Let $ {{\mathbb B}}^n $ be the unit ball in $ {{\mathbb C}}^N $, and denote by Rat$ ({{\mathbb B}}^n, {{\mathbb B}}^N) $ the collection of all proper holomorphic rational maps from $ {{\mathbb B}}^n $ to $ {{\mathbb B}}^N $. Let $ {{\mathbb H}}_n = \{(z, w)\in{{\mathbb C}}^{n-1}\times {{\mathbb C}}\ |\ \text{Im}(w)>|z|^2 \} $ be the Siegel upper half space and denote by Rat$ ({{\mathbb H}}_n, {{\mathbb H}}_N) $ the collection of all proper holomorphic rational maps from $ {{\mathbb H}}_n $ to $ {{\mathbb H}}_N $. By the Cayley transform, we can identify $ {{\mathbb B}}^n $ with $ {{\mathbb H}}_n $, and identify Rat$ ({{\mathbb B}}^n, {{\mathbb B}}^N) $ with Rat$ ({{\mathbb H}}_n, {{\mathbb H}}_N) $. We say that $ f, g\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^N) $ are spherically equivalent (or equivalent, for short) if there are $ \sigma\in {\hbox{Aut}}({{\mathbb B}}^n) $ and $ \tau\in {\hbox{Aut}}({{\mathbb B}}^N) $ such that $ f = \tau\circ g\circ \sigma $.

The study of proper holomorphic maps can date back to the work of Poincaré [17]. Alexander [1] proved that for equal dimensional case, the map must be an automorphism. For the different dimensional case, by the combining efforts of [5, 8, 19], we know that for $ N\in (n, 2n-1) $ with $ n\ge 2 $, any map $ F\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^N) $ is spherically equivalent to the map $ (z, 0) $. This is now called the first gap theorem.

For the second gap theorem, Huang-Ji-Xu [11] proved that any map $ F\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^N) $ with $ N\in (2n, 3n-3) $ and $ n\ge 4 $, must equivalent to a map of the form $ (G, 0) $ with $ G\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{2n}) $. Together with the classification theorems Huang-Ji [10] and Hamada [7], $ {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{2n}) $ must equivalent to $ (z, 0) $ or $ (G_1, 0) $ or $ (G_2, 0) $, where $ G_1\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{2n-1}) $ is the Whitney map and $ G_2\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{2n}) $ is in the D'Angelo family.

In [12], Huang-Ji-Yin proved the third gap theorem. Namely, when $ N\in (3n, 4n-7) $ and $ n\ge 8 $, any map $ F\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^N) $ must be equivalent to a map of the form $ (G, 0) $ with $ G\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{3n}) $. The classification problem for $ N = 3n-3 $ is achieved by [2]. When $ N\geq 3n-2 $, the map is no longer monomials and can be very complicated. In fact, [6] constructed a family of maps in Rat$ ({{\mathbb B}}^n, {{\mathbb B}}^{3n-2}) $, which can not be equivalent to any polynomial maps. Recently, Gul-Ji-Yin [14] gave a characterization of maps in Rat$ ({{\mathbb B}}^n, {{\mathbb B}}^{3n-2}) $. The interesting reader can refer to [3, 4, 9, 12, 15, 16, 18] for other related mapping problems between balls.

It seems to be quite less in known for $ N\geq 3n-2 $. The present paper is devoted to a characterization of proper holomorphic maps from $ \mathbb{B}^n $ to $ \mathbb{B}^{N} $ with geometric rank $ \kappa_0 $ and $ N = n+\frac{(2n-\kappa_0-1)\kappa_0}{2} $. We now state our main theorem, with some terminology to be defined in the next section.

Theorem 1.1 (1)   Let $ F\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{N}) $ with the geometric rank of $ F $ being $ \kappa_0 $ and $ N = n+\frac{(2n-\kappa_0-1)\kappa_0}{2} $. Suppose that $ \frac{\kappa_0(\kappa_0+1)}{2}<n $ and $ F $ is normalized, then it takes the following form

$ \begin{equation} \begin{split} f_j& = \frac{1}{q}\big(z_jq+\frac{i}{2}\mu_jz_jw\big)\ \ \ \text{for}\ 1\leq j\leq \kappa_0, \ f_k = z_k\ \ \ \text{for}\ \kappa_0+1\leq k\leq n-1, \\ \phi_{jj}& = \frac{1}{q}\big(\mu_{jj}z_j^2+e_{j, jj}z_jw\big)\ \ \ \text{for}\ 1\leq j\leq \kappa_0, \\ \phi_{jk}& = \frac{1}{q}\big(\mu_{jk}z_jz_k+e_{j, jk}z_kw+e_{k, jk}z_jw\big)\ \ \ \text{for}\ 1\leq j<k\leq \kappa_0, \\ \phi_{j\alpha}& = \frac{1}{q}\mu_{j\alpha}z_jz_{\alpha}\ \ \ \text{for}\ 1\leq j\leq \kappa_0<\alpha\leq n-1, \\ g& = w, \end{split} \end{equation} $

(1.1)

where

$ \begin{array}{l} q = q(z, w) = 1 + {q^{(1)}}(z) + Ew, {q^{(1)}}(z) = - 2i\sum\limits_{j = 1}^{{\kappa _0}} {\frac{{{\mu _{1j}}\overline {{e_{1, 1j}}} }}{{{\mu _1}}}} {z_j}, {e_{j, ji}} = \frac{{{\mu _j}{\mu _{1i}}}}{{{\mu _1}{\mu _{ij}}}}{e_{1, 1i}}, \\ 0 < {\mu _1} \le {\mu _2} = \cdots = {\mu _{{\kappa _0}}}, |{e_{1, 11}}{|^2} = \frac{1}{4}(\mu _2^2 - \mu _1^2), \;{\rm{Im}}(E) = - \frac{{\mu _2^2}}{{4{\mu _1}}}. \end{array} $

(1.2)

(2) Conversely, if $ F $ is defined by (1.1) and (1.2), then the map $ F $ is in $ {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^{N}) $ with $ N = n+\frac{(2n-\kappa_0-1)\kappa_0}{2} $.

Let $ F = (f, \phi, g) = (\widetilde{f}, g) = (f_1, \cdots $, $ f_{n-1} $, $ \phi_1, \cdots $, $ \phi_{N-n}, g) $ be a non-constant rational CR map from an open subset $ M $ of $ \partial{{{\mathbb H}}}_n $ into $ \partial{{{\mathbb H}}}_N $ with $ F(0) = 0 $. For each $ p\in M $ close to 0, we write $ \sigma^0_p\in \hbox{Aut}({{\mathbb H}}_n) $ for the map sending $ (z, w) $ to $ (z+z_0, w+w_0+2i \langle z, \overline{z_0} \rangle ) $ and write $ \tau^F_p\in\hbox{Aut}({{\mathbb H}}_N) $ for the map

$ \tau^F_p(z^*, w^*) = (z^*-\widetilde{f}(z_0, w_0), w^*-\overline{g(z_0, w_0)}- 2i \langle z^*, \overline{\widetilde{f}(z_0, w_0)} \rangle ). $

Then $ F $ is equivalent to $ F_p = \tau^F_p\circ F\circ \sigma^0_p = (f_p, \phi_p, g_p). $ Notice that $ F_0 = F $ and $ F_p(0) = 0 $. The following theorem is important for understanding the geometric properties of Prop$ ({{\mathbb H}}_n, {{\mathbb H}}_N) $.

Lemma 2.1   Let $ F\in {\hbox{Prop}}_2({{\mathbb H}}_n, {{\mathbb H}}_N) $ with $ 2\le n\le N $. For each $ p\in \partial {{\mathbb H}}_n $, there is an automorphism $ \tau^{**}_p\in {\hbox{Aut}}_0({{{\mathbb H}}}_N) $ such that $ F_{p}^{**}: = \tau^{**}_p\circ F_p $ satisfies the following normalization

$ f^{**}_{p} = z+{\frac{i}{ 2}}a^{**(1)}_{p}(z)w+o_{wt}(3), \ \phi_p^{**} = {\phi_p^{**}}^{(2)}(z)+o_{wt}(2), \ g^{**}_{p} = w+o_{wt}(4) $

with

$ \langle \overline{z}, a_{p}^{**(1)}(z)\rangle |z|^2 = |{\phi_p^{**}}^{(2)}(z)|^2. $

Here we use the notation $ h^{(k)}(z) $ to denote a polynomial $ h $ which has degree $ k $ in $ z $, and a function $ h(z, \overline z , u) $ is said to be quantity $ o_{wt}(m) $ if $ h(tz, t\overline z , {t^2}u)/|t{|^m} \to 0 $ uniformly for $ (z, u) $ on any small compact subset of $ 0 $ as $ t(\in \mathbb{R}) \rightarrow 0 $.

Now, we are in a position to the definition of the geometric rank. Write $ \mathcal{A}(p): = -2i(\frac{\partial^2(f_p)^{\ast\ast} _l}{\partial z_j\partial w}|_0)_{1\leq j, l\leq (n-1)} $. Then the geometric rank of $ F $ at $ p $ is defined to be the rank of the $ (n-1)\times (n-1) $ matrix $ \mathcal{A}(p) $, which is denoted by $ Rk_F(p) $. Notice that $ Rk_F(p) $ is a lower semi-continuous function on $ p $, it is independent of the choice of $ \tau^{**}_p(p) $, and depends only on $ p $ and $ F $. Define the geometric rank of $ F $ to be $ \kappa_0(F) = \max\limits_{p\in \partial{{\mathbb H}}_n} Rk_F(p) $. Define the geometric rank of $ F\in \text{Prop}_2({{{\mathbb B}}}^n, {{{\mathbb B}}}^N) $ to be the one for the map $ \rho_N^{-1}\circ F \circ \rho_n\in \text{Prop}_2({{\mathbb H}}_n, {{\mathbb H}}_N) $.

When $ 1\le \kappa_0\le n-2 $, a nice normalization was achieved by [9] and [11].

Theorem 2.2   Suppose that $ F\in {\hbox{Prop}}_3({{{\mathbb H}}}_n, {{{\mathbb H}}}_N) $ has geometric rank $ 1\le\kappa_0\le n-2 $ with $ F(0) = 0 $. Then there are $ \sigma\in \hbox{Aut}({{{\mathbb H}}}_n) $ and $ \tau\in \hbox{Aut}({{{\mathbb H}}}_N) $ such that $ \tau\circ F\circ \sigma $ takes the following form, which is still denoted by $ F = (f, \phi, g) $ for convenience of notation

$ \begin{equation} \left\{ \begin{array}{l} f_l = \sum\limits_{j = 1}^{\kappa_0}z_jf_{lj}^*(z, w), \ l\le\kappa_0, \\ f_j = z_j, \ \text{for} \ \kappa_0+1\leq j\leq n-1, \\ \phi_{lk} = \mu_{lk}z_lz_k+\sum\limits_{j = 1}^{\kappa_0}z_j\phi^*_{lkj}\ \text{for } \ \ (l, k)\in {\cal S}_0, \\ \phi_{lk} = O_{wt}(3), \ \ (l, k)\in {\cal S}_1, \\ g = w, \\ f_{lj}^*(z, w) = \delta_l^j+\frac{i\delta_{l}^j\mu_l}{2}w+b_{lj}^{(1)}(z)w+O_{wt}(4), \\ \phi^*_{lkj}(z, w) = O_{wt}(2), \ \ (l, k)\in {\cal S}_0, \\ \phi_{lk} = \sum\limits_{j = 1}^{\kappa_0}z_j\phi_{lkj}^* = O_{wt}(3)\ \ {\hbox{for}}\ (l, k)\in {\cal S}_1. \end{array}\right. \end{equation} $

(2.1)

Here, for $ 1\le \kappa_0\le n-2 $, we write $ {\cal S} = {\cal S}_0\cup {\cal S}_1 $, the index set for all components of $ \phi $, where

$ \begin{eqnarray*} && {\cal S}_{0} = \{(j, l): 1\le j\leq \kappa_0, 1\leq l\leq n-1, j\leq l\}, \\ && {\cal S}_1 = \{(j, l): j = \kappa_0+1, \kappa_0+1\le l \le \kappa_0+N-n-\frac{(2n-\kappa_0-1)\kappa_0}{2} \}, \end{eqnarray*} $

and

$ \begin{equation} \mu_{jl} = \begin{cases}\sqrt{\mu_j+\mu_l} &\ {\hbox{for}}\ j<l\le \kappa_0; \\ \sqrt{\mu_j} & {\hbox{if}} \ j\le \kappa_0 < l\ {\hbox{or}}\ {\hbox{if}}\ j = l\le \kappa_0. \end{cases} \end{equation} $

(2.2)

Here we can assume $ \mu_1\leq \mu_2\leq\cdots\leq \mu_{\kappa_0} $. By [2], we can further assume that

$ \begin{equation} e_{1, 1\alpha} = 0\; \; \ \text{for}\ \kappa_0+1\leq \alpha\leq n-1. \end{equation} $

(2.3)

A map $ F\in {\hbox{Rat}}({{\mathbb H}}_n, {{\mathbb H}}_N) $ is called a normalized map if $ F $ takes form (2.1) with (2.3). Notice that any $ F\in {\hbox{Rat}}({{\mathbb H}}_n, {{\mathbb H}}_N) $ can not be further normalized except some rotations in $ \phi_{lk} $ with $ (l, k)\in \mathcal{S}_1 $.

Next we introduce some notations that will be use during the proof of our main theorem. Let $ F = (f, \phi, g)\in {\hbox{Rat}}({{\mathbb B}}^n, {{\mathbb B}}^N) $ be as in Theorem 1.1. Denote by $ \sharp(A) $ the number of elements in the set $ A $. Then we have $ N = \sharp(f)+\sharp(\phi)+\sharp(g) $ and $ \sharp(\phi) = \sharp({\cal S}_0)+\sharp({\cal S}_1) $. Notice that $ \sharp(f) = n-1 $, $ \sharp(g) = 1 $, $ \sharp ({\cal S}_0) = n\kappa_0-\frac{(\kappa_0+1)\kappa_0}{2} $. Hence $ \sharp({\cal S}_1) = N-n-\sharp ({\cal S}_0) = 0 $, which means we do not have the $ {\cal S}_1 $ part in the map.

For any rational holomorphic map $ H = \frac{(P_1, \cdots, P_m)}{Q} $ on $ {{\mathbb C}}^n $, where $ \{P_j, Q\} $ are relatively prime holomorphic polynomials, the degree of $ H $ is defined to be

$ {\hbox{deg}}(H): = \max\{\deg(P_j), {\hbox{deg}}(Q), 1\le j\le m\}. $

For any holomorphic map $ P(z, w) $, we will use the following notations

$ P(z, w) = \sum\limits_{|\alpha | = j, k \ge 0} {{P^{(\alpha , k)}}} {z^\alpha }{w^k} = \sum\limits_{j, k} {{P^{(j, k)}}} (z) \cdot {w^k}, $

where $ P^{(j, k)}(z) $ is a homogeneous polynomial with respect to $ z $ of degree $ j $.

This section is devoted to the proof of Theorem 1.1.

By [15, Theorem 1.1], we know deg$ F = 2 $. From (4.2)–(4.4) of [15], we obtain

$ \begin{align} &e_{i, jk} = 0, \ \ \ \forall j, k\neq i\ \text{or }\ \kappa_0+1\leq j\leq n-1\ \text{or }\ \kappa_0+1\leq k\leq n-1, \end{align} $

(3.1)

$ \begin{align} &\mu_j\mu_{il} e_{l, il} = \mu_l \mu_{ij} e_{j, ij}\ \text{for}\ 1\leq i, j, l\leq \kappa_0. \end{align} $

(3.2)

From (5.3) of [15], we get

$ \begin{array}{l} {f_\mu }(z, 0) = {z_\mu }\;{\rm{for}}\;1 \le \mu \le n - 1, \\ {\phi _{jj}}(z, 0) = \frac{{{\mu _{jj}}z_j^2}}{{1 - 2i\sum\limits_{j = 1}^{{\kappa _0}} {\overline {{A_j}} {z_j}} }}\;{\rm{for}}\;1 \le j \le {\kappa _0}, \\ {\phi _{jk}}(z, 0) = \frac{{{\mu _{jk}}{z_j}{z_k}}}{{1 - 2i\sum\limits_{j = 1}^{{\kappa _0}} {\overline {{A_j}} {z_j}} }}{\rm{for}}\;1 \le j < k \le {\kappa _0}, \\ {\phi _{j\alpha }}(z, 0) = \frac{{{\mu _{j\alpha }}{z_j}{z_\alpha }}}{{1 - 2i\sum\limits_{j = 1}^{{\kappa _0}} {\overline {{A_j}} {z_j}} }}{\rm{for}}\;1 \le j \le {\kappa _0} < \alpha \le n - 1. \end{array} $

(3.3)

Here we have set $ A_j = \frac{\mu_{1j}e_{1, 1j}}{\mu_1} $.

Notice that the degree of $ F $ must be $ 2 $. Together with the normalization properties in (2.1), the expression of $ F $ must have the following form

$ \begin{equation} \begin{split} f_j& = \frac{1}{q}\big(z_jq+\frac{\mu_j}{2}iz_jw\big)\ \ \ \text{for}\ 1\leq j\leq \kappa_0, \ f_k = z_k\ \ \ \text{for}\ \kappa_0+1\leq k\leq n-1, \\ \phi_{jj}& = \frac{1}{q}\big(\mu_{jj}z_j^2+e_{j, jj}z_jw\big)\ \ \ \text{for}\ 1\leq j\leq \kappa_0, \\ \phi_{jk}& = \frac{1}{q}\big(\mu_{jk}z_jz_k+e_{j, jk}z_kw+e_{k, jk}z_jw\big)\ \ \ \text{for}\ 1\leq j<k\leq \kappa_0, \\ \phi_{j\alpha}& = \frac{1}{q}\mu_{j\alpha}z_jz_{\alpha}\ \ \ \text{for}\ 1\leq j\leq \kappa_0<\alpha\leq n-1, \\ g& = w. \end{split} \end{equation} $

(3.4)

Here we have set $ q = 1 - 2i\sum\limits_{j = 1}^{{\kappa _0}} {\overline {{A_j}} {z_j}} + Ew $. In what follows, we write $ {q^{(1)}}(z) = - 2i\sum\limits_{j = 1}^{{\kappa _0}} {\overline {{A_j}} {z_j}} $.

Since $ F $ maps $ {\partial} \mathbb{B}^n $ to $ {\partial} \mathbb{B}^N $, we have the basic equation

$ \begin{equation} \text{Im}w = |f|^2+|\phi|^2\ \text{for}\ \text{Im}w = |z|^2. \end{equation} $

(3.5)

Substituting (3.4) into this equation, we obtain, for $ \text{Im}w = |z|^2 $, the following

$ \begin{equation} \begin{split} |z|^2|q|^2& = \sum\limits_{j = 1}^{\kappa_0}|z_jq+\frac{i}{2}\mu_jz_jw|^2+\sum\limits_{j = \kappa_0+1}^{n-1}|z_jq|^2 +\sum\limits_{j = 1}^{\kappa_0}|\mu_{jj}z_j^2+e_{j, jj}z_jw|^2\\ &+\sum\limits_{1\leq j<k\leq \kappa_0}|\mu_{jk}z_jz_k+e_{j, jk}z_kw+ e_{k, kj}z_jw|^2+\sum\limits_{1\leq j\leq \kappa_0<\alpha\leq n-1}|\mu_{j\alpha}z_jz_\alpha|^2. \end{split} \end{equation} $

(3.6)

After a quick simplification, we obtain

$ \begin{array}{l} \sum\limits_{j = 1}^{{\kappa _0}} | {z_j}{|^2}\{ 2{\rm{Re}}(\frac{i}{2}{\mu _j}\overline w (1 + {q^{(1)}}(z) + Ew)) - \frac{{\mu _j^2}}{4}({u^2} + |z{|^4})\} = \sum\limits_{j = 1}^{{\kappa _0}} | {\mu _{jj}}z_j^2 + {e_{j, jj}}{z_j}w{|^2}\\ + \sum\limits_{1 \le j < k \le {\kappa _0}} | {\mu _{jk}}{z_j}{z_k} + {e_{j, jk}}{z_k}w + {e_{k, kj}}{z_j}w{|^2} + \sum\limits_{j = 1}^{{\kappa _0}} | {\mu _j}|{z_j}{|^2} \cdot (|z{|^2} - \sum\limits_{j = 1}^{{\kappa _0}} | {z_j}{|^2}). \end{array} $

(3.7)

Consideration of the Degree 4 Terms By considering the degree 4 terms, we get

$ \begin{equation} \begin{split} &\sum\limits_{j = 1}^{\kappa_0}|z_j^2|\Big\{2\text{Re}\Big(\frac{i}{2}\mu_j (-i|z|^2)+\frac{i}{2}\mu_ju(q^{(1)}(z)+Eu)\Big) -\frac{\mu_j^2}{4}u^2\Big\}\\ = &\sum\limits_{j = 1}^{\kappa_0}|\mu_{jj}z_j^2+e_{j, jj}z_ju|^2+\sum\limits_{1\leq j<k\leq \kappa_0}|\mu_{jk}z_jz_k+e_{j, jk}z_ku+ e_{k, kj}z_ju|^2\\ &+\sum\limits_{j = 1}^{\kappa_0}\mu_{j}z_j|^2\cdot \Big(|z|^2-\sum\limits_{j = 1}^{\kappa_0}\mu_{j}|z_j|^2\Big). \end{split} \end{equation} $

(3.8)

Collecting $ {z^\alpha }{\overline z ^\beta }{u^2} $ term with $ |\alpha| = |\beta| = 1 $, we obtain

$ \begin{equation} \begin{split} &\sum\limits_{j = 1}^{\kappa_0}|z_j^2|\Big\{2\text{Re}(\frac{i}{2}\mu_ju\cdot Eu)-\frac{\mu_j^2}{4}u^2\Big\} = \sum\limits_{j = 1}^{\kappa_0}|e_{j, jj}z_ju|^2+\sum\limits_{1\leq j<k\leq \kappa_0}|e_{j, jk}z_ku+ e_{k, kj}z_ju|^2. \end{split} \end{equation} $

(3.9)

By considering the coefficients of $ |z_j|^2u $ terms with $ 1\leq j\leq \kappa_0 $ and $ z_j{\overline{z_k}}u $ terms with $ 1\leq j< k\leq \kappa_0 $, respectively, we get

$ \begin{equation} \begin{split} \text{Re}(i\mu_jE)-\frac{1}{4}\mu_j^2 = \sum\limits_{k = 1}^{\kappa_0}|e_{j, jk}|^2, \ e_{j, jk}\cdot e_{k, jk} = 0\ \text{for}\ 1\leq j< k\leq \kappa_0. \end{split} \end{equation} $

(3.10)

Combining this with (3.2), we know only one of $ e_{1, 1j} $ $ 1\leq j\leq \kappa_0 $ is non zero. From (3.2) and (3.10), we obtain

$ \begin{equation*} \begin{split} \frac{1}{4}\mu_j = \text{Re}(i\mu_jE)-\sum\limits_{k = 1}^{\kappa_0}\frac{1}{\mu_j}\big|\frac{\mu_j\mu_{1k}}{\mu_1\mu_{kj}}e_{1, 1k}\big|^2. \end{split} \end{equation*} $

Together with $ \mu_1\leq \cdots\leq \mu_{\kappa_0} $, we get $ e_{1, 12} = \cdots = e_{1, 1\kappa_0} = 0 $ and $ \mu_2 = \cdots = \mu_{\kappa_0} $. Furthermore, we obtain

$ \begin{equation} \begin{split} &\frac{1}{4}(\mu_1-\mu_2) = \frac{1}{\mu_2}|e_{2, 12}|^2-\frac{1}{\mu_1}|e_{1, 11}|^2 = \frac{\mu_2-(\mu_1+\mu_2)}{\mu_1(\mu_1+\mu_2)}|e_{1, 11}|^2 = -\frac{1}{\mu_1+\mu_2}|e_{1, 11}|^2.\\ &\text{Im}(E) = -\frac{1}{4}\mu_1-\frac{1}{\mu_1}\cdot \frac{\mu_2^2-\mu_1^2}{4} = -\frac{\mu_2^2}{4\mu_1}. \end{split} \end{equation} $

(3.11)

Thus

$ \begin{equation} \begin{split} &|e_{1, 11}|^2 = \frac{1}{4}(\mu_2^2-\mu_1^2), \ \ \text{Im}(E) = -\frac{\mu_2^2}{4\mu_1}. \end{split} \end{equation} $

(3.12)

Collecting $ z^\alpha {\overline{z}}^{\beta} $ term with $ |\alpha| = |\beta| = 2 $, we obtain

$ \begin{equation} \begin{split} \sum\limits_{j = 1}^{\kappa_0}|z_j^2|\text{Re}\Big(\mu_j|z|^2\Big) = &\sum\limits_{j = 1}^{\kappa_0}\mu_j|z_j|^4 +\sum\limits_{1\leq j<k\leq \kappa_0}(\mu_j+\mu_k)|z_jz_k|^2\\ &+\sum\limits_{j = 1}^{\kappa_0}\mu_j|z_j|^2\cdot (|z|^2-\sum\limits_{k = 1}^{\kappa_0}|z_k|^2). \end{split} \end{equation} $

(3.13)

This equation is a trivial formula.

Collecting $ z^\alpha {\overline{z}}^{\beta}u $ term with $ |\alpha|+|\beta| = 3 $, we obtain

$ \begin{equation} \begin{split} &\sum\limits_{j = 1}^{\kappa_0}|z_j^2|\cdot 2\text{Re}\Big(\frac{i}{2}\mu_jq^{(1)}(z)\Big)\\ = &\sum\limits_{j = 1}^{\kappa_0}2\text{Re}\Big(\mu_{jj}{\overline{z_j^2}}\cdot e_{j, jj}z_j\Big)+ \sum\limits_{1\leq j<k\leq \kappa_0}2\text{Re}\Big(\mu_{jk}{\overline{z_jz_k}}\cdot \big(e_{j, jk}z_k+e_{k, jk}z_j\big)\Big). \end{split} \end{equation} $

(3.14)

We can easily derive from this equation the following

$ \begin{equation} \begin{split} -\frac{i}{2}\mu_j\cdot 2i\frac{\mu_{ij}}{\mu_j}e_{1, 1j} = \mu_{jj}e_{j, jj}, \ \ e_{j, jk}\cdot e_{k, jk} = 0\ \text{for}\ 1\leq j<k\leq \kappa_0. \end{split} \end{equation} $

(3.15)

These equations were achieved before and we do not get new equations.

Consideration of the Degree 5 Terms By considering the degree 5 terms, we get

$ \begin{equation} \begin{split} &\sum\limits_{j = 1}^{\kappa_0}|z_j^2|\cdot 2\text{Re}\Big\{\frac{i}{2}\mu_j(-i|z|^2)\cdot q^{(1)}(z)\Big\} = \sum\limits_{j = 1}^{\kappa_0} 2\text{Re}\Big\{\mu_{jj}{\overline{z_j}}^2\cdot e_{j, jj}z_j(i|z|^2)\Big\}\\ &+\sum\limits_{1\leq j<k\leq \kappa_0}2\text{Re} \Big\{\mu_{jk}{\overline{z_jz_k}}\cdot \Big(e_{j, jk}z_j(i|z|^2)+e_{k, kj}z_k(i|z|^2)\Big)\Big\}. \end{split} \end{equation} $

(3.16)

Deleting $ |z|^2 $ from both sides and consider the terms $ z_j{\overline{z_j}}^2 $ and $ |z_j^2{\overline{z_k}}| $, respectively, we can easily derive from this equation the following

$ \begin{equation} \begin{split} -\frac{i}{2}\mu_j\cdot 2i\frac{\mu_{ij}}{\mu_j}e_{1, 1j} = \mu_{jj}e_{j, jj}, \ \ e_{j, jk}\cdot e_{k, jk} = 0\ \text{for}\ 1\leq j<k\leq \kappa_0. \end{split} \end{equation} $

(3.17)

As before, we do not get new equations.

Consideration of the Degree 6 Terms By considering the degree 6 terms, we get

$ \begin{equation} \begin{split} &\sum\limits_{j = 1}^{\kappa_0}|z_j^2|\cdot 2\text{Re}\Big\{\frac{i}{2}\mu_j\cdot E|z|^4 -\frac{\mu_j^2}{4}|z|^4\Big\}\\ = &\sum\limits_{j = 1}^{\kappa_0} |e_{j, jj}z_j|^2\cdot|z|^4+\sum\limits_{1\leq j<k\leq \kappa_0} |e_{j, jk}z_j+e_{k, kj}z_k|^2\cdot |z|^4. \end{split} \end{equation} $

(3.18)

Notice that this equation is equivalent to (3.10).

In conclusion, the map $ F $ must be of form (1.1) with relations in (1.2). The arguments above also showed that this map is indeed a map in Rat$ (\mathbb{B}^n, \mathbb{B}^{N}) $. This completes the proof of Theorem 1.1.