:

$$\begin{aligned} \det \begin{pmatrix} \frac{\partial G_1}{\partial w_1} &{} \dots &{} \frac{\partial G_1}{\partial w_m} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_m}{\partial w_1} &{} \dots &{} \frac{\partial G_m}{\partial w_m} \end{pmatrix}\cdot \frac{\partial P_{\beta }}{\partial z_\nu } =\sum _{j=1}^{m}\det \begin{pmatrix} \frac{\partial G_1}{\partial w_1} &{} \dots &{} \frac{\partial G_1}{\partial w_m} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_{j-1}}{\partial w_1} &{} \dots &{} \frac{\partial G_{j-1}}{\partial w_m} \\ \frac{\partial P_{\beta }}{\partial w_1} &{} \dots &{} \frac{\partial P_{\beta }}{\partial w_m} \\ \frac{\partial G_{j+1}}{\partial w_1} &{} \dots &{} \frac{\partial G_{j+1}}{\partial w_m} \\ \vdots &{} \ddots &{} \vdots \\ \frac{\partial G_m}{\partial w_1} &{} \dots &{} \frac{\partial G_m}{\partial w_m} \end{pmatrix}\cdot \frac{\partial G_j}{\partial z_\nu }\text {.} \end{aligned}$$

A calculation then shows that the following holds on ${\mathbb {C}}^n$ for all $\beta \in B$, $\nu \in \{1,\dots ,l\}$:

$$\begin{aligned} (k-D)(2d_\nu -\beta _\nu ) q_\beta (w_1 ,\dots ,w_m )=d_\nu \sum _{j=1}^{m} w_j\frac{\partial q_\beta }{\partial w_j}(w_1 ,\dots ,w_m )\text {,} \end{aligned}$$

but since $q_\beta \not \equiv 0$ is homogeneous, this simplifies to

$$\begin{aligned}&(k-D)(2d_\nu -\beta _\nu ) q_\beta (w_1 ,\dots ,w_m )\\&\quad = d_\nu \cdot (2k-(2d_1 +\dots +2d_l +\beta _{l+1}+\dots +\beta _n ))\cdot q_\beta (w_1 ,\dots ,w_m )\text {.} \end{aligned}$$

For $\beta \in B$ we have $q_\beta \not \equiv 0$, so we get for all $\beta \in B$, $\nu \in \{1,\dots ,l\}$:

$$\begin{aligned} (k-D)(2d_\nu -\beta _\nu )=&d_\nu \cdot (2k-(2d_1 +\dots +2d_l +\beta _{l+1}+\dots +\beta _n ))\\ =&d_\nu \cdot M_\beta \text {,} \end{aligned}$$

where $M_\beta$ is defined in the obvious way. Since $d=\gcd (d_1 ,\dots ,d_l ,k)$, there exist $c,c_1 ,\dots ,c_l\in {\mathbb {Z}}$, such that $d=c\cdot (k-D)+\sum _{\nu =1}^{l}c_\nu \cdot d_\nu$. But then we have for all $\beta \in B$:

$$\begin{aligned} \frac{d\cdot M_\beta }{k-D}=c\cdot M_\beta + \sum _{\nu =1}^{l}c_\nu d_\nu \frac{M_\beta }{k-D}=c\cdot M_\beta +\sum _{\nu =1}^{l}c_\nu (2d\nu -\beta _\nu )\text {,} \end{aligned}$$

so $d\cdot M_\beta /(k-D)$ is an integer for all $\beta \in B$. If $M_\beta$ was 0 for some $\beta \in B$, then $2d_\nu -\beta _\nu$ would be 0 for $\nu \in \{1,\dots ,l\}$, implying that $|\beta |=2k$, in contradiction to $\beta \in B$. Hence $d\cdot M_\beta /(k-D)$ is a positive integer for all $\beta \in B$. This implies that for all $\beta \in B$ we have

$$\begin{aligned} P_\beta (z_1 ,\dots ,z_n )=\left( {z_1}^{\frac{d_1}{d}}\cdots {z_l}^{\frac{d_l}{d}}\right) ^{\frac{d\cdot M_\beta }{k-D}}\cdot q_\beta (z_{l+1},\dots ,z_n )\text {,} \end{aligned}$$

where all occurring exponents are positive integers. If $\beta \in B$, then $M_\beta \le 2(k-D)$. If this was an equality, then, similarly to above, we would get $|\beta |=0$, in contradiction to $\beta \in B$. Hence we have for all $\beta \in B$:

$$\begin{aligned} \frac{d\cdot M_\beta }{k-D}, 2d-\frac{d\cdot M_\beta }{k-D}\in \{1,2,\dots ,2d-1\}\text {.} \end{aligned}$$

For $\beta \in B$, $\nu \in \{1,\dots ,l\}$ we have

$$\begin{aligned} \beta _\nu =2d_\nu -\frac{d_\nu \cdot M_\beta }{k-D}=\frac{d_\nu }{d}\cdot \left( 2d-\frac{d\cdot M_\beta }{k-D} \right) \text {,} \end{aligned}$$

and both factors in the latter equality are positive integers. We now calculate, noting that all occurring exponents are positive integers:

$$\begin{aligned} P(z)&=\sum _{\beta \in B}{{\overline{z}}^{\beta }P_{\beta }(z)}\\&=\sum _{\beta \in B}\left( {z_1}^{\frac{d_1}{d}}\cdots {z_l}^{\frac{d_l}{d}}\right) ^{\frac{d\cdot M_\beta }{k-D}}\cdot {\overline{\left( {z_1}^{\frac{d_1}{d}}\cdots {z_l}^{\frac{d_l}{d}}\right) }^{2d-\frac{d\cdot M_\beta }{k-D}}}\\&\quad \cdot \overline{z_{l+1}}^{\beta _{l+1}}\cdots \overline{z_n}^{\beta _n}q_{\beta }(z_{l+1},\dots ,z_n )\\&=\sum _{j=1}^{2d-1}\left( {z_1}^{\frac{d_1}{d}}\cdots {z_l}^{\frac{d_l}{d}}\right) ^{j}\cdot {\overline{\left( {z_1}^{\frac{d_1}{d}}\cdots {z_l}^{\frac{d_l}{d}}\right) }^{2d-j}}\\&\quad \cdot \sum _{\beta \in B:\frac{d\cdot M_\beta }{k-D}=j} \overline{z_{l+1}}^{\beta _{l+1}}\cdots \overline{z_n}^{\beta _n}q_{\beta }(z_{l+1},\dots ,z_n )\text {.} \end{aligned}$$

Now if $\beta \in B$ and $j\in \{1,\dots ,2d-1\}$ satisfy $d\cdot M_\beta /(k-D)=j$, then every term occurring in $\overline{z_{l+1}}^{\beta _{l+1}}\cdots \overline{z_n}^{\beta _n}q_{\beta }(z_{l+1},\dots ,z_n )$ has holomorphic degree $M_\beta =j\cdot (k-D)/d$ and anti-holomorphic degree $\beta _{l+1} +\dots +\beta _n =(2d-j)\cdot (k-D)/d$. This implies that the polynomial $Q:{\mathbb {C}}^m\rightarrow {\mathbb {R}}$,

$$\begin{aligned} Q(w_1 ,\dots ,w_m )=\sum _{j=1}^{2d-1}\sum _{\beta \in B:\frac{d\cdot M_\beta }{k-D}=j} \overline{w_{1}}^{\beta _{1+l}}\cdots \overline{w_m}^{\beta _{m+l}}q_{\beta }(w_1 ,\dots ,w_m )\text {,} \end{aligned}$$

has the property that both the holomorphic and the anti-holomorphic degree of every term appearing in Q are divisible by the integer $(k-D)/d$. Furthermore $Q(w_1 ,\dots ,w_m )=P(1,\dots ,1,w_1 ,\dots ,w_m )$, so Q is (indeed real-valued and) plurisubharmonic. It is now easy to see that Q has all the other desired properties. $\square$