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Distribution of zeros of functions with exponential growth B. Ya. Kazarnovskii Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
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     § 1. Introduction1.1. The asymptotic behaviour of the distribution of zerosConsider the set $X$ of common zeros of entire functions of exponential growth $f_1,\dots,f_m$ in $\mathbb C^n$. Let $X_{\mathrm{reg}}$ denote the set of points $z\in X$ such that the codimension of the analytic set $X$ in a neighborhood of the point $z$ is equal to $m$. In the generic situation ${X_{\mathrm{reg}}=X}$. Further, we consider $X$ as a current $\mathfrak M(f_1,\dots,f_m)$, that is, as integration of differential forms of degree $2n-2m$ with compact support over the analytic set $X_{\mathrm{reg}}$. For example, if $m=n$, then $\mathfrak M(f_1,\dots,f_n)=\sum_x\delta(x)$, where $\delta(x)$ is the delta function with support at the isolated root $x$ of the system $f_1=\dots=f_n=0$. For fixed $t>0$ we denote the function $f_j(tz)$ by $f_{t,j}(z)$. If the limit of the currents $\frac{1}{t^m}\mathfrak M(f_{t,1},\dots,f_{t,m})$ exists as $t\to+\infty$, then we we call it the asymptotic density of the variety of common zeros of the entire functions of exponential growth $f_1,\dots,f_m$. Example 1. Let $f$ be a quasipolynomial in one variable $z$ with spectrum $K$, that is, a finite sum of the form where the $p_\xi$ are polynomials in $z$. Then the asymptotic density of zeros of the function $f$ exists. In this case, the current of the asymptotic density is a distribution in $\mathbb C$ . If $K\subset\operatorname{Re}\mathbb C$, we denote by $[\alpha,\beta]$ the minimal interval in $\operatorname{Re}\mathbb C$ containing all points of the finite set $K$. If the polynomials $p_\alpha$ and $p_\beta$ are nonzero, then the currect of asymptotic density of the zero set of $f$ exists, and its value at a compactly supported function $\varphi$ is equal to $\displaystyle \frac{\beta-\alpha}{2\pi}\int_{\operatorname{Im}\mathbb C}\varphi\, dy$. 1.2. Functions with exponential growthRecall that a linear functional on a space of holomorphic functions on a complex manifold $M$ is called an analytic functional on $M$; see [1]. Let ${\mathbb C^n}^*$ be the space of linear functionals in $\mathbb C^n$, and let $\mu$ be an analytic functional on ${\mathbb C^n}^*$, that is, a linear functional on a space of entire holomorphic functions on ${\mathbb C^n}^*$. For fixed $z\in\mathbb C^n$ we regard $\mathrm{e}^{\xi(z)}$ as an entire function of $\xi\in{\mathbb C^n}^*$. Set $\widehat\mu(z)=\mu(\mathrm{e}^{\xi(z)})$. Recall that the entire function $\widehat\mu$ on $\mathbb C^n$ is called the Fourier–Borel (or Laplace; see [2]) transform of the analytic functional $\mu$. The Fourier–Borel transformation is a bijective map of the space of analytic functionals on ${\mathbb C^n}^*$ onto the space of entire functions of exponential growth in $\mathbb C^n$, that is, of entire functions $g\colon\mathbb C^n\to\mathbb C$ such that
$$
\begin{equation*}
\exists\, (C,a)\colon \quad \forall\, z\in\mathbb C^n \quad| g(z)|\leqslant C\mathrm{e}^{a| z|}.
\end{equation*}
\notag
$$
Given an analytic functional $\mu$, there exists a compact set $K\subset M$ satisfying the following. For each open neighbourhood $U$ of $K$ there exists a constant $C_U$ such that
$$
\begin{equation}
\forall\, f \quad |\mu(f)| \leqslant C_U\sup_U| f(\xi)|.
\end{equation}
\tag{1.1}
$$
The compact set $K$ is called a carrier1 of the $\mu$. Let $\mathcal S(K)$ be the space of analytic functional with carrier $K$ on ${\mathbb C^n}^*$. We let $\widehat{\mathcal S}(K)$ denote the space of entire functions in $\mathbb C^n$ formed by the Fourier–Borel transforms of functionals $\mu\in\mathcal S(K)$. For a convex compact set $K\subset\operatorname{Re}{\mathbb C^n}^*$ a precise description of the functions in $\widehat{\mathcal S}(K)$ is provided by the Paley–Wiener theorem (for instance, see [3], Theorem 1.7.7). Example 2. If $K\subset{\mathbb C^n}^*$ is a finite set, then the space $\widehat{\mathcal S}(K)$ contains all quasipolynomials with spectrum $K$, that is, all entire functions in $\mathbb C^n$ of the form where the coefficients $p_\xi$ are polynomials of $z_1,\dots,z_n$. Let $h_K\colon\mathbb C^n\to\mathbb R$ be the function defined by $h_K(z)=\max_{\xi\in K}\operatorname{Re}\xi(z)$. Recall that $h_K$ is called the support function of the compact set $K$. It follows from (1.1) (see § 3) that for each function $f\in\widehat{\mathcal S}(K)$
$$
\begin{equation}
\forall\, z\in\mathbb C^n \quad \varlimsup_{t\to+\infty}\frac{\log| f(tz)|}{t}\leqslant h_K(z).
\end{equation}
\tag{1.3}
$$
1.3. The averaged distribution of the rootsFor compact sets $K_i$, $i=1,\dots,m$, let ${V_1\subset\widehat{\mathcal S}(K_1),\dots,V_m\subset\widehat{\mathcal S}(K_m)}$ be finite-dimensional spaces. Recall that we regard the root variety of the system as the current $\mathfrak M(f_1,\dots,f_m)$. Using arbitrary Hermitian scalar products $\langle\,*\,{,}\,*\,\rangle_i$ in the spaces $V_i$ we define the averaging $\mathfrak M(V_1,\dots,V_m)$ of the current $\mathfrak M(f_1,\dots,f_m)$ over all systems (1.4); see Definition 3 in § 2. Definition 1. For a finite-dimensional space $V\subset\widehat{\mathcal S}(K)$ set $V(t)=\{f_t(z)= f(tz)$: $f\in V\}$. For an Hermitian scalar product $\langle\,*\,{,}\,*\,\rangle$ in $V$ set $\langle f_t,g_t\rangle=\langle f,g\rangle$. If the limit 1.4. The main resultRecall that by definition, given a function $f$ on a complex manifold $M$, the value of the 1-form $d^cf$ at a tangent vector $\xi$ is $df(-i\xi)$. If the continuous functions $h_1,\dots,h_m\colon M\to\mathbb R$ are plurisubharmonic on $M$, then $dd^ch_1\wedge\cdots\wedge dd^ch_m$ is a well-defined positive current of degree $2m$; for instance, see [1]. Remark 1. The currents $dd^ch_1\wedge\dots\wedge dd^ch_n$ are called Monge–Ampère measures. Such measures arise routinely in problems concerning the distributions of the zeros of holomorphic systems of equations: see [1], [4] and [5]. The support functions $h_i$ of the compact sets $K_i\subset{\mathbb C^n}^*$ are convex and therefore plurisubharmonic. In particular, the flow $dd^ch_1\wedge\dots\wedge dd^ch_n$ is a nonnegative measure in $\mathbb C^n$. Below we define the concept of regular finite-dimensional subspace of $\widehat{\mathcal S}(K)$ (see Definition 5 in § 3 and the example at the end of § 1.4). Theorem 1. Let $V_1\subset\widehat{\mathcal S}(K_1),\dots,V_m\subset\widehat{\mathcal S}(K_m)$ be regular spaces. Then the current of averaged asymptotic distribution of roots of system (1.4) exists and is defined by where $h_i$ is the support function of $K_i$. Definition 2. Let $h$ be the support function of a convex compact set $K\subset{\mathbb C^n}^*$. Set $\operatorname{pvol}(K)={\displaystyle\int_B(dd^ch)^n}$, where $B$ is a ball of radius $1$ with centre at the origin. The quantity $\operatorname{pvol}(K)$ is called the pseudovolume of the convex body $K$. If $h_1,\dots,h_n$ are the support functions of convex compact sets $A_1,\dots,A_n$, then is called the mixed pseudovolume of the convex bodies $A_1,\dots,A_n$. Remark 2. The pseudovolume is a unitarily invariant valuation on convex bodies in a complex vector space; see [6]. Theorem 1 has the following corollaries. Corollary 1. Let $V_1,\dots,V_n$ be regular subspaces of respective spaces $\widehat{\mathcal S}(K_1),\dots, \widehat{\mathcal S}(K_n)$. Then the number of the roots in the ball of increasing radius $r$ with centre at the origin averaged over all systems (1.4) is asymptotically equal to where $\operatorname{conv}(K)$ denotes the convex hull of the compact set $K$. If $K \subset \operatorname{Re}{\mathbb C^n}^*$, then the pseudovolume $K$ is equal to the volume of $K$ as a full-dimensional body in the $n$-dimensional vector space $\operatorname{Re}{\mathbb C^n}^*$. Hence the following holds (see [7]). Corollary 2. Let $K_1,\dots,K_n\subset\operatorname{Re}{\mathbb C^n}^*$. Then where $\operatorname{vol}(*,\dots,*)$ is the mixed volume of convex bodies in the $n$-dimensional real vector space $\operatorname{Re}{\mathbb C^n}^*$ and $\mu_n$ is the measure in $\mathbb C^n$ with support $\operatorname{Im}\mathbb C^n$ that is defined as the integral of a function with respect to the Lebesgue measure in $\operatorname{Im}\mathbb C^n$. Now we present an example which is the source of the notion of regular space. Assume that $K$ is a finite set. Let $\widehat {\mathcal S}_N(K)$ denote the space of quasipolynomials of the form (1.2) such that $\operatorname{deg}(p_\xi)\leqslant N$ for all $\xi\in K$. For example, $\widehat {\mathcal S}_0(K)$ consists of the exponential sums with spectrum $K$, that is, of the functions $\sum_{\xi\in K,\,c_\xi\in\mathbb C} c_\xi\mathrm{e}^{\xi(z)}$. Theorem 2. Each finite-dimensional space $V\subset\widehat{\mathcal S}(K)$ such that $V\supset\widehat {\mathcal S}_0(K)$ is regular. In particular, for each $N\geqslant0$ the space of quasipolynomials $\widehat {\mathcal S}_N(K)$ is regular. This yields the well-known result on the asymptotic density of the root variety of systems of exponential sums [4], [8]. § 2. Averaged distribution on a complex manifoldLet $V_1,\dots,V_m$ be finite-dimensional spaces of holomorphic functions on an $n$-dimensional manifold $X$ and $\langle\,*\,{,}\,*\,\rangle_i$ be an Hermitian scalar product in $V_i$. The roots of the system depend only on the projections $p_i=\pi_i(f_i)$ of the points $f_i\in V_i\setminus0$ onto the projectivizations $\mathbb P_i$ of the spaces $V_i$. We regard the root variety of system (2.1) as the current $\mathfrak M(f_1,\dots,f_m)$. In what follows, when necessary, we identify system (2.1) with the point $(p_1,\dots,p_m)\in\mathbb P_1\times\dots\times\mathbb P_m$. Using the Hermitian metrics $\langle\,*\,{,}\,*\,\rangle_i$ we define and calculate the averaging $\mathfrak M(V_1,\dots,V_m)$ of the current $\mathfrak M(f_1,\dots,f_m)$ over all systems (2.1). To do this we use the Fubini–Study metrics on the spaces $\mathbb P_i$ corresponding to the scalar products $\langle\,*\,{,}\,*\,\rangle_i$: for instance, see [9]. Let $\Omega_i$ denote the corresponding volume form in $\mathbb P_i$ normalized by $\displaystyle \int_{\mathbb P_i}\Omega_i=1$. Definition 3. Let $\pi_i(f_i)=p_i$, where $\pi_i\colon V_i\setminus0\to\mathbb P_i$ is the projection. We identify system (2.1) with the point $(p_1,\dots,p_m)\in \mathbb P_1\times\dots\times\mathbb P_m$. The current on the manifold $X$ defined by Definition 4. Assume that Let $V^*_i$ be the space of linear functionals on $V_i$. We define a map $\Theta_i\colon X\to V^*_i\setminus0$ by $\Theta_i(x)\colon f\mapsto f(x)$. In the space $V^*_i$ consider the Hermitian product $\langle\,*\,{,}\,*\,\rangle^*_i$ dual to $\langle\,*\,{,}\,*\,\rangle_i$. For $x\in X$ set $\|x\|_i=\sqrt{\langle \Theta_i(x),\Theta_i(x)\rangle^*_i}$. The assertion of the theorem is local: to prove it we can replace $X$ by an arbitrarily small neighbourhood of a point $x\in X$. Hence by Sard’s theorem the result reduces to the case when the map
$$
\begin{equation*}
X\xrightarrow{\Theta_1\times\dots\times\Theta_m}(V^*_1\setminus0)\times\dots\times (V^*_m\setminus0)\xrightarrow{\pi^*_1\times\dots\times\pi^*_m}\mathbb P^*_1\times\dots\times \mathbb P^*_m
\end{equation*}
\notag
$$
is a closed embedding, where $\mathbb P^*_i$ is the dual projective space of $\mathbb P_i$ and $\pi^*_i\colon\mkern-1mu V^*_i\mkern-1mu\to\mathbb P^*_i$ is the projectivization map. Let $\omega^*_i$ be a Kähler form in $\mathbb P^*_i$ whose integral over a projective line is equal to one. Then $\frac{1}{2\pi}\,dd^c\log(\|f\|^*_i)^2$ in $V^*_i\setminus0$ is the pullback of $\omega^*_i$ from $\mathbb P^*_i$ under the projectivization map. Hence Theorem 3 reduces to the following result. Theorem 4. Let $X$ be a closed complex manifold with boundary in ${\mathbb P^*_1\times\dots\times\mathbb P^*_m}$, let $p_i\in\mathbb P_i$, let $H(p_i)$ be the projective hyperplane $p_i(*)=0$ in $\mathbb P^*_i$, and let $H(p_1,\dots,p_m)=H(p_1)\times\dots\times H(p_m)$. Then the following equality holds for each differential form $\varphi$ of degree $2n-2m$ on $\mathbb P^*_1\times\dots\times\mathbb P^*_m$: This result can be identified as Crofton’s formula for products of projective spaces (which was announced in [8]). Proof of Theorem 4. We limit ourselves to a brief argument.
As is usual in integral geometry (for instance, see [10]), Theorem 4 can also be stated in the language of double fibre bundles. Let
$$
\begin{equation*}
\Gamma(\mathbb P)=\{(q,p)\colon p\in\mathbb P,q\in\mathbb P^*, p\in H(q)\},
\end{equation*}
\notag
$$
and let $\gamma(\mathbb P)\colon (p,q)\mapsto q$ and $\delta(\mathbb P)\colon(p,q)\mapsto p$ be the two projections $\Gamma(\mathbb P)\to\mathbb P^*$ and $\Gamma(\mathbb P)\to\mathbb P$. Consider the double fibre bundle
$$
\begin{equation}
\mathbb P^*_1\times\dots\times\mathbb P^*_m\xleftarrow{\gamma}\Gamma(\mathbb P_1)\times\dots\times\Gamma(\mathbb P_m)\xrightarrow{\delta}\mathbb P_1\times\dots\times\mathbb P_m,
\end{equation}
\tag{2.2}
$$
where $\gamma=\gamma(\mathbb P_1)\times\dots\times\gamma(\mathbb P_m)$ and $\delta=\delta(\mathbb P_1)\times\dots\times\delta(\mathbb P_m)$. Let $\delta^*$ and $\gamma_*$ be the pullback and pushdown maps acting on differential forms and corresponding to $\delta$ and $\gamma$. Then the proof of the theorem reduces to establishing the equality
$$
\begin{equation}
\gamma_*\delta^*\Omega_1\wedge\dots\wedge\Omega_m=\omega^*_1\wedge\dots\wedge\omega^*_m.
\end{equation}
\tag{2.3}
$$
The fibre bundle (2.2) is the Cartesian product of the $m$ double fibre bundles
$$
\begin{equation*}
\mathbb P^*_i\xleftarrow{\gamma(\mathbb P_i)}\Gamma(\mathbb P_i)\xrightarrow{\delta(\mathbb P_i)}\mathbb P_i.
\end{equation*}
\notag
$$
Hence (2.3) reduces to the case when $m=1$, that is, to the following statement.
Proposition 1. Let $\Gamma=\{(p,q)\in\mathbb P^*\times\mathbb P\colon \langle p,q\rangle=0\}$, and let $\gamma=\gamma(\mathbb P)$ and $\delta=\delta(\mathbb P)$. Then $\gamma_*\delta^*\Omega=\omega^*$. This is a well-known Crofton-type formula: see [10]. Its proof is based on the fact that the action of the unitary group on $\mathbb P$, $\mathbb P^*$ and $\Gamma$ commutes with $\gamma$ and $\delta$. Theorem 4 is proved. § 3. Regular subspacesLet $V$ be a finite-dimensional subspace of $\widehat{\mathcal S}(K)$ with fixed Hermitian scalar product $\langle\,*\,{,}\,*\,\rangle$. Let $B\subset V$ be a ball of radius $1$ with centre at the origin. Below we define the notion of regular subspace $V$. This definition uses the scalar product $\langle\,*\,{,}\,*\,\rangle$, but it is easy to see that the regularity of $V$ is independent of the choice of $\langle\,*\,{,}\,*\,\rangle$. Consider the function $\mathbb C^n\to\mathbb R$ depending on a parameter t and defined as $\max_{f\in B} | f(tz)|$. By Definition 1 the quantity $\max_{f\in B} | f(tz)|$ is equal to the norm of the linear functional $f_t\mapsto f_t(z)$ in the space $V(t)$. Definition 5. We say that the space $V$ is regular if, as $t\to+\infty$, the function depending on the parameter $t$, converges locally uniformly to the support function $h_K$ of the compact set $K$. Example 3. Let $L\subset K$. If $\operatorname{conv}(K)\ne\operatorname{conv}(L)$, then no subspace of $\widehat{\mathcal S}(K)$ consisting of functions in $\widehat{\mathcal S}(L)$ is regular. Conversely, if $\operatorname{conv}(K)=\operatorname{conv}(L)$, then each regular subspace of $\widehat{\mathcal S}(L)$ is also a regular subspace of $\widehat{\mathcal S}(K)$. Proposition 2. For an arbitrarily small positive $\varepsilon$ there exists a constant $C$ such that for each $t>0$ Proposition 2 reduces the question of regularity to a lower bound for the function $\frac{\log\max_{f\in B} | f(tz)|}{t}$, depending on the increasing parameter $t$. Corollary 3. Assume that for each nonzero $z\in\mathbb C^n$ there exist a neighbourhood $U_z$ of $z$ and a function $F_z\in V$ such that Then the space $V$ is regular. Proof. Let $K_z$ denote the open cone in $\mathbb C^n$ consisting of the points $\{\tau w\colon w\in U_z\}$. Then condition $(*)$ holds after we replace the neighbourhood $U_z$ by $K_z$. Fixing a finite cover of $\mathbb C^n$ by cones of the form $K_z$ we apply condition $(*)$ to each of them.
The proof is complete. Corollary 4. Let $V_1$ be a regular space, and let $V_1\subset V\subset\widehat{\mathcal S}(K)$. Then $V$ is a regular space too. Proof. By Corollary 3 it suffices to find functions $F_z \in V$ satisfying condition $(*)$. As regularity shows, these functions can be found in the subspace $V_1$.
The proof is complete. In the rest of § 3 we give the proof of Proposition 2. To this end we use the topology of uniform convergence on compact sets in the space of entire functions and the corresponding weak topology in the space of analytic functionals in ${\mathbb C^n}^*$. Lemma 1. Let $A$ be a compact set of analytic functionals with carrier $K$. Then for any open neighbourhood $U$ of the compact set $K$ there exists a constant $C_U$ such that for all $\mu\in A$ Since $A$ is compact, this result is a consequence of (1.1). Lemma 2. Let $A$ be a compact set of analytic functionals with carrier $K$. Then for each arbitrarily small positive $\varepsilon$ there exists a constant $C$ such that Proof. Let $U=K+\varepsilon B_1$, where $B_1\subset{\mathbb C^n}^*$ is a ball with centre $0$ and radius $1$.
Since we have (1) $\sup_{\xi\in U}|\mathrm{e}^{\xi(z)}|=\sup_{\xi\in U}\mathrm{e}^{\operatorname{Re} \xi(z)}= \mathrm{e}^{h_U(z)}$; (2) $h_U(z)=h_K(z)+\varepsilon| z|$; (3) $\widehat\mu(z)=\mu(\mathrm{e}^{\xi(z)})$ by definition; the required estimate is a consequence of Lemma 1. Lemma 3. Let $A$ be a compact set of analytic functionals with carrier $K$. Then for each arbitrarily small positive $\varepsilon$ there exists a constant $C$ such that for each ${t>0}$ Proof. By Lemma 2 we have $| \widehat\mu(tz)| \leqslant C \mathrm{e}^{h_K(tz)+\varepsilon | tz|}$. Taking the logarithm and using the fact that $| z|$ and $h_K$ are homogeneous we obtain the required result. Proof of Proposition 2. Let $A$ denote the set of analytic functionals on ${\mathbb C^n}^*$ whose Fourier–Borel transform belongs to the unit ball $B\subset V$. The Fourier–Borel transform is continuous and bijective, so $A$ is a compact set. Now the required result follows from Lemma 3.
§ 4. Proof of Theorem 1Recall that by Theorem 3 the current of averaged distribution of roots $\mathfrak M(V_1,\dots,V_m)$ (see Definition 3) satisfies
$$
\begin{equation*}
\mathfrak M(V_1,\dots,V_m)=\frac{1}{(2\pi)^m}dd^c\log\|x\|^2_1\wedge\dots\wedge dd^c\log\|x\|^2_m,
\end{equation*}
\notag
$$
where $\|x\|_i(z)$ is the norm of the linear functional $f\mapsto f(z)$ on the space $V_i$ (see Definition 4). Let $\|x\|_{t,i}(z)$ denote the norm of the linear functional $f_t\mapsto f_t(z)$ on $V_i(t)$. Then by the definition of the current of asymptotic density (Definition 1)
$$
\begin{equation*}
\mathfrak M^\infty(V_1,\dots,V_m)=\lim\frac{1}{t^m}\frac{1}{(2\pi)^m}dd^c\log\|x\|_{t,1}^2\wedge\dots\wedge dd^c\log\|x\|_{t,m}^2.
\end{equation*}
\notag
$$
The space $V_i$ being regular means that, as $t$ grows, the function $\frac{1}{t}\log\|x\|_{t,i}^2(z)$ converges to $2\log h_{K_i}(z)$ locally uniformly; see Definition 5. Now the required result (Theorem 1) follows from the continuity of the complex Monge–Ampère operator
$$
\begin{equation*}
(g_1,\dots,g_m)\mapsto dd^cg_1\wedge\dots\wedge dd^cg_m
\end{equation*}
\notag
$$
in the topology of locally uniform convergence of continuous plurisubharmonic arguments $g_i$; for instance, see [11] or [7]. Theorem 1 is proved. § 5. Proof of Theorem 2By Corollary 4 it is sufficient to prove Theorem 5 in the case when $V=\widehat{\mathcal S}_0(K)$, that is, when $V$ consists of exponential sums. Let $0\ne z\in\mathbb C^n$. Corollary 3 reduces Theorem 2 to the construction of an exponential sum $F_z\in V$ such that, as $t$ grows, the function $\frac{\log| F_z(tw)|}{t}$ converges uniformly to $h_K(w)$ in a neighbourhood of the point $z\in\mathbb C$. We present the construction of $F_z$ below. Definition 6. For $0\ne z\in\mathbb C^n$ consider the real linear functional $\varphi_z(\xi)=\operatorname{Re} \xi(z)$ on the space ${\mathbb C^n}^*$. To $ z\in\mathbb C^n$ we assign the face $\Delta(z)$ of the polytope $\operatorname{conv}(K)$ which consists of the points $\xi\in\operatorname{conv} (K)$ at which $\varphi_z$ attains its maximum on $\operatorname{conv} (K)$. We call $\Delta(z)$ the support face of $z$. Lemma 4. If the point $x\in\mathbb C$ is sufficiently close to $z$, then the support face $\Delta(x)$ is a face of the polytope $\Delta(z)$. This follows from the definition of support faces. Lemma 5. Let $\Delta_1=\Delta(z),\Delta_1,\dots,\Delta_N$ be the set of all faces of the polytope $\Delta(z)$. Let $F_z(w)=\sum_{\xi\in\Delta(z)\cap K,\,c_\xi\ne0}c_\xi\mathrm{e}^{\xi(w)}$ be a function such that $\sum_{\xi\in\Delta_i\cap K}c_\xi\mathrm{e}^{\xi(z)}\ne0$ for each $i$. Then condition $(*)$ in Corollary 3 is satisfied. Proof. Let $\Delta_i$ be the support face of a point $x\in\mathbb C^n$ close to $z$. Then by construction, where the quantities $A(t)$ and $B(t)$ are bounded for all $t>0$. Hence the ratio $\frac{\log| F_z(tx)|}{t}$ converges to $h_K(x)$ locally uniformly, so that $(*)$ holds.
The proof is complete. Now Theorem 2 is proved. 
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