Explicit Reconstruction of Riemann Surface with Given Boundary in Complex Projective Space

Abstract

In this paper we propose a numerically realizable method for reconstruction of a complex curve with known boundary and without compact components in complex projective space.

1 Introduction

Let us denote by ${\mathbb {C}} {\mathrm {P}}^2$ the complex projective space with homogeneous coordinates $(w_0 : w_1 : w_2)$. Let a real closed rectifiable, oriented curve $\gamma$ in ${\mathbb {C}} {\mathrm {P}}^2$ be the boundary of a complex curve $X \subset {\mathbb {C}} {\mathrm {P}}^2$ with notation $\gamma = bX$. Without restriction of generality we suppose that the following conditions of general position hold:

$$\begin{aligned} (0:1:0) \not \in X, \quad w_0 |_\gamma \ne 0. \end{aligned}$$

Put ${\mathbb {C}}^2 = \{ w \in {\mathbb {C}} {\mathrm {P}}^2 :w_0 \ne 0 \}$ with coordinates $z_1 = \frac{w_1}{w_0}$, $z_2 = \frac{w_2}{w_0}$. For almost all $\xi = (\xi _0,\xi _1) \in ({\mathbb {C}}^2)^*$ the points of intersection of $X$ with complex line ${\mathbb {C}}^1_\xi = \{ z \in {\mathbb {C}}^2 :\xi _0 + \xi _1 z_1 + z_2 = 0 \}$ form a finite set of points

$$\begin{aligned} \bigl ( z_1^{(j)}(\xi ), z_2^{(j)}(\xi ) \bigr ) = \bigl ( h_j(\xi _0,\xi _1), -\xi _0 - \xi _1 h_j(\xi _0,\xi _1) \bigr ), \quad j=1,\ldots ,N_+(\xi ). \end{aligned}$$

By Darboux’s lemma [3, 5] functions $\{h_j\}$ satisfy the equations

$$\begin{aligned} \frac{\partial h_j(\xi _0,\xi _1)}{\partial \xi _1} = h_j(\xi _0,\xi _1) \frac{\partial h_j(\xi _0,\xi _1)}{\partial \xi _0}, \quad j = 1,\ldots ,N_+(\xi ), \end{aligned}$$

(1)

which are often called shock-wave equations or Riemann–Burgers equations. In this interpretation $\xi _1$ is the time variable and $\xi _0$ is the space variable.

The following Cauchy-type formula from [5] plays the essential role in reconstruction of $X$ through $\gamma$:

$$\begin{aligned}&G_m(\xi _0,\xi _1) {\mathop {=}^{\mathrm{def}}} \frac{1}{2\pi i} \int \limits _\gamma z_1^m (\xi _0+\xi _1 z_1 + z_2)^{-1} d(\xi _0+\xi _1 z_1 + z_2) \nonumber \\&\qquad \qquad \qquad =\sum \limits _{j=1}^{N_+(\xi )} h_j^m(\xi _0,\xi _1) + P_m(\xi _0,\xi _1), \quad m=0,1,\ldots , \end{aligned}$$

(2)

where $N_+(\xi ) = N_+(\xi _0,\xi _1)$ is the number of points of intersection (multiplicities taken into account) of $X$ with complex line ${\mathbb {C}}^1_\xi$, $P_m(\xi _0,\xi _1)$ is a polynomial of degree at most $m$ with respect to $\xi _0$. In addition, $P_0(\xi _0,\xi _1) = -N_{-}$, where $N_{-}$ is the number of points of intersection of $X$ with infinity $\{ w \in {\mathbb {C}} {\mathrm {P}}^2 :w_0 = 0\}$,

$$\begin{aligned} P_1(\xi _0,\xi _1) = \sum \nolimits _{k=1}^{N_{-}(\xi )} \frac{a_k \xi _0 - b_k}{a_k \xi _1 + 1}, \end{aligned}$$

(3)

$a_k = w_2(q_k)$, $b_k = \frac{dw_2}{dw_0}(q_k)$, where $q_k$, $k=1$, ..., $N_{-}$, are the points of intersection of $X$ with infinity $\{w \in {\mathbb {C}} {\mathrm {P}}^2 :w_0 = 0\}$. In particular, the following corollary of (2) holds:

$$\begin{aligned} G_0(\xi _0,\xi _1) = \frac{1}{2\pi i} \int \limits _\gamma \frac{d(\xi _0+\xi _1 z_1 + z_2)}{\xi _0 + \xi _1 z_1 + z_2} = N_+(\xi )-N_-. \end{aligned}$$

(4)

Let further $\xi _1 = 0$ and let $\pi _2 :{\mathbb {C}}^2 \rightarrow {\mathbb {C}}$ be the projection on the second factor: $\pi _2(z_1,z_2)=-z_2$. We have $\pi _2 \gamma \subset {\mathbb {C}}$, ${\mathbb {C}}\setminus \pi _2 \gamma = \cup _{l=0}^L \Omega _l$, where $\{\Omega _l\}$ are the connected components of ${\mathbb {C}}\setminus \pi _2 \gamma$. For every component $\Omega _l$ the number of points of intersection of $X$ with line $z_2 = -\xi _0$, $\xi _0 \in \Omega _l$, multiplicities taken into account, will be denoted by $\mu _l = N_+(\xi _0,0)$. Let $\Omega _0$ denote the unbounded component of set ${\mathbb {C}}\setminus \pi _2 \gamma$. From the definition of $N_{\pm }$ it follows that

$$\begin{aligned} \mu _0 = N_+(\xi _0,0) = N_{-}, \quad \xi _0 \in \Omega _0. \end{aligned}$$

(5)

Assume that complex curve $X$ does not contain compact components, or equivalently, satisfies the following condition of minimality:

Condition of minimality (*) is a condition of general position and is fulfilled for $X$ if, for example, every irreducible component of $X$ is a transcendental complex curve. Note that from theorems of Chow [2] and Harvey, Shiffman [7] it follows that an arbitrary complex curve $\widetilde{X} \subset {\mathbb {C}} {\mathrm {P}}^2$ with condition $b\widetilde{X} = bX$ admits the unique representation $\widetilde{X} = X \cup V$, where $X$ is a curve with condition of minimality (*), and $V$ is a compact algebraic curve, possibly with multiple components.

The main result of [4] gives a solution of the important problem of J. King [9], when a real curve $\gamma \subset {\mathbb {C}} {\mathrm {P}}^2$ is the boundary of a complex curve $X \subset {\mathbb {C}} {\mathrm {P}}^2$. Let $\gamma \subset {\mathbb {C}}^2 \subset {\mathbb {C}} {\mathrm {P}}^2$. Then $\gamma = bX$ for some open connected complex curve $X$ in ${\mathbb {C}} {\mathrm {P}}^2$ if and only if on a neighborhood $W_{\xi ^*}$ of some point $\xi ^* \in ({\mathbb {C}}^2)^*$ one can find mutually distinct holomorphic functions $h_1$, ..., $h_p$ satisfying shock-wave Eq. (1) and also the equation

$$\begin{aligned} \frac{\partial ^2}{\partial \xi _0^2} \bigl ( G_1(\xi _0,\xi _1)-\sum \nolimits _{j=1}^p h_j(\xi _0,\xi _1) \bigr ) = 0, \quad \xi = (\xi _0,\xi _1) \in W_{\xi ^*}. \end{aligned}$$

In this work in development of [4, 5] we obtained a numerically realizable algorithm for reconstruction of complex curve $X\subset {{\mathbb {C}}}P^2$ with known boundary and with condition of minimality. This algorithm permits, in particular, making applicable the result of [8] about the principal possibility to reconstruct topology and conformal structure of a two-dimensional bordered surface $X$ in ${\mathbb {R}}^3$ with constant scalar conductivity from measurements on $bX$ of electric current densities, being created by three potentials in general position.

Our algorithm depends on parameter $\mu _0 = N_{\pm }(\xi _0,0)$, $\xi _0 \in \Omega _0$. It was tested on many examples and admits simple and complete justification for $\mu _0 = 0$, $1$, $2$. Despite a cumbersome description for $\mu _0 \ge 3$, the algorithm shows that there are no obstacles for its justification and numerical realization for any $\mu _0 \ge 0$. Moreover, in Theorem 3.2 we propose a method for finding parameter $\mu _0$ in terms of $\gamma$. This makes the algorithm much more applicable.

The preliminary version [1] of this work appeared in HAL (http://hal.archives-ouvertes.fr/hal-00912925) 2013, 2014.

2 Cauchy-Type Formulas and Riemann–Burgers Equations

Let us give at first a new proof of the Cauchy-type formula (2) from [5], permitting us to obtain explicit expressions for functions $P_m(\xi _0,\xi _1)$.

Theorem 2.1

Let $X \subset {\mathbb {C}} {\mathrm {P}}^2 \setminus [0:1:0]$ be a complex curve without compact components, $\gamma = bX \subset {\mathbb {C}}^2$ be a real rectifiable oriented curve. Suppose that for almost all $\xi \in ({\mathbb {C}}^2)^*$ all the points of intersection of $X$ with ${\mathbb {C}} {\mathrm {P}}^1_\xi$ have multiplicity at most one. Then the following equalities are fulfilled:

$$\begin{aligned} G_m(\xi _0,\xi _1) = \sum \nolimits _{j=1}^{N_+(\xi )} h_j^m(\xi _0,\xi _1) + P_m(\xi _0,\xi _1), \quad \xi = (\xi _0,\xi _1) \in ({\mathbb {C}}^2)^*, \; m \ge 1, \end{aligned}$$

(6)

where $P_m(\xi _0,\xi _1)$ is a polynomial of degree at most $m$ with respect to $\xi _0$ of the following form:

$$\begin{aligned} P_m(\xi _0,\xi _1)&= \sum \nolimits _{s=1}^{\mu _0}\sum \nolimits _{k=0}^{m-1} \sum \nolimits _{i_1+\cdots +i_m=k} \frac{\frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s)\cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s)}{(m-k-1)} \frac{d^{m-k}}{dw_0^{m-k}}\\&\times \ln (\xi _0 w_0 + \xi _1 w_1 + w_2)|_{q_s}\\&-\sum \nolimits _{s=1}^{\mu _0}\sum \nolimits _{i_1+\cdots +i_m = m} \frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s) \cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s), \end{aligned}$$

where $q_s \in X \cap \{w \in {\mathbb {C}} {\mathrm {P}}^2 :w_0 = 0\}$. In particular, if $\mu _0 = 0$ then $P_m \equiv 0$.

Remark 2.1

In the exceptional case, when $[0:1:0] \in X$, the term $P_m(\xi _0,\xi _1)$ in (6) need not be a polynomial with respect to $\xi _0$ in general.

Proof

Put $\widetilde{g} = \xi _0 w_0 + \xi _1 w_1 + w_2$ and $g = \frac{\widetilde{g}}{w_0} = \xi _0 + \xi _1 z_1 + z_2$. Consider differential forms

Then $G_m(\xi ) = \frac{1}{2\pi i} \int _\gamma \omega _m$. Let us compute this integral explicitly. Denote by $p_j$, $j=1$, ..., $N_+(\xi )$ the points of intersection of $X$ with ${\mathbb {C}} {\mathrm {P}}^1_\xi$, and by $q_s$, $s=1$, ..., $\mu _0$ the points of intersection of $X$ with infinity $\{ w \in {\mathbb {C}} {\mathrm {P}}^2 :w_0 = 0 \}$. Denote by $B_j^\varepsilon$ the intersection of $X$ with the ball of radius $\varepsilon$ in ${\mathbb {C}} {\mathrm {P}}^2$ centered at $p_j$ and by $D_s^\varepsilon$ the intersection of $X$ with the ball of radius $\varepsilon$ centered at $q_s$. The restriction of form $\omega _m$ on $X$ is meromorphic with poles at points $p_j$ and $q_s$. Thus the following equality is valid:

$$\begin{aligned} G_m(\xi ) = \frac{1}{2\pi i}\int \limits _\gamma \omega _m = \sum \nolimits _{j=1}^{N_+(\xi )} \frac{1}{2\pi i}\int \limits _{bB_j^\varepsilon } \omega _m + \sum \nolimits _{s=1}^{\mu _0} \frac{1}{2\pi i} \int \limits _{bD_s^\varepsilon } \omega _m. \end{aligned}$$

If $\mu _0 = 0$, then the second group of terms is absent. The integral $\int _{bB_j^\varepsilon } \omega _m$ can be calculated as a residue at the first order pole:

$$\begin{aligned} \int \limits _{bB_j^\varepsilon } \omega _m = \int \limits _{bB_j^\varepsilon } z_1^m \frac{d\widetilde{g}}{\widetilde{g}} - \int \limits _{bB_j^\varepsilon } \frac{w_1^m}{w_0^{m+1}}dw_0 = \int \limits _{bB_j^\varepsilon } z_1^m \frac{d\widetilde{g}}{\widetilde{g}} = 2\pi i \, h_j^m(\xi ). \end{aligned}$$

Let $\mu _0 > 0$. Computation of integral $\int _{bD_s^\varepsilon } \omega _1$ will be done in two steps. Let us calculate first $\int _{bD_s^\varepsilon } \frac{w_1^m}{w_0^{m+1}} dw_0$. Consider the expansion of $w_1(w_0)$ into power series in $w_0$ in the neighborhood of point $q_s$:

$$\begin{aligned} w_1(w_0) = w_1(q_s)+\frac{dw_1}{dw_0}(q_s) w_0 + \frac{d^2 w_1}{dw_0^2}(q_s) w_0^2 + \cdots . \end{aligned}$$

Note further that

$$\begin{aligned} w_1^m(w_0) = \sum \nolimits _{k=0}^\infty \sum \nolimits _{i_1+\cdots +i_m=k} \frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s) \cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s) w_0^k. \end{aligned}$$

The coefficient near $w_0^m$ can be presented in the form

$$\begin{aligned} \int \limits _{bD_s^\varepsilon } \frac{w_1^m}{w_0^{m+1}} dw_0 = 2 \pi i \sum \nolimits _{i_1+\cdots +i_m = m} \frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s) \cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s). \end{aligned}$$

Now we can calculate the integral $\int _{bD_s^\varepsilon } \frac{w_1^m}{w_0^m} \frac{d\widetilde{g}}{\widetilde{g}}$. Using relation $d\widetilde{g} = \frac{d\widetilde{g}}{dw_0} dw_0$ and expansion of $w_1(w_0)$ into power series in $w_0$ we obtain:

$$\begin{aligned} \int \limits _{bD_s^\varepsilon } \frac{w_1^m}{w_0^m}\frac{d\widetilde{g}}{\widetilde{g}}&= \!\int _{bD_s^\varepsilon } \frac{1}{w_0^m} \biggl ( w_1(q_s)\!+\!\frac{dw_1}{dw_0}(q_s)w_0+\frac{d^2 w_1}{dw_0^2}(q_s)w_0^2+\cdots \biggr )^m \frac{d\widetilde{g}}{dw_0} \frac{1}{\widetilde{g}} dw_0\\&= \sum \nolimits _{k=0}^{\infty } \int _{bD_s^\varepsilon } \sum \nolimits _{i_1+\cdots +i_m=k} \frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s)\cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s) w_0^{k-m} \frac{d\widetilde{g}}{dw_0}\frac{1}{\widetilde{g}} dw_0\\&= \sum \nolimits _{k=0}^{m-1} \int _{bD_s^\varepsilon } \sum \nolimits _{i_1+\cdots +i_m=k} \frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s)\cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s) w_0^{k-m} \frac{d\widetilde{g}}{dw_0}\frac{1}{\widetilde{g}} dw_0\\&= \sum \nolimits _{k=0}^{m-1} \frac{2\pi i}{(m-k-1)} \sum \nolimits _{i_1+\cdots +i_m=k} \frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s)\cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s)\\&\times \lim _{w_0\rightarrow 0} \frac{d^{m-k-1}}{dw_0^{m-k-1}} \biggl ( \frac{d\widetilde{g}}{dw_0} \frac{1}{\widetilde{g}} \biggr ). \end{aligned}$$

From here, taking into account the relation $\frac{d\widetilde{g}}{dw_0}\frac{1}{\widetilde{g}} = \frac{d\ln \widetilde{g}}{dw_0}$, we obtain, finally

$$\begin{aligned} \int \limits _{bD_s^\varepsilon } \frac{w_1^m}{w_0^m} \frac{d\widetilde{g}}{\widetilde{g}}&= 2\pi i \sum \nolimits _{k=0}^{m-1} \sum \nolimits _{i_1+\cdots +i_m=k} \frac{\frac{d^{i_1}w_1}{dw_0^{i_1}}(q_s)\cdots \frac{d^{i_m}w_1}{dw_0^{i_m}}(q_s)}{(m-k-1)} \frac{d^{m-k}}{dw_0^{m-k}}\\&\times \ln (\xi _0 w_0 + \xi _1 w_1 + w_2)|_{q_s}. \end{aligned}$$

It is a polynomial of degree at most $m$ with respect to $\xi _0$. $\square$

We will also use the following result, giving the effective characterization of functions $\{ h_j(\xi _0,\xi _1) \}$ and $\{P_m(\xi _0,\xi _1)\}$ satisfying Riemann–Burgers equations for $\{h_j\}$ and system (6) for $\{h_j\}$ and $\{P_m\}$. Denote by $\Omega _l^{(k)}$ the infinitesimal neighborhood of order $k$ of the set $\Omega _l$ in $\{(z_1,z_2) :z_1 \in \Omega _l, \; z_2 \in \mathbb C\}$.

Theorem 2.2

Let $X \subset {\mathbb {C}} {\mathrm {P}}^2 \setminus [0:1:0]$ be a complex curve without compact components, $\gamma = bX \subset {\mathbb {C}}^2$. Fix $l \in \{0,\ldots ,L\}$. Suppose that functions $\widehat{h}_j$, $j = 1$, ..., $\mu _l$, are mutually distinct and analytic in $\Omega ^{(1)}_l$ and satisfy the Riemann–Burgers equation in $\xi \in \Omega ^{(1)}_l$:

$$\begin{aligned} \frac{\partial \widehat{h}_j}{\partial \xi _1}(\xi ) = \widehat{h}_j(\xi ) \frac{\partial \widehat{h}_j}{\partial \xi _0}(\xi ), \quad j = 1,\ldots , \mu _l. \end{aligned}$$

(7)

Then the functions $\widehat{h}_j$, $j = 1$, $\ldots$, $\mu _l$, satisfy the system

$$\begin{aligned} G_m(\xi ) = \sum \nolimits _{j=1}^{\mu _l} \widehat{h}_j^m(\xi ) + \widehat{P}_m(\xi ), \quad \xi \in \Omega ^{(1)}_l, \quad m = 1, 2, \ldots , \end{aligned}$$

(8)

where $\widehat{P}_m$, $m = 1$, $2$, ..., are some analytic functions in $\Omega ^{(1)}_l$, being polynomials of degree at most $m$ with respect to $\xi _0$, if and only if the functions $\widehat{h}_j$, $j=1$, ..., $\mu _l$, satisfy the equation

$$\begin{aligned} 0 = \frac{\partial ^2}{\partial \xi _0^2} \biggl ( G_1(\xi ) - \sum \nolimits _{j=0}^{\mu _l} \widehat{h}_j(\xi ) \biggr ), \quad \xi \in \Omega ^{(1)}_l. \end{aligned}$$

(9)

Moreover, for minimal $\{\mu _l\}$ with properties (7)–(9) there exists the unique set of functions $\widehat{h}_j$, $j=1$, ..., $\mu _l$, satisfying the equivalent conditions (8)–(9) and the unique set of functions $\widehat{P}_m$, $m = 1$, $2$, ..., from condition (8). Furthermore, $\widehat{h}_j = h_j$, $\widehat{P}_m = P_m$ for $j = 1$, ..., $\mu _l$, $m = 1$, ..., where functions $h_j$ and $P_m$ are defined in Theorem (2.1).

Proof

Necessity. From (8) it follows that

$$\begin{aligned} G_1(\xi ) - \sum \nolimits _{j=1}^{\mu _l} \widehat{h}_j(\xi ) = \widehat{P}_1(\xi ), \quad \xi \in \Omega ^{(1)}_l. \end{aligned}$$

Differentiating the latter equality two times with respect to $\xi _0$ and taking into account that $\widehat{P}_1$ is a polynomial in $\xi _0$ of degree at most $1$ we obtain (9).

Sufficiency. Suppose that mutually distinct functions $\{\widehat{h}_j(\xi )\}$ on $\Omega ^{(1)}_l$, $l=0$, $1$, ..., $L$, are holomorphic and satisfy the Eqs. (7), (9). In particular, for any $\xi _0 \in \Omega ^{(0)}_l$ we have

$$\begin{aligned} \frac{\partial \widehat{h}_j}{\partial \xi _1}(\xi _0,0) = \widehat{h}_j(\xi _0,0) \frac{\partial \widehat{h}_j}{\partial \xi _0}(\xi _0,0). \end{aligned}$$

By Cauchy–Kowalewski’s theorem in a neighborhood of arbitrary $\xi ^*\in \Omega ^{(1)}_l$ there exist unique holomorphic functions $\{\widetilde{h}_j(\xi _0,\xi _1)\}$, satisfying the Riemann–Burgers equation (7) and such that $\widetilde{h}_j|_{\Omega _l^{(1)}} = \widehat{h}_j$.

From here and from Proposition 3.3.3 of [5] we obtain existence and uniqueness of holomorphic functions $\{\widehat{P}_m(\xi )\}$, being polynomials of degree at most $m$ in $\xi _0$, such that $\{\widehat{h}_j(\xi )\}$ and $\{\widehat{P}_m(\xi )\}$ satisfy the system (8) for $m = 1$, $2$, ..., $j = 1$, ..., $\mu _l$, $\xi \in \Omega _l^{(1)}$.

Existence and uniqueness. Existence of functions $\{\widehat{h}_j\}$ and $\{\widehat{P}_m\}$ with necessary properties follows from Theorem 2.1. More precisely, $\widehat{h}_j = h_j$ and $\widehat{P}_m = P_m$, $j = 1$, ..., $\mu _l$, $m \ge 1$. Uniqueness of functions $\{\widehat{h}_j\}$ for minimal $\{\mu _l\}$ with properties (8)–(9) follows from Theorem II of [5] and from Theorem 3 of [8]. Uniqueness of polynomials $\{\widehat{P}_m\}$ for minimal $\{\mu _l\}$ with properties (8)–(9) follows from the proof of sufficiency. $\square$

3 Reconstruction Algorithm

Consider now the reconstruction algorithm of complex curve $X \subseteq {\mathbb {C}} {\mathrm {P}}^2$ with given boundary $bX$ and with condition of minimality (*). Let us consider the cases $\mu _0 = 0$, $1$, $2$.

The reconstruction algorithm is based on formulas (6) with polynomials $P_m$, $m=0$, $1$, .... The next theorem permits calculating these polynomials. If $i$, $j$, $k$, $l$ are non-negative integers we will use the notation

$$\begin{aligned} \ae ^{ij}_{kl} = \frac{1}{2\pi i} \int \limits _\gamma \bigl ( z_1^i z_2^j \, dz_1 + z_1^k z_2^l \, dz_2 \bigr ). \end{aligned}$$

(10)

Theorem 3.1

Let $X\subset {{\mathbb {C}}}P^2$ be a complex curve without algebraic subdomains, $\gamma \subset {\mathbb {C}}^2$ be its boundary. Let mutually distinct holomorphic in $\xi \in \Omega _l^{(1)}$, $l=0$, $1$, ..., $L$, functions $\{h_j(\xi )\}$ and holomorphic in $\xi \in \Omega _l^{(1)}$, $l=0$, ..., $L$, functions $P_m(\xi _0,\xi _1)$, being polynomials of degree at most $m$ in $\xi _0$, satisfy the system (1), (6) for $\xi \in \Omega _l^{(1)}$, $j=1$, ..., $N_+(\xi )$ with minimal $N_+(\xi )$ (existence and uniqueness of such functions follow from Theorem 2.2). Then the following statements are valid:

• 1. If $\mu _0 = 0$, then $P_m(\xi _0,0) \equiv 0$ for all $m$. Besides, $G_1(\xi _0,\xi _1) = 0$, if $|\xi _0| \ge \mathrm {const}(X)(1+|\xi _1|)$.

• 2. If $\mu _0 = 1$, then $P_1(\xi _0,0) = c_{11}+c_{12}\xi _0$, where constants $c_{11}$ and $c_{12}$ satisfy the identity in $\xi _0 \in \Omega _0$:

  $$\begin{aligned}&c_{11} \frac{\partial G_1}{\partial \xi _0}(\xi _0,0) + c_{12} \biggl ( \xi _0 \frac{\partial G_1}{\partial \xi _0}(\xi _0,0) + G_1(\xi _0,0) \biggr ) \nonumber \\&\quad = G_1(\xi _0,0)\frac{\partial G_1}{\partial \xi _0}(\xi _0,0)-\frac{\partial G_1}{\partial \xi _1}(\xi _0,0). \end{aligned}$$

  (11)

• 3. If $\mu _0 = 2$, then $P_1(\xi _0,0) = c_{11}+c_{12}\xi _0$, $P_2(\xi _0,0) = c_{21}+c_{22}\xi _0 + c_{23}\xi _0^2$, where constants $c_{11}$, $c_{12}$, $c_{21}$, $c_{22}$, $c_{23}$ satisfy the identity in $\xi _0 \in \Omega _0$:

  $$\begin{aligned} \ae ^{00}_{10} (c_{12}^2 + c_{23})&= \frac{\partial G_2}{\partial \xi _1} - 2 \frac{\partial G_1}{\partial \xi _1} ( G_1 - c_{11} - c_{12} \xi _0) + G_1(c_{22}+2c_{23} \xi _0)\nonumber \\&+ \frac{\partial G_1}{\partial \xi _0} \cdot \bigl ( (G_1 - c_{11} -c_{12}\xi _0)^2 - G_2 + c_{21} + c_{22}\xi _0 + c_{23}\xi _0^2 \bigr ) \nonumber \\&+ \bigl ( G_1^2 - 2c_{11} G_1 - 2c_{12} G_1 \xi _0 - G_2 \bigr )\cdot (-c_{12}), \end{aligned}$$

  (12)

  where all the functions are evaluated at point $(\xi _0,0)$.

Proof

By Theorem 2.2 functions $P_m$ from the condition of this theorem are defined in Theorem 2.1.

• 1. By Theorem 2.1 $P_m \equiv 0$, if $\mu _0 = 0$.

• 2. For $\xi \in \Omega _l^{(1)}$ we have equality $P_1(\xi _0,\xi _1)=C_{11}(\xi _1)+C_{12}(\xi _1)\xi _0$. We need to find constants $c_{11}=C_{11}(0)$ and $c_{12} = C_{12}(0)$. Differentiate equation (6) with respect to $\xi _0,\xi _1$ and restrict this equation and its differentiated versions to $\xi \in \Omega ^{(0)}_l$:

  $$\begin{aligned} h_1(\xi _0,0)&= G_1(\xi _0,0)-c_{11}-c_{12}\xi _0, \\ \frac{\partial h_1}{\partial \xi _1}(\xi _0,0)&= \frac{\partial G_1}{\partial \xi _1}(\xi _0,0) - \dot{C}_{11}(0)-\dot{C}_{12}(0)\xi _0,\\ \frac{\partial h_1}{\partial \xi _0}(\xi _0,0)&= \frac{\partial G_1}{\partial \xi _0}(\xi _0,0) - c_{12}, \end{aligned}$$

  where $\xi _0 \in \Omega _0$. By (1) for $\xi _0 \in \Omega _0$ function $h_1(\xi _0,0)$ satisfies the equality $\frac{\partial h_1}{\partial \xi _1}(\xi _0,0) = h_1(\xi _0,0)\frac{\partial h_1}{\partial \xi _0}(\xi _0,0)$. If we substitute in this equality the expressions for $h_1(\xi _0,0)$, $\frac{\partial h_1}{\partial \xi _1}(\xi _0,0)$ and $\frac{\partial h_1}{\partial \xi _0}(\xi _0,0)$, we will obtain the equation

  $$\begin{aligned}&\frac{\partial G_1}{\partial \xi _1}(\xi _0,0) - \dot{C}_{11}(0)-\dot{C}_{12}(0)\xi _0 \nonumber \\&\quad = \biggl ( G_1(\xi _0,0)-c_{11}-c_{12}\xi _0 \biggr )\biggl ( \frac{\partial G_1}{\partial \xi _0}(\xi _0,0) - c_{12} \biggr ). \end{aligned}$$

  (13)

  This equation is valid for $\xi _0 \in \Omega _0$. Let us divide it into $\xi _0$ and tend $\xi _0 \rightarrow \infty$. We obtain equality $\dot{C}_{12}(0) = -c_{12}^2$. Taking into account this equality, we can rewrite the Eq. (13) in the form

  $$\begin{aligned}&\frac{\partial G_1}{\partial \xi _1}(\xi _0,0) - \dot{C}_{11}(0)\nonumber \\&\quad =\biggl (G_1(\xi _0,0)-c_{11}-c_{12}\xi _0\biggr ) \frac{\partial G_1}{\partial \xi _0}(\xi _0,0) - \biggl (G_1(\xi _0,0)-c_{11}\biggr )c_{12}.\qquad \end{aligned}$$

  (14)

  Taking into account that $\xi _0 \frac{\partial G_1}{\partial \xi _0}(\xi _0,0) \rightarrow 0$ as $\xi _0 \rightarrow \infty$ and passing $\xi _0 \rightarrow \infty$ in (14), we obtain the equality $\dot{C}_{11}(0) = -c_{11}c_{12}$. Due to the just-obtained equality, Eq. (14) takes the desired form.

• 3. By (1) functions $h_1(\xi )$ and $h_2(\xi )$ satisfy the Riemann–Burgers equation for $\xi \in \Omega _l^{(1)}$. So, the following equalities are valid:

  $$\begin{aligned} \frac{\partial (h_1h_2)}{\partial \xi _1}&= h_1 \frac{\partial h_2}{\partial \xi _1}+\frac{\partial h_1}{\partial \xi _1} h_2 = h_1 h_2 \frac{\partial (h_1+h_2)}{\partial \xi _0}, \end{aligned}$$

  (15)
  $$\begin{aligned} \frac{\partial (h_1^2+h_2^2)}{\partial \xi _0}&= 2 h_1\frac{\partial h_1}{\partial \xi _0}+2h_2 \frac{\partial h_2}{\partial \xi _0} = 2 \frac{\partial (h_1+h_2)}{\partial \xi _1}. \end{aligned}$$

  (16)

Note that $h_1 h_2 = \frac{1}{2} \bigl ( h_1+h_2 \bigr )^2 - \frac{1}{2} \bigl ( h_1^2+h_2^2 \bigr )$. Therefore the system  (15)–(16) is equivalent to the system

$$\begin{aligned} \frac{\partial (h_1+h_2)^2}{\partial \xi _1} - \frac{\partial (h_1^2+h_2^2)}{\partial \xi _1}&= \biggl ( \bigl (h_1+h_2\bigr )^2-\bigl (h_1^2+h_2^2\bigr ) \biggr ) \frac{\partial (h_1+h_2)}{\partial \xi _0}, \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial (h_1^2+h_2^2)}{\partial \xi _0}&= 2 \frac{\partial (h_1+h_2)}{\partial \xi _1}. \end{aligned}$$

(18)

We substitute into this system $h_1^2+h_2^2$ and $h_1+h_2$ from Eq. (6), using the notation $P_1(\xi _0,\xi _1)=C_{11}(\xi _1)+C_{12}(\xi _1)\xi _0$, $P_2(\xi _0,\xi _1)=C_{21}(\xi _1)+C_{22}(\xi _1)\xi _0+C_{23}(\xi _1)\xi _0^2$. Equation (18) restricted to $\Omega _l^{(0)}$ takes the form

$$\begin{aligned} \frac{\partial G_2}{\partial \xi _0}(\xi _0,0) - c_{22}-2c_{23}\xi _0 = 2 \biggl ( \frac{\partial G_1}{\partial \xi _1}(\xi _0,0)-\dot{C}_{11}(0)-\dot{C}_{12}(0) \xi _0 \biggr ). \end{aligned}$$

(19)

Divide this equation into $\xi _0$ and tend $\xi _0 \rightarrow \infty$. We obtain the equality $\dot{C}_{12}(0) = c_{23}$. Taking this equality into account and passing $\xi _0 \rightarrow \infty$ in (19) we obtain the equality $\dot{C}_{11}(0) = \frac{1}{2} c_{22}$.

Now substitute the expressions for $h_1^2+h_2^2$ and $h_1+h_2$ into (17) and restrict the obtained formula to $\Omega _l^{(0)}$. We obtain the equality

$$\begin{aligned}&2\bigl ( G_1-c_{11}-c_{12}\xi _0 \bigr )\biggl ( \frac{\partial G_1}{\partial \xi _1} - \dot{C}_{11}(0) - \dot{C}_{12}(0) \xi _0 \biggr )\nonumber \\&\qquad - \frac{\partial G_2}{\partial \xi _1} + \dot{C}_{21}(0)+\dot{C}_{22}(0)\xi _0 + \dot{C}_{23}(0) \xi _0^2 \nonumber \\&\quad = \biggl ( \bigl (G_1-c_{11}-c_{12}\xi _0 \bigr )^2 - G_2 + c_{21}+c_{22}\xi _0+c_{23}\xi _0^2 \biggr ) \biggl ( \frac{\partial G_1}{\partial \xi _0} - c_{12} \biggr ).\qquad \end{aligned}$$

(20)

Divide this equation into $\xi _0^2$ and pass $\xi _0 \rightarrow \infty$. This leads to the equality

$$\begin{aligned} 2c_{12}\dot{C}_{12}(0) + \dot{C}_{23}(0) = -\bigl (c_{12}^2+c_{23}\bigr )c_{12}. \end{aligned}$$

Using the latter equality, divide (20) into $\xi _0$ and pass $\xi _0 \rightarrow \infty$ to obtain the equality

$$\begin{aligned} 2c_{11}\dot{C}_{12}(0)+2c_{12}\dot{C}_{11}(0) + \dot{C}_{22}(0) = -\bigl ( 2c_{11}c_{12}+c_{22} \bigr ) c_{12}. \end{aligned}$$

Taking into account the obtained equalities one can rewrite (20) in the form

$$\begin{aligned}&2\bigl ( G_1-c_{11}-c_{12}\xi _0 \bigr ) \frac{\partial G_1}{\partial \xi _1} - 2G_1 \bigl ( \dot{C}_{11}(0)+\dot{C}_{12}(0) \xi _0 \bigr )\nonumber \\&\qquad +2c_{11}\dot{C}_{11}(0) - \frac{\partial G_2}{\partial \xi _1} + \dot{C}_{21}(0)\nonumber \\&\quad = \biggl ( \bigl (G_1-c_{11}-c_{12}\xi _0 \bigr )^2 - G_2 + c_{21}+c_{22}\xi _0+c_{23}\xi _0^2 \biggr ) \frac{\partial G_1}{\partial \xi _0}\nonumber \\&\qquad + \biggl ( \bigl (G_1-c_{11}\bigr )^2 - 2G_1 c_{12}\xi _0 -G_2 + c_{21} \biggr )(-c_{12}). \end{aligned}$$

(21)

Pass $\xi _0 \rightarrow \infty$ in this equality and note that the following relations are valid:

$$\begin{aligned} \lim \nolimits _{\xi _0\rightarrow \infty } \xi _0 \frac{\partial G_1}{\partial \xi _1}&= \lim \nolimits _{\xi _0\rightarrow \infty } \xi _0 \frac{1}{2\pi i} \!\int \limits _\gamma \frac{z_1 \, dz_1}{\xi _0+z_2} \!=\! \frac{1}{2\pi i}\!\int \limits _\gamma z_1 \, d z_1 =\langle b\gamma , \frac{1}{4\pi i} z_1^2 \rangle \!=\! 0, \\ \lim \nolimits _{\xi _0 \rightarrow \infty } \xi _0 G_1&= \lim \nolimits _{\xi _0\rightarrow \infty } \xi _0 \frac{1}{2\pi i} \int \limits _\gamma \frac{z_1 \, dz_2}{\xi _0+z_2} = \frac{1}{2\pi i} \int \limits _\gamma z_1 \, dz_2 = \ae ^{00}_{10}, \\ \lim \nolimits _{\xi _0 \rightarrow \infty } \xi _0^2 \frac{\partial G_1}{\partial \xi _0}&= - \lim \nolimits _{\xi _0\rightarrow \infty } \xi _0^2 \frac{1}{2\pi i} \int \limits _\gamma \frac{z_1 \, dz_2}{(\xi _0+z_2)^2} = - \frac{1}{2\pi i} \int \limits _\gamma z_1 \, dz_2 = -\ae ^{00}_{10}. \end{aligned}$$

We obtain

$$\begin{aligned} -2\ae ^{00}_{10} \dot{C}_{12}(0) + 2c_{11} \dot{C}_{11}(0) + \dot{C}_{21}(0) = -\bigl ( c_{12}^2 + c_{23}\bigr )\ae ^{00}_{10} - \bigl (c_{11}^2 - 2c_{12}\ae ^{00}_{10}+c_{21}\bigr )c_{12}. \end{aligned}$$

Express constants $\dot{C}_{ij}(0)$ through $c_{ij}$ in the obtained equations:

$$\begin{aligned} \begin{aligned} \dot{C}_{11}(0)&= \frac{1}{2} c_{22}, \\ \dot{C}_{12}(0)&= c_{23}, \\ \dot{C}_{23}(0)&= -c_{12}^3 - 3c_{12}c_{23}, \\ \dot{C}_{22}(0)&= -2\bigl ( c_{11} c_{12}^2 + c_{12} c_{22} + c_{11} c_{23} \bigr ), \\ \dot{C}_{21}(0)&= \ae ^{00}_{10}(c_{12}^2+c_{23}) - c_{12}(c_{11}^2 + c_{21}) - c_{11} c_{22}. \end{aligned} \end{aligned}$$

(22)

Substituting these constants into (21), we obtain the third statement of Theorem 3.1.

Complement 3.1

Statement of Theorem 3.1 admits a development for the case $\mu _0 \ge 3$. In this case

$$\begin{aligned} P_k(\xi _0,\xi _1) = C_{k1}(\xi _1) + C_{k2}(\xi _1)\xi _0 + \cdots + C_{k,k+1}(\xi _1)\xi _0^k, \quad k = 1,\ldots ,\mu _0. \end{aligned}$$

Define $\dot{C}_{ij}(0) = \frac{\partial C_{ij}}{\partial \xi _1}(0)$ and $c_{ij} = C_{ij}(0)$ for $i = 1$, ..., $\mu _0$ and $j = 1$, ..., $i+1$.

Let us indicate the following general procedure for finding constants $c_{ij}$. Due to the Riemann–Burgers equations (1) the following identities in $\xi _0 \in \Omega _0$ hold for $k = 1,\ldots ,\mu _0-1$:

$$\begin{aligned}&- \frac{\partial G_k}{\partial \xi _1}(\xi _0,0) + \dot{C}_{k1}(0) + \dot{C}_{k2}(0) \xi _0 + \cdots + \dot{C}_{k,k+1}(0) \xi _0^k \\&\quad = \frac{k}{k+1}\left( -\frac{\partial G_{k+1}}{\partial \xi _0}(\xi _0,0) + c_{k+1,2} + 2c_{k+1,3} \xi _0 + \cdots + (k+1) c_{k+1,k+2} \xi _0^k\right) . \end{aligned}$$

Taking into account that $\frac{\partial G_k}{\partial \xi _1}(\xi _0,0) \rightarrow 0$ and $\frac{\partial G_{k+1}}{\partial \xi _0}(\xi _0,0) \rightarrow 0$ as $\xi _0 \rightarrow + \infty$ we obtain the equalities

$$\begin{aligned} \dot{C}_{k,m}(0) = \frac{km}{k+1} c_{k+1,m+1}, \quad k = 1,\ldots , \mu _0-1, \quad m = 1,\ldots ,k+1. \end{aligned}$$

Due to the Riemann–Burgers equations (1) the following identity in $\xi _0 \in \Omega _0$ holds:

$$\begin{aligned} \frac{\partial e_{\mu _0}}{\partial \xi _1}(\xi _0,0) = e_{\mu _0}(\xi _0,0) \frac{\partial p_1}{\partial \xi _0}(\xi _0,0), \end{aligned}$$

(23)

where functions $e_k$ are given by the following formulas:

$$\begin{aligned} \begin{aligned} k e_k(\xi _0,\xi _1)&= \sum \nolimits _{i=1}^{k-1} (-1)^{i+1} e_{k-i}(\xi _0,\xi _1) p_i(\xi _0,\xi _1) + (-1)^{k+1} p_k(\xi _0,\xi _1),\\ p_k(\xi _0,\xi _1)&= G_k(\xi _0,\xi _1) - C_{k1}(\xi _1) - C_{k2}(\xi _1)\xi _0 - \cdots - C_{k,k+1}(\xi _1) \xi _0^k, \end{aligned} \end{aligned}$$

(24)

where $k = 1$, ..., $\mu _0$.

Equality (23) allows us to represent constants $\{ \dot{C}_{\mu _0, j}(0) \}$ as functions of constants $\{ c_{ij} \}$. Finally, substituting the obtained expressions for constants $\{ \dot{C}_{ij}(0) \}$ via constants $\{ c_{ij}\}$ into Eq. (23) we obtain the identity in $\xi _0 \in \Omega _0$ for computation of constants $\{ c_{ij} \}$.

For example, in the case $\mu _0 = 3$ the identity (23) in $\xi _0 \in \Omega _0$ for finding constants $c_{ij}$ takes the form

$$\begin{aligned}&\dot{C}_{31}(0) + \dot{C}_{32}(0) \xi _0 + \dot{C}_{33}(0)\xi _0^2 + \dot{C}_{34}(0) \xi _0^3 \\&\quad = \frac{\partial G_3}{\partial \xi _1}+ \frac{3}{4} (p_1^2 - p_2) \frac{\partial p_2}{\partial \xi _0} - p_1 \frac{\partial p_3}{\partial \xi _0} - \frac{1}{2} \bigl ( p_1^3 - 3p_1 p_2 + 2p_3\bigr ) \frac{\partial p_1}{\partial \xi _0}, \end{aligned}$$

where all functions are evaluated at point $(\xi _0,0)$, the functions $p_k$ are defined in formula (24) and the constants $\dot{C}_{31}(0)$, $\dot{C}_{32}(0)$, $\dot{C}_{33}(0)$, $\dot{C}_{34}(0)$ are given by formulas

$$\begin{aligned} \dot{C}_{31}(0)&= \frac{1}{2} \ae ^{00}_{10} \bigl ( 3c_{11} c_{12}^2 + 3c_{12} c_{22} + 3c_{11} c_{23} + 2 c_{33} \bigr ) \\&\quad - \frac{1}{2} \ae ^{00}_{11} \bigl ( c_{12}^3 + 3 c_{12} c_{23} + 2 c_{34} \bigr ) - \frac{3}{2} \ae ^{11}_{00} \bigl ( c_{12}^2 + c_{23} \bigr ) \\&\quad - \frac{1}{2} c_{11}^3 c_{12} - \frac{3}{2} c_{11} c_{12} c_{21} - \frac{3}{4} c_{11}^2 c_{22} - \frac{3}{4} c_{21} c_{22} - c_{12}c_{31}-c_{11}c_{32},\\ \dot{C}_{32}(0)&= \ae ^{00}_{10} \bigl ( c_{12}^3 + 3 c_{12}c_{23} + 2 c_{34} \bigr ) - \frac{3}{2} c_{11}^2 c_{12}^2 - \frac{3}{2} c_{12}^2 c_{21} - 3 c_{11} c_{12} c_{22} \\&\quad -\frac{3}{4} c_{22}^2 - \frac{3}{2} c_{11}^2 c_{23} - \frac{3}{2} c_{21} c_{23} - 2 c_{12} c_{32} - 2 c_{11} c_{33},\\ \dot{C}_{33}(0)&= -\frac{3}{2} c_{11} c_{12}^3 - \frac{9}{4} c_{12}^2 c_{22} - \frac{9}{2} c_{11} c_{12} c_{23} - \frac{9}{4} c_{22} c_{23} - 3 c_{12} c_{33} - 3 c_{11} c_{34} \\ \dot{C}_{34}(0)&= - \frac{1}{2} c_{12}^4 - 3 c_{12}^2 c_{23} - \frac{3}{2} c_{23}^2 - 4 c_{12} c_{34}, \end{aligned}$$

where $\ae ^{00}_{10}$, $\ae ^{11}_{00}$ and $\ae ^{00}_{11}$ are defined in formula (10).

The next theorem permits us to find $\mu _0$ through $\gamma$.

Theorem 3.2

Let $X \subset {\mathbb {C}} {\mathrm {P}}^2 \setminus [0:1:0]$ be a complex curve without algebraic subdomains, $\gamma = bX \subset {\mathbb {C}}^2$ be the boundary of $X$. Let functions $G_m(\xi _0,\xi _1)$, $m \ge 1$, be defined by formula (2) and number $\mu _0$ defined by formula (5). Then the following statements are valid:

• 1. If $G_1(\xi _0,\xi _1) = 0$ for $|\xi _0| \ge \mathrm {const}(X)(1+|\xi _1|)$, then $\mu _0 =0$.

• 2. If there exist such complex constants $c_{11}$, $c_{12}$ that for any $\xi \in \Omega ^{(1)}_0$ the following equality is valid:

  $$\begin{aligned} c_{11} \frac{\partial G_1}{\partial \xi _0}(\xi ) + c_{12} \biggl ( \xi _0 \frac{\partial G_1}{\partial \xi _0}(\xi ) + G_1(\xi ) \biggr ) = G_1(\xi )\frac{\partial G_1}{\partial \xi _0}(\xi )-\frac{\partial G_1}{\partial \xi _1}(\xi ), \end{aligned}$$

  (25)

  then $\mu _0 \le 1$.

• 3. If there exist such complex constants $c_{11}$, $c_{12}$, $c_{21}$, $c_{22}$, $c_{23}$ that the following identity in $\xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0$ is valid:

  $$\begin{aligned} \ae ^{00}_{10} (c_{12}^2 + c_{23})&= \frac{\partial G_2}{\partial \xi _1} - 2 \frac{\partial G_1}{\partial \xi _1} ( G_1 - c_{11} - c_{12} \xi _0) + G_1(c_{22}+2c_{23} \xi _0) \nonumber \\&+ \frac{\partial G_1}{\partial \xi _0} \cdot \bigl ( (G_1 - c_{11} -c_{12}\xi _0)^2 - G_2 + c_{21} + c_{22}\xi _0 + c_{23}\xi _0^2 \bigr ) \nonumber \\&+ \bigl ( G_1^2 - 2c_{11} G_1 - 2c_{12} G_1 \xi _0 - G_2 \bigr )\cdot (-c_{12}), \end{aligned}$$

  (26)

  where all functions are evaluated at point $\xi$, then $\mu _0 \le 2$.

Complement 3.2

The statement of Theorem 3.2 for $\mu _0 \ge 3$ in the spirit of cases $\mu _0 \le 2$ will be developed in a separate paper together with statement of Theorem 3.1 for $\mu _0 \ge 3$, indicated in Complement 3.1.

Proof

• 1. Equality $G_1(\xi _0,\xi _1) = 0$ for $|\xi _0| \ge \mathrm {const}(X)(1+|\xi _1|)$ implies according to [5] the moment condition

  $$\begin{aligned} \int \limits _\gamma z_1^{k_1} z_2^{k_2} \, dz_2 = 0 \quad \text {for all } k_1, k_2 \in \mathbb N. \end{aligned}$$

  From here according to [10] and [6] it follows that for an appropriate choice of orientation $\gamma$ is the boundary of a complex curve in $\mathbb C^2$. Hence either $\mu _0 = 0$ or $X$ is a domain on an algebraic curve in ${\mathbb {C}} {\mathrm {P}}^2$. But $X$ cannot be a domain on an algebraic curve in ${\mathbb {C}} {\mathrm {P}}^2$ because $X$ does not contain algebraic subdomains.

• 2. Let the conditions of general position be fulfilled. Put

  $$\begin{aligned} \begin{aligned} h(\xi _0,\xi _1)&= G_1(\xi _0,\xi _1) - C_{11}(\xi _1)-C_{12}(\xi _1)\xi _0,\\ C_{11}(\xi _1)&= c_{11} - c_{11}c_{12}\xi _1,\\ C_{12}(\xi _1)&= c_{12} - c_{12}^2 \xi _1,\qquad \end{aligned} \end{aligned}$$

  (27)

  where $\xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0$. Taking into account (27), we can rewrite equality (25) in the form

  $$\begin{aligned} \frac{\partial h}{\partial \xi _1}(\xi _0,\xi _1) = h(\xi _0,\xi _1) \frac{\partial h}{\partial \xi _0}(\xi _0,\xi _1), \quad \xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0. \end{aligned}$$

  (28)

  From definition (27) we obtain the following equality:

  $$\begin{aligned} \frac{\partial ^2}{\partial \xi _0^2} \bigl ( G_1(\xi _0,\xi _1)-h(\xi _0,\xi _1) \bigr ) = 0, \quad \xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0. \end{aligned}$$

  (29)

  From equalities (28), (29) due to Theorems 2.1, 2.2 and Theorem 3 from [8] we obtain

  $$\begin{aligned} X \cap \bigl \{ z_2 = -\xi _0 \bigr \} = \bigl \{ \bigl ( h(\xi _0,0), -\xi _0 \bigr ) \bigr \}, \quad \xi _0 \in \Omega _0. \end{aligned}$$

  (30)

  From here it follows that $\mu _0 \le 1$.

• 3. Let the conditions of general position be fulfilled. Let us define functions $h_1$ and $h_2$ by the following relations:

  $$\begin{aligned} \begin{aligned} h_1(\xi _0,\xi _1)+h_2(\xi _0,\xi _1)&= G_1(\xi _0,\xi _1)-C_{11}(\xi _1)-C_{12}(\xi _1)\xi _0,\\ h_1^2(\xi _0,\xi _1)+h_2^2(\xi _0,\xi _1)&= G_2(\xi _0,\xi _1)-C_{21}(\xi _1)-C_{22}(\xi _1)\xi _0-C_{23}(\xi _1)\xi _0^2, \end{aligned} \qquad \end{aligned}$$

  (31)
  $$\begin{aligned} C_{ij}(\xi _1) = c_{ij}+ \dot{C}_{ij}(0)\xi _1, \quad \text {i=1, j=1, 2 and i=2, j=1, 2, 3}, \qquad \end{aligned}$$

  (32)

  where $\xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0$ and constants $\dot{C}_{ij}(0)$ are defined by formulas (22).

Taking into account definitions (32), identity (26) is equivalent to identity (20), where $\xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0$.

Taking into account definitions (31), identity (20) for $\xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0$ is equivalent to identity (17) for $\xi \in \Omega ^{(1)}_0$.

By Lemma 3.2.1 from paper [4] for $\xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0$ the following equality is valid:

$$\begin{aligned} \frac{\partial G_2}{\partial \xi _0}(\xi _0,\xi _1) = 2 \frac{\partial G_1}{\partial \xi _1}(\xi _0,\xi _1). \end{aligned}$$

(33)

From definitions (31), (32) and from equality (33) we obtain equality (18) for $\xi \in \Omega ^{(1)}_0$.

Equalities (17), (18) mean that functions $h_1$ and $h_2$ satisfy the Riemann–Burgers equations:

$$\begin{aligned} \frac{\partial h_j}{\partial \xi _1}(\xi _0,\xi _1) = h_j(\xi _0,\xi _1)\frac{\partial h_j}{\partial \xi _0}(\xi _0,\xi _1), \quad \xi = (\xi _0,\xi _1) \in \Omega ^{(1)}_0. \end{aligned}$$

(34)

Further, because of definition (31) the following equality is valid:

$$\begin{aligned} \frac{\partial ^2}{\partial \xi _0} \bigl ( G_1(\xi _0,\xi _1)-h_1(\xi _0,\xi _1)-h_2(\xi _0,\xi _1) \bigr ) = 0. \end{aligned}$$

(35)

From equalities (34), (35) and using Theorems 2.1, 2.2 and Theorem 3 of the paper [8] we obtain:

$$\begin{aligned} X \cap \bigl \{ z_2 = -\xi _0 \bigr \} = \bigl \{ \bigl ( h_j(\xi _0,0), -\xi _0 \bigr ) \mid j=1,2 \bigr \}, \quad \xi _0 \in \Omega _0. \end{aligned}$$

From here it follows that $\mu _0 \le 2$. $\square$

Let us describe the algorithm of reconstruction of a complex curve $X$ in ${\mathbb C}P^2$ without compact components (satisfying minimality condition (*)). As above, $\gamma = bX$ is a compact real curve.

The algorithm for reconstruction of curve $X$ permits us to find a curve coinciding with the original curve in the given finite number of points and obtained by interpolation in other points. Let $\{ \xi ^k_0 \}_{k=1}^N$, $\xi ^k_0 \in {\mathbb {C}}$ be an arbitrary grid on $\mathbb C$, $\xi ^i_0 \ne \xi ^j_0$, $i \ne j$, and $\xi ^k_0 \notin \pi _2 \gamma$, $k=1$, ..., $N$. Complex curve $X$ intersects complex line $\{z_2 = -\xi ^k_0\}$ in $N_+(\xi ^k_0,0)$ points. The algorithm allows us to find these points.

The algorithm takes as input points $\{\xi ^k_0\}_{k=1}^N$ and a curve $\gamma$ (for example, represented as a finite number of points belonging to $\gamma$). On the output of the algorithm we obtain a set of points $(h_s(\xi ^k_0,0),-\xi ^k_0)$, $k=1$, ..., $N$; $s=1$, ..., $N_+(\xi ^k_0,0)$, belonging to the complex curve $X$.

3.1 The Case of $\mu _0 = 0$

• 1. Calculation of $\mu _l$. By formula (4) for every domain $\Omega _l$, $l=1$, ..., $L$, the number $\mu _l$ is equal to the winding number of curve $\pi _2 \gamma$ with respect to point $\xi _0 \in \Omega _l$:

  $$\begin{aligned} \mu _l \equiv N_{+}(\xi _0,0) = \frac{1}{2\pi i} \int \limits _{\gamma } \frac{\mathrm dz_2}{z_2+\xi _0}\equiv \frac{1}{2\pi i} \int \limits _{\pi _2 \gamma } \frac{\mathrm dz}{z-\xi _0}, \quad \xi _0 \in \Omega _l. \end{aligned}$$

• 2. Computation of power sums. If $\mu _0 = 0$ then for every point $\xi ^k_0 \in \Omega _l$, $l = 1$, ..., $L$, by Theorem 3.1 we have equalities $P_m(\xi ^k_0,0) \equiv 0$. By formula (6) we have the following formulas for the power sums:

  $$\begin{aligned} s_m(\xi ^k_0) \equiv h_1^m(\xi ^k_0,0)+\cdots +h_{\mu _l}^m(\xi ^k_0,0) = \frac{1}{2\pi i} \int \limits _\gamma \frac{z_1^m \, d z_2}{z_2 + \xi ^k_0}, \quad m = \overline{1,\mu _l}. \end{aligned}$$

  By Theorem 2.1 the points $(h_s(\xi ^k_0,0),-\xi ^k_0)$, $s=1$, ..., $N_+(\xi ^k_0,0)$; $k=1$, ..., $N$, are the desired points of $X$.

• 3. Computation of symmetric functions. For every point $\xi ^k_0 \in \Omega _l$, $l=1$, ..., $L$, the Newton identities

  $$\begin{aligned} k \sigma _k(\xi ^k_0) = \sum \nolimits _{i=1}^k (-1)^{i-1} \sigma _{k-i}(\xi ^k_0) s_{i}(\xi ^k_0), \quad k=1,\ldots ,N_+(\xi ^k_0,0). \end{aligned}$$

  allow us to reconstruct the elementary symmetric functions:

  $$\begin{aligned} \sigma _1(\xi ^k_0)&= h_1(\xi ^k_0,0)+\cdots +h_{\mu _l}(\xi ^k_0,0),\\ \cdots&= \cdots \\ \sigma _{\mu _l}(\xi ^k_0)&= h_1(\xi ^k_0,0)\times \cdots \times h_{\mu _l}(\xi ^k_0,0). \end{aligned}$$

• 4. Desymmetrization. For every point $\xi ^k_0 \in \Omega _l$ using Vieta formulas one can find complex numbers $h_1(\xi ^k_0,0)$, ..., $h_{\mu _l}(\xi ^k_0,0)$. The points $\bigl (h_s(\xi ^k_0,0),-\xi ^k_0)$, $s=1$, ..., $N_+(\xi ^k_0,0)$; $k = 1$, ..., $N$, are the required points of complex curve $X$.

3.2 The Cases of $\mu _0 = 1$, $2$

These cases are reduced to the case $\mu _0 = 0$ in the following way. Since $\pi _2 \gamma \subset {\mathbb {C}}$ is a compact real curve, there exists such $R>0$, such that the set $B_R^c(0) = \{ z \in {\mathbb {C}}\mid |z| \geqslant R \}$ belongs to $\Omega _0$. Without restriction of generality, one can suppose that $|\xi ^k_0| < R$ for all $k=1$, ..., $N$. Otherwise one can increase $R$.

Let us define the auxiliary complex curve $X_R = \{ (z_1,z_2) \in X \mid |z_2| \leqslant R \}$. Its boundary $\gamma _R$ consists of two disjoint parts (possibly, multiconnected): the first part is $\gamma$ and the second is a real curve obtained by lifting the circle $S_R = \{ z \in {\mathbb {C}}\mid |z| = R\}$ on surface $X$ by inversion of projection $\pi _2 :X \rightarrow {\mathbb {C}}$. Complex curve $X_R$ does not intersect infinity and, as a consequence $\mu _0(X_R)= 0$. Moreover, every point $\bigl (z_1(\xi ^k_0),-\xi ^k_0\bigr )$, $k=1$, ..., $N$, belongs to $X$ if and only if it belongs to $X_R$. Therefore, in order to reconstruct the complex curve $X_R$ it is sufficient to reconstruct the real curve obtained by lifting $S_R$ on $X$ and to solve the reconstruction problem for surface $X_R$, being in the conditions of the case of $\mu _0=0$. Finally, we come to the following algorithm:

• 1. New boundary. Choose a sufficiently large constant $R$, so that the exterior of the disk of radius $R$ centered at origin belongs to $\Omega _0$ and all $\xi ^k_0$ belong to this disk. Denote the boundary of this disk by $S_R$. In the case of $\mu _0 = 1$ by virtue of formulas (6) the points $\xi _0 \in S_R$ satisfy the equality

  $$\begin{aligned} h_1(\xi _0,0) = \frac{1}{2 \pi i}\int \limits _{\gamma } \frac{z_1 \, dz_2}{z_2 + \xi _0} - P_1(\xi _0,0). \end{aligned}$$

  In the case of $\mu _0 = 2$ we have two equalities:

  $$\begin{aligned} h_1(\xi _0,0)+h_2(\xi _0,0)&= \frac{1}{2\pi i}\int \limits _{\gamma } \frac{z_1 \, dz_2}{z_2 + \xi _0} - P_1(\xi _0,0),\\ h_1^2(\xi _0,0)+h_2^2(\xi _0,0)&= \frac{1}{2 \pi i}\int \limits _{\gamma } \frac{z_1^2 \, dz_2}{z_2 + \xi _0} - P_2(\xi _0,0), \end{aligned}$$

  where the polynomials can be found using Theorem 3.1. In the case of $\mu _0 = 1$ by lifting the curve $S_R$ on $X$ we obtain at once the real curve of the form $\{ (h_1(\xi _0,0),-\xi _0) \mid \xi _0 \in S_R \}$. In the case of $\mu _0 = 2$ we have to apply Newton identities and Vieta formulas in order to obtain $h_1$ and $h_2$ from functions $h_1+h_2$ and $h_1^2+h_2^2$.

• 2. Reduction. In order to find the complex curve $X_R$ with boundary $bX_R = \gamma _R$ we apply the algorithm of reconstruction for the case of $\mu _0=0$. The discussion before the description of the algorithm shows that we will obtain the desired points.

4 Visualization

Let us describe in a few words the algorithm of visualization of complex curves that we have used in our examples. Denote by $\pi _1 :\mathbb C^2 \rightarrow \mathbb C$ the projection into the first factor: $\pi _1(z_1,z_2) = z_1$. Suppose that $X$ is a complex curve in $\mathbb C^2$ such that the covering $\pi _1 :X \setminus \{ \text {ramification points} \} \rightarrow \mathbb C$ has multiplicity $L$. Consider, for simplicity, a rectangular grid $\Lambda$ in $\mathbb C$:

$$\begin{aligned} \Lambda = \bigl \{ z_1^{ij} :\mathop {\mathrm {Re}}z_1^{ij} = \frac{i}{N}, \; \mathop {\mathrm {Im}}z_1^{ij} = \frac{j}{N}, \; i,j = 0,\ldots ,N \bigr \}, \end{aligned}$$

where $N$ is a natural number. Suppose now that we are given the set $X_\Lambda = \pi _1^{-1}(\Lambda ) \cap X$ and we need to visualize the part of $X$ lying above the rectangle $0 \le \mathop {\mathrm {Re}}z_1 \le 1$, $0 \le \mathop {\mathrm {Im}}z_1 \le 1$.

Let us introduce some terminology. We define a path in $\Lambda$ as a map $\gamma :\{1,\ldots ,M\} \!\rightarrow \! \Lambda$ such that $|\gamma (k+1)-\gamma (k)|=\frac{1}{N}$ for all admissible $k$, where $M$ is some natural number.

Let $\gamma :\{1,\ldots ,M\} \rightarrow \Lambda$ be a path in $\Lambda$ and let $i :\{1,\ldots ,M\} \rightarrow [1,M]$ be the inclusion map. Define the function $i_*\gamma :[1,M] \rightarrow \mathbb C$ such that $i_*\gamma (k) = \gamma (k)$ for integer $k$ and $i_*\gamma |_{[k,k+1]}$ is linear for all admissible $k$. It is clear that $i_*\gamma$ is a continuous function and hence it can be lifted to $X$ by the map $\pi _1$.

We define a path in $X_\Lambda$ as a map $\Gamma :\{1,\ldots ,M\} \rightarrow X_\Lambda$ such that $\gamma = \pi _1 \circ \Gamma$ is a path in $\Lambda$ and $\Gamma = i^*L(i_*\gamma )$, where $i^*$ is the pullback map with respect to $i$ and $L(i_*\gamma )$ is some lift of $i_*\gamma$ to $X$ by $\pi _1$, i.e., $L(i_*\gamma )$ is a continuous map from $[1,M]$ to $X$ such that $\pi _1 \circ L(i_*\gamma ) = i_*\gamma$. We also say that $\Gamma$ is obtained by lifting $\gamma$.

We will call subsets of $\Lambda$ and $X_\Lambda$ path-connected if every two points of these sets can be connected by a path in $\Lambda$ and $X_\Lambda$, respectively.

Let us describe the practical way to lift paths in $\Lambda$ to paths in $X_\Lambda$. Suppose that $N$ is sufficiently large. Let $\gamma :\{1,\ldots ,M\} \rightarrow \Lambda$ be a path in $\Lambda$ and let $\Gamma (1) \in \pi _1^{-1}(\gamma (1)) \cap X$ be an arbitrary point. We select $\Gamma (k) \in \pi _1^{-1}(\gamma (k)) \cap X$ in such a way that

$$\begin{aligned} |\Gamma (k)-\Gamma (k-1)| = \min \bigl \{ |z-\Gamma (k-1)| :z \in \pi _1^{-1}(\gamma (k)) \cap X \bigr \}, \quad k = 2, \ldots , M. \end{aligned}$$

Then $\Gamma$ is a path in $X_\Lambda$ obtained by lifting $\gamma$. All possible lifts of $\gamma$ may be obtained by varying $\Gamma (1)$. Note that if $\gamma$ is closed, i.e., $\gamma (1) = \gamma (M)$, $\Gamma$ need not to be closed.

Finding Ramification Points and Making Branch Cuts. The first step in the visualization procedure consists in finding ramification points of $X$ with respect to projection $\pi _1$. Since we have only a finite number of points on $X$ we can find ramification points only approximately. More precisely, we will localize them in small circles.

Without restriction of generality we suppose that all ramification points are projected by $\pi _1$ into interior points of $\Lambda$. Take any interior point $z_1 \in \Lambda$ and select a small closed path $\gamma :\{1,\ldots ,M\} \rightarrow \Lambda$ around $z_1$ so that there is at most one ramification point inside the polygon $\overline{\gamma (1)\ldots \gamma (M)}$. For example, one can take as $\gamma$ the following path:

$$\begin{aligned} z_1+\frac{1}{N} \rightarrow z_1+\frac{1+i}{N} \rightarrow z_1+ \frac{i}{N} \rightarrow \cdots \rightarrow z_1+\frac{1-i}{N} \rightarrow z_1+\frac{1}{N}, \end{aligned}$$

where $i$ is the imaginary unit.

Now consider different lifts of $\gamma$ to $X_\Lambda$. If at least one lift is not closed, mark $z_1$ as a possible ramification point (meaning that it is situated near the projection of some ramification point of $X$). Now vary $z_1$ and mark all possible ramification points. The resulting set will consist of several path-connected components each of which localizes the position of one ramification point of $X$ with respect to $\pi _1$.

Now connect each of the obtained connected components of possible ramification points by path with boundary of the grid $\Lambda$ in such a way that different paths do not intersect. Denote the union of the set of possible ramification points with images of these paths by $\Lambda _c$. An important observation is that every closed path in $\Lambda \setminus \Lambda _c$ always lifts to a closed path in $X_\Lambda$ since it does not contain $\pi _1$-projections of ramification points inside.

Visualization. Now denote $\Lambda \setminus \Lambda _c = \cup _{s=1}^S \Lambda _s$, where $\Lambda _s$ are different path-connected components. Take any $z_1^s \in \Lambda _s$ and $z_2^s \in \pi _1^{-1}(z_1^s) \cap X$. Now take other $z_1 \in \Lambda _s$ and connect $z_1^s$ with $z_1$ by some path $\gamma$. Then $\gamma$ lifts to a path $\Gamma$ with $\Gamma (1) = (z_1^s,z_2^s)$ and $\Gamma (2) = (z_1,z_2)$ for some $z_2 \in \pi _1^{-1}(z_1) \cap X$ and $z_2$ does not depend on $\gamma$. Varying $z_1$ we thus obtain the map $\Sigma (z_1^s,z_2^s) :\Lambda _s \rightarrow X_\Lambda$ which allows us to visualize the part of $X$.

Varying $z_2^s \in \pi _1^{-1}(z_1^s) \cap X$ (the latter is the finite set, namely, it consists of $L$ elements) we obtain the other maps $\Sigma (z_1^s,z_2^s)$ which allow us to visualize other parts of $X$. Clearly, the set of obtained maps does not depend on the choice of $z_1^s \in \Lambda _s$. Hence we can denote the obtained maps by $\Sigma _s^l$, $l = 1$, ..., $L$. It is clear that $\cup _{l=1}^L \Sigma _s^l(\Lambda _s) = \pi ^{-1}(\Lambda _s) \cap X$. Now vary $s$ to visualize

$$\begin{aligned} \cup _{s=1}^S \cup _{l=1}^L \Sigma _s^l(\Lambda _s) = \pi ^{-1}(\cup _{s=1}^S \Lambda _s) \cap X = X_\Lambda \setminus \pi ^{-1}(\Lambda _c). \end{aligned}$$

The part $\pi ^{-1}(\Lambda _c) \cap X_\Lambda$ consists of cuts and preimages of possible ramification points. The cuts can be visualized as the already-visualized part of the surface. The only problem is the visualization of $\pi _1$-preimages of possible ramification points. But the latter take a little part of the surface when $N$ is large and one can just forget about their visualization. On the other hand, in our examples they were visualized using a low-level graphics approach.

Examples of application of this algorithm are given in Figs. 1 and 2. The visualization algorithm can be easily generalized to the case of general grids. For instance, in our examples we have used a modification with periodic grid.

Fig. 1

Riemann surface of function $f(z) = \sqrt{\exp \left( \frac{z}{4}\right) +\sqrt{z^2+1}}$, $|z| \leqslant 2$, obtained by the visualization algorithm. Red and green curves represent two connected components of the surface boundary, colored small balls represent ramification points (Color figure online)

Fig. 2

Riemann surfaces of functions $f(z) = \sqrt{\sin (z)}$, $|z| \leqslant 2$ (left) and $f(z) = \sqrt{z^4+1}$, $|z|\leqslant 2$ (right) obtained by the visualization algorithm

5 Examples

5.1 The Case of $\mu _0 = 1$

Consider an example of reconstruction of a Riemann surface with given boundary. The simplest case is the case of $\mu _0 = 0$ but it follows from the discussion of the reconstruction algorithm that this case is directly included in any other case. Therefore, we begin with the next simplest case, namely, the case of $\mu _ 0 = 1$.

Let us reconstruct the surface

$$\begin{aligned} X_1 = \left\{ (z_1,z_2) \in {\mathbb {C}}^2 \mid (z_1-1)z_2 = \exp (z_1^2), \quad |z_1| \leqslant 2 \right\} . \end{aligned}$$

We suppose that the boundary $\gamma _1 = bX_1$ is given in the form of a discrete number of points (see further Fig. 3).

Fig. 3

Boundary $\gamma _1 = bX_1$ of the surface $X_1$

Note that if point $(z_1,z_2) \in X_1$ is such that $z_1$ approaches $1$, then $z_2$ approaches infinity. Choose $R$ large enough, e.g., $R = 60$, and reconstruct the real curve $\Gamma = \left\{ (z_1,z_2) \in X_1 \mid |z_2| = R \right\}$. At first, compute for two different points $\xi _0^1$, $\xi _0^2$, $|\xi _0^1| = |\xi _0^2| = R$, the values of functions

$$\begin{aligned} G_1(\xi _0,0)&= \frac{1}{2\pi i} \int \limits _{\gamma _1} \frac{z_1 dz_2}{z_2 + \xi _0}, \\ \frac{\partial G_1}{\partial \xi _0}(\xi _0,0)&= -\frac{1}{2\pi i} \int \limits _{\gamma _1} \frac{z_1^2 dz_2}{(z_2+\xi _0)^2}, \\ \frac{\partial G_1}{\partial \xi _1}(\xi _0,0)&= \frac{1}{2\pi i}\int \limits _{\gamma _1} \left( \frac{z_1 dz_1}{z_2+\xi _0} - \frac{z_1^2 dz_2}{(z_2+\xi _0)^2} \right) \end{aligned}$$

and find from the linear system for $c_{11}$ and $c_{12}$

$$\begin{aligned}&c_{11} \frac{\partial G_1}{\partial \xi _0}(\xi _0^1,0) + c_{12} \biggl ( \xi _0^1 \frac{\partial G_1}{\partial \xi _0}(\xi _0^1,0) + G_1(\xi _0^1,0) \biggr )\\&\quad = G_1(\xi _0^1,0)\frac{\partial G_1}{\partial \xi _0}(\xi _0^1,0)-\frac{\partial G_1}{\partial \xi _1}(\xi _0^1,0),\\&c_{11} \frac{\partial G_1}{\partial \xi _0}(\xi _0^2,0) + c_{12} \biggl ( \xi _0 \frac{\partial G_1}{\partial \xi _0}(\xi _0^2,0) + G_1(\xi _0^2,0) \biggr )\\&\quad = G_1(\xi _0^2,0)\frac{\partial G_1}{\partial \xi _0}(\xi _0^2,0)-\frac{\partial G_1}{\partial \xi _1}(\xi _0^2,0) \end{aligned}$$

values $c_{11} = 1$, $c_{12} = 0$. Now calculate the values of functions $G_1(\xi _0,0)$ on the circle $|\xi _0| = R$ and find function $h_1(\xi _0,0) = G_1(\xi _0,0)-c_{11}-c_{12}\xi _0$, $|\xi _0| = R$. This allows us to reconstruct the real curve $\Gamma = \left\{ (h_1(\xi _0,0),-\xi _0) \mid |\xi _0| = R \right\} \subseteq X_1$ (see Fig. 4).

Fig. 4

The contour of surface $X_1$ (black), boundary $\gamma _1$ of $X_1$ (blue), reconstructed curve $\Gamma$, belonging to $X_1$ (red) (Color figure online)

We apply further the reconstruction algorithm for the case of $\mu _0 = 0$ to the surface $X_1^R = \{ z \in X_1 \mid |z_2| \leqslant 60\}$ with boundary $bX_1^R = \gamma _1 + \Gamma$. We can compute the values of function

$$\begin{aligned} \sigma _0(\xi _0) = \frac{1}{2\pi i}\int \limits _{\gamma _1+\Gamma } \frac{dz_2}{z_2+\xi _0}. \end{aligned}$$

The value of function $\sigma _0(\xi _0)$ at point $\xi _0$ is equal to the number $N_+(\xi _0,0)$ of points of surface $X_1^R$ projected onto the point $\xi _0$ under projection $(z_1,z_2) \mapsto -z_2$. Further, for every point with $N_+(\xi _0,0) > 0$ we compute functions

$$\begin{aligned} s_k(\xi _0) = \frac{1}{2\pi i}\int \limits _{\gamma _1+\Gamma } \frac{z_1^k dz_2}{z_2+\xi _0}, \quad k=1,\ldots ,N_+(\xi _0,0). \end{aligned}$$

From functions $s_k(\xi _0)$ we can find functions $\sigma _k(\xi _0)$ using Newton identities:

$$\begin{aligned} k \sigma _k(\xi _0) = \sum \nolimits _{i=1}^k (-1)^{i-1} \sigma _{k-i}(\xi _0) s_{i}(\xi _0), \quad k=1,\ldots ,N_+(\xi _0,0). \end{aligned}$$

After, we find roots $h_1(\xi _0,0)$, ..., $h_{\sigma _0(\xi _0)}(\xi _0,0)$ of polynomial

$$\begin{aligned} t^{N_+(\xi _0)} - \sigma _1(\xi _0) t^{N_+(\xi _0)-1} + \cdots + (-1)^{N_+(\xi _0)} \sigma _{N_+(\xi _0,0)}(\xi _0) = 0. \end{aligned}$$

The points $\{ (h_k(\xi _0,0), -\xi _0) \mid k=1,\ldots ,N_+(\xi _0,0) \}$ represent the set of all points of $X_1^R$ projected onto $\xi _0$ by projection $(z_1,z_2) \rightarrow -z_2$. Visualization of the obtained set of points $\{ (h_k(\xi _0,0), -\xi _0) \}$ corresponding to varying $\xi _0$ can be realized by the visualization algorithm described in the previous section. The reconstructed surface is represented in Fig. 5.

Fig. 5

The contour of surface $X_1$ (black), boundary $\gamma _1$ of $X_1$ (blue), reconstructed curve $\Gamma$, belonging to $X_1$ (red), colored domains represent the reconstructed leaves of surface $X_1$ (Color figure online)

5.2 The Case of $\mu _0 = 2$

Consider an example of the reconstruction of the Riemann surface for the case of $\mu _0 = 2$. We are going to reconstruct the surface

$$\begin{aligned} X_2 = \left\{ (z_1,z_2) \in {\mathbb {C}}^2 \mid z_2 (z_1^2-1) = z_1 \exp (z_1^2), \quad |z_1| \leqslant 2 \right\} \end{aligned}$$

given its boundary $\gamma _2$ represented as an array of a finite number of equidistributed points on $\gamma _2$ (see Fig. 6).

Fig. 6

Boundary $\gamma _2$ of the surface $X_2$

Choose $R$ large enough, e.g., $R=60$, and consider the circle $C_R$ of radius $R$ in the $z_2$-plane centered at the origin. Compute for $\xi _0 \in C_R$ the values of functions

$$\begin{aligned} G_1(\xi _0,0)&= \frac{1}{2\pi i} \int \limits _{\gamma _2} \frac{z_1dz_2}{z_2 + \xi _0}, \\ G_2(\xi _0,0)&= \frac{1}{2\pi i} \int \limits _{\gamma _2} \frac{z_1^2 dz_2}{z_2 + \xi _0}, \\ \frac{\partial G_1}{\partial \xi _0}(\xi _0,0)&= -\frac{1}{2\pi i} \int \limits _{\gamma _2} \frac{z_1 dz_2}{(z_2+\xi _0)^2}, \\ \frac{\partial G_1}{\partial \xi _1}(\xi _0,0)&= \frac{1}{2\pi i}\int \limits _{\gamma _2} \left( \frac{z_1 dz_1}{z_2+\xi _0} - \frac{z_1^2 dz_2}{(z_2+\xi _0)^2} \right) ,\\ \frac{\partial G_2}{\partial \xi _1}(\xi _0,0)&= \frac{1}{2\pi i} \int \nolimits _{\gamma _2} \left( \frac{z_1^2 dz_1}{z_2+\xi _0}-\frac{z_1^3 dz_2}{(z_2+\xi _0)^2} \right) \end{aligned}$$

and the value of constant $\ae ^{00}_{10} = \frac{1}{2\pi i}\int _{\gamma _2} z_1 dz_2$, for example, using the method of rectangles.

In order to find constants $c_{11}$, $c_{12}$, $c_{21}$, $c_{22}$, $c_{23}$ we solve numerically the problem of minimization of the $L^2(C_R)$-norm of function

$$\begin{aligned} f_\text {error}(\xi _0)&= \frac{\partial G_2}{\partial \xi _1} - 2 \frac{\partial G_1}{\partial \xi _1} ( G_1 - c_{11} - c_{12} \xi _0) + G_1(c_{22}+2c_{23} \xi _0)\\&+ \frac{\partial G_1}{\partial \xi _0} \cdot \bigl ( (G_1 - c_{11} -c_{12}\xi _0)^2 - G_2 + c_{21} + c_{22}\xi _0 + c_{23}\xi _0^2 \bigr )\\&+ \bigl ( G_1^2 - 2c_{11} G_1 - 2c_{12} G_1 \xi _0 - G_2 \bigr )\cdot (-c_{12}) - \ae ^{00}_{10} (c_{12}^2 + c_{23}), \end{aligned}$$

in variables $c_{11}$, $c_{12}$, $c_{21}$, $c_{22}$, $c_{23}$. As a result of solving of this minimization problem we find $c_{11}=0$, $c_{12} = 0$, $c_{21} = 2$, $c_{22} = 0$, $c_{23} = 0$ (see Fig. 7).

Fig. 7

Graph of function $\Vert f_\text {error}\Vert _{L^2(C_R)}(c_{11},c_{21})$ in a neighborhood of the point of global minimum

Then we compute the power sums $s_1 = G_1 - c_{11} - c_{12}\xi _0$, $s_2 = G_2 - c_{21} - c_{22}\xi _0 - c_{23}\xi _0^2$ and symmetric functions $\sigma _1$, $\sigma _2$ on the circle $C_R$. Further, we desymmetrize functions $\sigma _1$, $\sigma _2$ to obtain functions $h_1(\xi _0,0)$ and $h_2(\xi _0,0)$ on the circle $C_R$, which perform lifting of the circle $C_R$ to the surface $X_2$. Denote by $\Gamma _{1,2} = \{ (h_{1,2}(z_2,0),-z_2) \mid z_2 \in C_R\}$ the curves, obtained by the corresponding lifting of $C_R$ to $X_2$ (see Fig. 8).

Fig. 8

The contour of surface $X_2$ (black), boundary $\gamma _2$ of $X_2$ (blue), reconstructed curves $\Gamma _{1,2}$, belonging to $X_2$ (red) (Color figure online)

Now we consider the curve $\gamma _2+\Gamma _1+\Gamma _2$ as a new initial curve and we reconstruct the surface $X_2^R = \left\{ z \in X_2 \mid |z_2| \leqslant R \right\}$, $bX_2^R = \gamma _2 + \Gamma _1 + \Gamma _2$. Further, our considerations are similar to those for the case of $\mu _0 = 1$. At first, we compute functions $s_k$. Then, we find symmetric functions $\sigma _k$. Further, we solve the algebraic equation (numerically) and find functions $h_k(\xi _0,0)$, $k=1$, ..., $N_+(\xi _0,0)$. The reconstructed surface is given by Fig. 9.

Fig. 9

The contour of surface $X_2$ (black), boundary $\gamma _2$ of $X_2$ (blue), reconstructed curves $\Gamma _{1,2}$, belonging to $X_2$ (red), reconstructed leaves of the surface are represented by dark-blue, orange, red and blue domains (Color figure online)