# Tensor Monopoles in Superconducting Systems

Hannes Weisbrich  Fachbereich Physik, Universität Konstanz, D-78457 Konstanz,
Germany  Markus Bestler  Fachbereich Physik, Universität Konstanz, D-78457
Konstanz, Germany  Wolfgang Belzig  Fachbereich Physik, Universität Konstanz,
D-78457 Konstanz, Germany<EMAIL_ADDRESS>https://www.belzig.uni-konstanz.de

###### Abstract

Topology in general but also topological objects such as monopoles are a
central concept in physics. They are prime examples for the intriguing physics
of gauge theories and topological states of matter. Vector monopoles are
already frequently discussed such as the well-established Dirac monopole in
three dimensions. Less known are tensor monopoles giving rise to tensor gauge
fields. Here we report that tensor monopoles can potentially be realized in
superconducting multi-terminal systems using the phase differences between
superconductors as synthetic dimensions. In a first proposal we suggest a
circuit of superconducting islands featuring charge states to realize a tensor
monopole. As a second example we propose a triple dot system coupled to
multiple superconductors that also gives rise to such a topological structure.
All proposals can be implemented with current experimental means and the
monopole readily be detected by measuring the quantum geometry.

##  1 Introduction

The concept of monopoles was initially introduced in the framework of
electrodynamics to prove the quantization of the electric charge [1]. This
idea of the electrodynamic monopoles was later translated to condensed matter
using the geometrical properties of Bloch states [2]. Here, the central object
describing the geometry of these quantum systems in parameter space is the
Berry connection with its physical observable: the geometric Berry phase [3].
As the Berry connection has the form of a gauge potential in parameter space
one can define a corresponding gauge invariant vector field, that is nothing
else than the Berry curvature. Similar as in the electrodynamic case,
monopoles can be defined in the parameter space of these quantum systems.
Monopoles are sources of the gauge invariant vector field, the Berry
curvature, and, hence, a quantized charge can be related to a topological
invariant. For instance, in the case of the 3D Dirac monopole the
corresponding topological invariant is the first Chern number determining the
quantized topological charge [4]. This topological invariant classifies also a
wide spectrum of topological states of matter as in topological insulators
[5], topological superconductors [6], or in the quantum Hall effect [7, 8].
Indeed the topology emerges in these cases from artificial Dirac monopoles in
momentum space. This idea of monopoles can be also extended to higher
dimension, for instance in 5D the non-Abelian Yang monopole [9] gives rise to
a non-Abelian Berry curvature [10] with its topological charge determined by
the second Chern number. Interestingly, such a Yang monopole was realized
experimentally in ultracold atoms [11]. More generally the charge of monopoles
in $2n+1$ dimensions can be described by the $n$-th Chern number [12]. As
these monopoles in odd dimensions give rise to vector gauge fields, they are
classified as vector monopoles.

Another class are tensor monopoles [13, 14, 15] being sources of tensor gauge
fields which are for instance crucial in string theory [16, 17, 18]. The
elementary case is the Abelian 4D tensor monopole with its topological charge
defined as Dixmier-Douady invariant [19, 20]. It was shown that similar as the
Berry curvature one can define a tensor Berry curvature [21, 22] to realize
fictitious tensor monopoles in quantum systems. As such tensor monopole
require an higher dimensional phase space they can only exist in synthetic
dimensions. For instance, it was shown that using the coupling parameters of a
superconducting qudrit as artificial dimensions the 4D tensor monopole could
be experimentally observed [23]. In a similar experiment a single NV center
was used to create this topological structure of a tensor monopole [24].
Another prominent approach to topology in synthetic dimensions are
multiterminal Josephson junctions [25, 26, 27, 28, 29, 30, 31, 32, 33, 34] or
topological superconducting circuits [35, 36, 37], where the topology is
defined in the space of superconducting phases. Due to the scalability,
superconducting Josephson circuits are the most promising candidate for
realistic implementations of quantum technologies, see for example the reviews
[38, 39]. In the context of topology various topological states of matter can
emerge in these systems without dimensional limitations, thus they provide a
promising tool to explore the physics of more exotic topological states of
matter such as the one of tensor monopoles.

The goal of this work is to show that tensor monopoles can emerge in such
superconducting systems with the monopole defined in the space of the
superconducting phases. In the first section we recall the concept of the
tensor Berry curvature and tensor monopoles in quantum systems following Ref.
[21, 22]. In the second section, we describe the superconducting circuit using
three superconducting islands to realize a synthetic tensor monopole in the
lowest charge states of the circuit. In the last part, we study a system with
three dots coupled through superconducting leads.

Figure 1: (a) Superconducting circuit with three superconducting islands
(black dots) that simulates a tensor monopole using the lowest charge states.
The middle superconductor is the reference node, whereas left and right island
are described by their offset charge $n_{gL/R}$. These islands are coupled by
several Josephson junctions $J_{L/R\,i}$ (black boxes). The four loops
appearing in this circuit are threaded by the magnetic fluxes
$\varphi_{x},\varphi_{y},\varphi_{z},\varphi_{w}$. (b) Parameter region where
tensor monopoles can appear depending on the Josephson energies of the left
Josephson junctions. The condition for the right Josephson junctions is
equivalent. (c) Energies in the topological region (upper graph at
$E_{J_{L3}}=0.5E_{J_{L1}}$ and $E_{J_{L2}}=1.3E_{J_{L1}}$) with the
topological charge of the emerging tensor monopoles $Q_{T}=\pm 1$ and in the
gapped region (lower graph at $E_{J_{L3}}=0.5E_{J_{L1}}$ and
$E_{J_{L2}}=1.6E_{J_{L1}}$) with $E_{J_{R1}}=E_{J_{R2}}=E_{J_{R3}}$ at
$\varphi_{z}=-1/2\varphi_{w}=2\pi/3$.

##  2 Concept of tensor monopoles

To explain the concept of tensor monopoles we repeat the idea of the better
known Dirac monopole in quantum systems. Here the most simple example has the
form $H_{\mathrm{3D}}=\bm{q}\bm{\sigma}$ with $\bm{q}=(q_{x},q_{y},q_{z})$ and
the Pauli matrices $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. This
Hamiltonian has two eigenstates with energy $E_{\pm}=\pm|\bm{q}|$. Analogously
to the electromagnetic gauge potential one can define the Berry connection
$A_{\mu}=i\bra{\psi(\bm{q})}\partial_{q_{\mu}}\ket{\psi(\bm{q})}$ with
$\ket{\psi(\bm{q})}$ being the lower eigenstate. The Berry connection here
plays the role of a gauge potential in parameter space $\bm{q}$, whereas the
gauge invariant vector field is then the Berry curvature
$F_{\mu\nu}=\partial_{q_{\mu}}A_{\nu}-\partial_{q_{\nu}}A_{\mu}$. To verify
this property of gauge invariance one can define the gauge transformation
$\ket{\psi}\rightarrow e^{i\gamma}\ket{\psi}$ which transforms the Berry
connection as
$A_{\mu}=i\bra{\psi}\partial_{q_{\mu}}\ket{\psi}-\partial_{q_{\mu}}\gamma$,
whereas the additional term cancels out in the Berry curvature. For the above
mentioned example the Berry curvature takes the analytical form
$F_{\mu\nu}=\epsilon_{\mu\nu\lambda}\frac{q_{\lambda}}{2|\bm{q}|^{3}}$. Here
the Berry curvature has a singularity at $\bm{q}=0$ when the two eigenstates
with $E_{\pm}=\pm|\bm{q}|$ are degenerate which defines the position of the
vector monopole in parameter space. Similar as in the electromagnetic case the
quantized charge of the monopole can be determined by a closed 2D surface
integral of the gauge invariant vector field defined as the first Chern number
$C^{(1)}=1/(2\pi)\int_{S^{2}}F$.
This idea of fictitious monopoles in parameter space can be expanded to even
dimension with a generalized tensor Berry connection/curvature [21, 22]. Again
the idea is best illustrated by a simple example

$\displaystyle
H_{\mathrm{4D}}=\bm{q}\bm{\lambda}=\begin{pmatrix}0&q_{x}-iq_{y}&0\\\
q_{x}+iq_{y}&0&q_{z}+iq_{w}\\\ 0&q_{z}-iq_{w}&0\end{pmatrix}$ (1)

with $\bm{q}=(q_{x},q_{y},q_{z},q_{w})$ and
$\bm{\lambda}=(\lambda_{1},\lambda_{2},\lambda_{6},\lambda_{7}^{*})$ being
four out of eight Gell-Mann-matrices [40] with
$\text{tr}(\lambda_{i},\lambda_{j})=2\delta_{ij}$ analogously to the lower
dimensional Pauli-matrices in the case of the Dirac monopole. This Hamiltonian
has in general three eigenstates with $E_{0}=0$, and $E_{\pm}=\pm|\bm{q}|$. We
can see that we have an exceptional Weyl-like point when $|\bm{q}|=0$. The
generalized (abelian) tensor Berry connection for the lowest eigenstate
$\ket{\psi_{E_{-}}}=1/\sqrt{2}(v_{1},-1,v_{2})$ with
$v_{1}=(q_{x}-iq_{y})/|\bm{q}|$ and $v_{2}=(q_{z}-iq_{w})/|\bm{q}|$ can be
defined as [21, 22]

$\displaystyle
B_{\mu\nu}=\frac{i}{3}\sum_{j,k,l=1}^{3}\epsilon_{jkl}\alpha_{j}\left(\partial_{q_{\mu}}\alpha_{k}\right)\left(\partial_{q_{\nu}}\alpha_{l}\right)$
(2)

with $\alpha_{1}=-i\log(v_{2})$ and $\alpha_{2}=\alpha_{3}^{*}=v_{1}^{*}$ and
the corresponding tensor Berry curvature

$\displaystyle\mathcal{H}_{\mu\nu\lambda}=\partial_{q_{\mu}}B_{\nu\lambda}+\partial_{q_{\nu}}B_{\lambda\mu}+\partial_{q_{\lambda}}B_{\mu\nu}\,.$
(3)

One can verify that under the gauge transformation
$\ket{\psi_{E_{-}}}\rightarrow e^{i\gamma}\ket{\psi_{E_{-}}}$ the tensor Berry
connection transforms as $B_{\mu\nu}\rightarrow B_{\mu\nu}+\Lambda_{\mu\nu}$
with $\Lambda_{\mu\nu}$ being a function of the fields $\alpha_{i}$ and
$\gamma$, whereas the tensor Berry curvature $\mathcal{H}_{\mu\nu\lambda}$ is
invariant under a gauge transformation. For the example of Eq. 1 the tensor
Berry curvature takes the analytical form
$\mathcal{H}_{\mu\nu\lambda}=\epsilon_{\mu\nu\lambda\gamma}\frac{q_{\gamma}}{|\bm{q}|^{4}}$
indicating the position of the tensor monopole at $|\bm{q}|=0$ when the three
states are degenerate. The quantized charge of the tensor monopole can be
calculated by the Dixmier-Douady invariant [19, 20, 21] by integrating the
tensor Berry curvature over a 3D hyper surface enclosing the tensor monopole

$\displaystyle
Q_{T}=\frac{1}{2\pi^{2}}\int_{S^{3}}\mathcal{H}_{\mu\nu\lambda}\,dq^{\mu}\wedge
dq^{\nu}\wedge dq^{\lambda}$ (4)

with $\wedge$ being the wedge product. This results in $Q_{T}=1$ for the
discussed example of a single tensor monopole.

##  3 Tensor monopoles in a superconducting circuit

Figure 2: (a) Energy of the three lowest energy levels at
$\varphi_{y}=\varphi_{w}=-\frac{4\pi}{3}$ with a tensor monopole ($Q_{T}=-1$)
at $\varphi_{x}=\varphi_{z}=\frac{2\pi}{3}$ for equal Josephson energies
$E_{J}=E_{J_{L/R\,i}}$. (b) Tensor Berry curvature $\mathcal{H}_{xyz}$ near
the the tensor monopole from (a).

The system illustrated in Fig. 1(a) is a superconducting circuit with three
superconducting islands (left, middle, and right), whereas the middle island
is grounded and connected by six Josephson Junction to the left and right
island. We assume that the Josephson Junctions $J_{L/R\,i}$ with $i=1,2,3$ are
characterized by their capacitance $C_{L/R\,i}$ and their respective Josephson
energy $E_{J_{L/R\,i}}$. Additionally the circuit loops are threaded by the
magnetic fluxes $\varphi_{x},\varphi_{y}$ (left) and $\varphi_{z},\varphi_{w}$
(right) which are also used to construct the artificial phase space for the
tensor monopoles. Following Ref. [41] the Hamiltonian for this system can be
in general expressed as

$\displaystyle H_{circ}$
$\displaystyle=\frac{(2e)^{2}}{2}(\hat{\bm{n}}-\bm{n}_{g})^{T}\bm{C}^{-1}(\hat{\bm{n}}-\bm{n}_{g})+E_{J_{L\,1}}\cos\left(\hat{\phi}_{L}\right)+E_{J_{L\,2}}\cos\left(\hat{\phi}_{L}-\varphi_{x}\right)+E_{J_{L\,3}}\cos\left(\hat{\phi}_{L}-\varphi_{x}-\varphi_{y}\right)$
$\displaystyle+E_{J_{R\,1}}\cos\left(\hat{\phi}_{R}\right)+E_{J_{R\,2}}\cos\left(\hat{\phi}_{R}+\varphi_{z}\right)+E_{J_{R\,3}}\cos\left(\hat{\phi}_{R}+\varphi_{z}+\varphi_{w}\right)$
(5)

with $\hat{\bm{n}}=(\hat{n}_{L},\hat{n}_{R})$ the number operator of Cooper
pairs on the left and right islands, $\bm{n}_{g}=(n_{gL},n_{gR})$ are the
offset charges of each island controlled by gate voltages, $\hat{\phi}_{L/R}$
are the superconducting phase operators of the left and right superconductors
island in respect to the grounded island in the middle, and $E_{J_{L/R\,i}}$
are the Josephson energies of the respective junctions. Note that a specific
gauge has been chosen here for the magnetic fluxes. The inverse capacitance
matrix $\bm{C}^{-1}$ takes the diagonal form
$\bm{C}^{-1}=\text{diag}\left(\frac{1}{C_{L1}+C_{L2}+C_{L3}},\frac{1}{C_{R1}+C_{R2}+C_{R3}}\right)$.
For the sake of simplicity we will assume in the following that all junctions
have the same capacitance $C\equiv C_{L/R\,i}$.
In the charging limit when $E_{C}=(2e)^{2}/(2C)\gg E_{J}$ the number of
charges on the islands is a well defined quantity. Hence, the lowest energy
states are the three charge states with zero Cooper pairs on the islands
$\ket{0}\equiv\ket{0,0}$ (with $\ket{n_{L},n_{R}}$), one Cooper pair on the
left island $\ket{L}\equiv\ket{1,0}$, and one Cooper pair on the right island
$\ket{R}\equiv\ket{0,1}$. These charge states can be used to construct the low
energy Hamiltonian in this limit

$\displaystyle H_{0}$
$\displaystyle=\frac{E_{C}}{3}(1-2n_{gL})\ket{L}\bra{L}+\frac{E_{C}}{3}(1-2n_{gR})\ket{R}\bra{R}$
$\displaystyle-
f_{L}(\varphi_{x},\varphi_{y})\ket{L}\bra{0}-f_{R}(\varphi_{z},\varphi_{w})\ket{0}\bra{R}+h.c.$

up to a constant and with
$f_{L}(\varphi_{x},\varphi_{y})=E_{J_{L1}}+E_{J_{L2}}e^{i\varphi_{x}}+E_{J_{L3}}e^{i(\varphi_{x}+\varphi_{y})}$
and
$f_{R}(\varphi_{z},\varphi_{w})=E_{J_{R1}}+E_{J_{R2}}e^{i\varphi_{z}}+E_{J_{R3}}e^{i(\varphi_{z}+\varphi_{w})}$.
See Appendix A for a more detailed derivation of the low energy Hamiltonian.
The energy of the lowest state in general is dominated by the diagonal
charging energy, however at the charge degenerate point when
$n_{gL}=n_{gR}=1/2$ all three states have the same charging energy, thus the
off-diagonal Josephson part dominates the energetic structure. In this case,
the Hamiltonian takes a similar form as the tensor monopole in Sec. 2 and can
be expressed with the Gell-Mann matrices $\lambda_{i}$ in the space
$(\ket{L},\ket{0},\ket{R})$ as

$\displaystyle H_{0}=$
$\displaystyle\Re(f_{L}(\varphi_{x},\varphi_{y}))\lambda_{1}-\Im(f_{L}(\varphi_{x},\varphi_{y}))\lambda_{2}$
$\displaystyle+\Re(f_{R}(\varphi_{z},\varphi_{w}))\lambda_{6}+\Im(f_{R}(\varphi_{z},\varphi_{w}))\lambda_{7}^{*}\,.$
(6)

The energies are $E_{0}=0$, and
$E_{\pm}=\pm\sqrt{|f_{L}(\varphi_{x},\varphi_{y})|^{2}+|f_{R}(\varphi_{z},\varphi_{w})|^{2}}$.
Thus it allows for triply degenerate points in the phase space
$(\varphi_{x},\varphi_{y},\varphi_{z},\varphi_{w})$ similar to the example in
Sec. 2. The degenerate points can be determined by the condition
$|f_{L}(\varphi_{x},\varphi_{y})|=|f_{R}(\varphi_{z},\varphi_{w})|=0$ which
depends on the Josephson energies of the individual junctions. In general the
condition for $E_{J_{L\,i}}$ and $(\varphi_{x},\varphi_{y})$ are independent
of the condition for $E_{J_{R\,i}}$ and $(\varphi_{z},\varphi_{w})$ as the two
functions $f_{L}$ and $f_{R}$ are independent of each other. As a result of
the periodicity of the functions $f_{L/R}$ four single degenerate points can
emerge in the phase space if
$E_{J_{L\,2}}+E_{J_{L\,3}}>E_{J_{L\,1}}>|E_{J_{L\,2}}-E_{J_{L\,3}}|$ and
$E_{J_{R\,2}}+E_{J_{R\,3}}>E_{J_{R\,1}}>|E_{J_{R\,2}}-E_{J_{R\,3}}|$, as
illustrated in Fig. 1. If one of these conditions is not fulfilled a gapped
phase appears and no tensor monopole can emerge in the system. However, as
shown in Fig. 1, a wide parameter region exists for which these degenerate
points occur.

Using the definition of the tensor Berry connection
$B_{\mu\nu}=\frac{i}{3}\sum_{jkl}\epsilon_{jkl}\alpha_{j}\left(\partial_{\varphi_{\mu}}\alpha_{k}\right)\left(\partial_{\varphi_{\nu}}\alpha_{l}\right)$
and the tensor Berry curvature
$\mathcal{H}_{\mu\nu\lambda}=\partial_{\varphi_{\mu}}B_{\nu\lambda}+\partial_{\varphi_{\nu}}B_{\lambda\mu}+\partial_{\varphi_{\lambda}}B_{\mu\nu}$
one can verify that in the topological region the degenerate points are indeed
tensor monopoles with charges $Q_{T}=\pm 1$. The charge was evaluated
numerically by a Monte-Carlo integration of the tensor Berry curvature around
a small hypercube around the degenerate points, see Appendix B for the
concrete integral. Note that the result is independent of the surface
enclosing the degenerate point, e.g. for a hyper spherical surface the
integral results in the same topological charge. In case of equal Josephson
energies $E_{J}=E_{J_{L/R\,i}}$, the position of these tensor monopoles are
independent of the Josephson energies $E_{J}$ and have the simple form
$\varphi_{x0}=s_{1}\frac{2\pi}{3}$, $\varphi_{y0}=-s_{1}\frac{4\pi}{3}$,
$\varphi_{z0}=s_{2}\frac{2\pi}{3}$, $\varphi_{w0}=-s_{2}\frac{4\pi}{3}$ with
$s_{1}=\pm 1$ and $s_{2}=\pm 1$. The topological charge of these four tensor
monopoles are $Q_{T}=-s_{1}s_{2}$. Another characteristic behavior is the
diverging tensor curvature at the monopoles position which can be seen in Fig.
2 for the example of $\mathcal{H}_{xyz}$, with similar results for the other
components of $\mathcal{H}_{\mu\nu\lambda}$. Here, the diverging peak
coincides with the position where the three states are degenerate confirming
that a tensor monopole is indeed observed.

##  4 Tensor monopoles in a triple dot system

In the following we will discuss another system using three dots coupled to
superconducting leads as illustrated in Fig. 4(a). The Hamiltonian of the dots
is
$H_{\mathrm{d}}=\sum_{\alpha=L,M,R}\sum_{\sigma}\left(\epsilon_{\alpha}d_{\alpha\sigma}^{\dagger}d_{\alpha\sigma}+Un_{\alpha\downarrow}n_{\alpha\uparrow}\right)$
with the fermionic operators $d_{\alpha}$ of the dots (left dot $\alpha=L$,
middle dot $\alpha=M$ and right dot $\alpha=R$), $\epsilon_{\alpha}$ the
energy of the dot levels, and $U$ the Coulomb energy taken to be same for all
dots. The superconducting leads with the phases $\phi_{j}$ are described by a
BCS Hamiltonian
$H_{\mathrm{s},j}=\sum_{\bm{k}\sigma}\epsilon_{\bm{k}}c_{j\bm{k}\sigma}^{\dagger}c_{j\bm{k}\sigma}+\sum_{\bm{k}}\left(\Delta
e^{i\phi_{j}}c_{j\bm{k}\uparrow}^{\dagger}c_{j-\bm{k}\downarrow}^{\dagger}+h.c.\right)$,
where $\Delta$ is the superconducting gap (assumed to be the same for all
leads), $\epsilon_{k}$ the normal state dispersion, and $c_{j\bm{k}\sigma}$ is
the creation operator of an electron with momentum $\bm{k}$ and spin $\sigma$
in the respective lead with phase $\phi_{j}$. Additionally a normal coupling
between the dots
$H_{\mathrm{c,d}}=\sum_{\sigma}v_{L}d_{L\sigma}^{\dagger}d_{M\sigma}+v_{R}d_{M\sigma}^{\dagger}d_{R\sigma}+h.c.$,
a coupling between neighboring dots and leads
$H_{\mathrm{c},\mathrm{d}_{\alpha}-\mathrm{s}_{j}}=\sum_{\sigma
k}v_{j}\left(d_{\alpha\sigma}^{\dagger}c_{jk\sigma}+h.c.\right)$, and a
coupling between neighboring superconducting leads
$H_{\mathrm{c},\mathrm{s}_{i}-\mathrm{s}_{j}}=\sum_{\sigma
k}w_{ij}\left(c_{ik\sigma}^{\dagger}c_{jk\sigma}+h.c.\right)$ is assumed. The
total Hamiltonian can then be summarized to

$\displaystyle H_{\mathrm{tot}}=$ $\displaystyle
H_{\mathrm{d}}+H_{\mathrm{c,d}}+\sum_{j}H_{\mathrm{s},j}+\sum_{\braket{\alpha,j}}H_{\mathrm{c},\mathrm{d}_{\alpha}-\mathrm{s}_{j}}$
$\displaystyle+\sum_{\braket{i,j}}H_{\mathrm{c},\mathrm{s}_{i}-\mathrm{s}_{j}}\,,$
(7)

with $\sum_{\braket{\alpha,j}}$ denoting the sum over the three dots
($\alpha=L,M,R$) and their respective nearest leads ($j$) and
$\sum_{\braket{i,j}}$ the sum over nearest neighbors of superconducting leads
as illustrated in Fig. 4.

Figure 3: (a) Three dots coupled via superconducting leads with the phases
$\phi_{x},\phi_{y},\phi_{z},\phi_{w}$ and $\phi_{0}=0$ as reference phase.
Black arrows indicate a normal coupling between dots, whereas blue arrows
indicate the indirect coupling via the superconducting leads. (b) This
indirect coupling process via the leads is a result of two crossed Andreev
reflections where in one lead a new Cooper pair is combined and in the other
lead a Cooper pair is split which results in an effective electron transport
between two dots. Figure 4: (a) Parameter region for $v_{L}/\Gamma$ and
$v_{R}/\Gamma$ with four tensor monopoles for equal non-local couplings
$\Gamma\equiv\Gamma_{i}$ in the triple dot system. (b) Energy in dependence of
$\phi_{x}$ and $\phi_{y}$ for $-\phi_{z}=\phi_{w}=\arccos(v/2\Gamma)$,
$\epsilon=0$, and for different ratios $v/\Gamma$ ($v=v_{L}=v_{R}$) marked in
(a) as black crosses. The topological charge of appearing monopoles are
indicated by $Q_{T}=\pm 1$.

As we are only interested in the low energy states within the gap, the limit
of $\Delta\rightarrow\infty$ is assumed. For a weak coupling between the
superconducting leads $w_{ij}\ll v_{j}$ one can derive an effective
Hamiltonian for the dot states by integrating out the superconductors. The
details of this calculation are given in the Appendix C. In general the
coupling between leads and dots can lead to Cooper pair injections on the dots
which is however excluded by a large local Coulomb interaction $U$ suppressing
double occupation on the dots. An additional effect of the coupling between
superconductors and dots is schematically illustrated in Fig.3(b), here the
coupling enables a non-local tunneling of electrons via two superconductors.
This non-local transport is possible due to two crossed Andreev reflections
where in one lead a new Cooper pair is combined and in the other lead a Cooper
pair is split which results in an effective transport between two dots. The
crucial point of this nonlocal transport is the dependence on the phase
difference of the two SC, which allows to construct an effective Hamiltonian
for the low energy state with artificial dimensions being the phase
differences of the leads.

As a result of these transport processes enabled by the coupling between the
dots and the superconductors the low-energy Hamiltonian is conserving electron
number and the spin. Hence, we obtain the effective single-particle
Hamiltonian

$\displaystyle H_{\mathrm{eff}}$
$\displaystyle=\sum_{\alpha=L,M,R}\epsilon_{\alpha}\ket{\alpha_{\sigma}}\bra{\alpha_{\sigma}}$
$\displaystyle+f_{LM}(\phi_{x},\phi_{y})\ket{M_{\sigma}}\bra{L_{\sigma}}+h.c.$
$\displaystyle+f_{MR}(\phi_{z},\phi_{w})\ket{R_{\sigma}}\bra{M_{\sigma}}+h.c\,.$
(8)

where we denoted the states $\ket{L_{\sigma}}\equiv
d_{L\sigma}^{\dagger}\ket{0}$ (with $\ket{0}$ the state without electrons on
the dots), $\ket{M_{\sigma}}\equiv d_{M\sigma}^{\dagger}\ket{0}$,
$\ket{R_{\sigma}}\equiv d_{R\sigma}^{\dagger}\ket{0}$, and the total couplings
$f_{LM}=v_{L}-\Gamma_{x}e^{-i\phi_{x}}+\Gamma_{y}e^{-i\phi_{y}}$,
$f_{MR}=v_{R}-\Gamma_{z}e^{i\phi_{z}}+\Gamma_{w}e^{i\phi_{w}}$. Note that
gauge invariance allows us to set one phase to zero ($\phi_{0}=0$) as a
reference. In addition $\Gamma_{j}=2\pi N_{0}v_{j}v_{0}w_{j0}$ are the non-
local tunneling amplitudes with $N_{0}$ the normal density of states at the
Fermi energy of the superconductors. For equal energies of the dots
$\epsilon_{\alpha}=\epsilon$ the energy of the lowest states read
$E_{0}=\epsilon$ and $E_{\pm}=\epsilon\pm\sqrt{|f_{LM}|^{2}+|f_{MR}|^{2}}$.
Hence in general for
$\Gamma_{x}+\Gamma_{y}>v_{L}>\big{|}\Gamma_{x}-\Gamma_{y}\big{|}$ and
$\Gamma_{z}+\Gamma_{w}>v_{R}>\big{|}\Gamma_{z}-\Gamma_{w}\big{|}$ (assuming
$\Gamma_{i},v_{i}>0$) four isolated triply degenerate points appear.

The topological charges of these points can be again determined by an
integration of the tensor Berry curvature $\mathcal{H}_{\mu\nu\lambda}$ over a
closed surface around the degeneracies which yields the charges $Q_{T}=\pm 1$.
Otherwise, if one of the condition is not fulfilled, a gapped phase emerges as
illustrated in Fig. 4. In the limit of symmetric couplings $v\equiv v_{L/R}$
and $\Gamma\equiv\Gamma_{j}$ and $0<v<2\Gamma$ the four triply degenerate
point can be observed at $\phi_{x0}=s_{1}\arccos(v/2\Gamma)$,
$\phi_{y0}=-s_{1}\arccos(v/2\Gamma)$, $\phi_{z0}=s_{2}\arccos(v/2\Gamma)$, and
$\phi_{w0}=-s_{2}\arccos(v/2\Gamma)$ with $s_{1}=\pm 1$, $s_{2}=\pm 1$ and the
topological charge $Q_{T}=s_{1}s_{2}$. In the case of $v=0$ degenerate lines
in the phase space of $(\phi_{x},\phi_{y},\phi_{z},\phi_{w})$ appear at
$\phi_{x}=\phi_{y}+\pi$ and $\phi_{z}=\phi_{w}+\pi$ which cannot be classified
as tensor monopoles as there are no isolated degenerate points in this case.
For $2\Gamma<v$ a gapped phase appears and, thus, no monopoles with
degeneracies can emerge, as depicted in Fig.4(b). We note, that both a direct
coupling between the dots and comparable non-local coupling via the
superconductors are necessary to have tensor monopoles in the triple dot
chain.

In the regime $0<\frac{v}{\Gamma}<2$ the Hamiltonian can be linearized close
to the degenerate points $(\phi_{x0},\phi_{y0},\phi_{z0},\phi_{w0})$ to prove
a similar form as in the example of Sec. 2. The linearized couplings near
these degeneracies are

$\displaystyle f_{LM}\approx$ $\displaystyle
s_{1}\Gamma\sqrt{1-\left(\frac{v}{2\Gamma}\right)^{2}}\left(\delta\phi_{x}-\delta\phi_{y}\right)$
$\displaystyle-is_{1}\frac{v}{2}\left(\delta\phi_{x}+\delta\phi_{y}\right)$
(9) $\displaystyle f_{MR}\approx$ $\displaystyle
s_{2}\Gamma\sqrt{1-\left(\frac{v}{2\Gamma}\right)^{2}}\left(\delta\phi_{z}-\delta\phi_{w}\right)$
$\displaystyle+is_{2}\frac{v}{2}\left(\delta\phi_{z}+\delta\phi_{w}\right)$
(10)

with $\delta\phi_{j}=\phi_{j}-\phi_{j0}$ the detuning of the phases from the
position of the tensor monopoles. Thus it defines the same Hamiltonian of Eq.1
with $q_{x}\equiv
s_{1}\Gamma\sqrt{1-\left(\frac{v}{2\Gamma}\right)^{2}}\left(\delta\phi_{x}-\delta\phi_{y}\right)$,
$q_{y}\equiv s_{1}\frac{v}{2}\left(\delta\phi_{x}+\delta\phi_{y}\right)$,
$q_{z}\equiv
s_{2}\Gamma\sqrt{1-\left(\frac{v}{2\Gamma}\right)^{2}}\left(\delta\phi_{z}-\delta\phi_{w}\right)$,
and $q_{w}\equiv s_{2}\frac{v}{2}\left(\delta\phi_{z}+\delta\phi_{w}\right)$
ultimately with the same analytical form of the tensor Berry curvature
$\mathcal{H}_{\mu\nu\lambda}=\epsilon_{\mu\nu\lambda\gamma}\frac{q_{\gamma}}{|\bm{q}|^{4}}$
in the vicinity of the tensor monopoles.

##  5 Discussion

In this work we have theoretically studied tensor monopoles in superconducting
systems. These synthetic tensor monopoles which are defined in the 4D-space of
the superconducting phases can be characterized by a tensor Berry curvature
inspired by the better-known Berry curvature. We have shown that there is a
wide variety of approaches in superconducting systems to realize this exotic
type of topology. Our first proposal uses the Josephson-coupled charge states
of superconducting islands to realize the necessary topological structure. The
second example realizes the monopoles in the low energy states of three dots
coupled to several superconducting terminal. Remarkably both systems allow to
engineer nontrivial topological tensor monopoles in a large parameter region
implying robustness against certain fluctuations of the systems parameters. In
the example of the superconducting circuit however larger charge fluctuations
could be detrimental for the degenerate points in parameter space, as it
requires having three degenerate charge states. Similar problems might occur
in the second proposal in respect to large asymmetric fluctuations in the dots
energies.

In a setup using the coupling parameters of a superconducting qudrit to
construct the required artificial 4D space these tensor monopoles were already
studied experimentally [23], as well as in [24] where a single NV center was
observed to have the same topological structure. In both cases the tensor
Berry curvature was measured indirectly via the quantum geometric tensor which
can be used to reconstruct the tensor Berry curvature [22]. In the first
experiment the quantum geometric tensor was measured by quenching the system
[23]. In the second example the quantum geometric tensor was determined by
evaluating the Rabi frequencies for small modulations of the coupling
parameters [24]. The same approaches should be also feasible in the proposed
systems. A similar manipulation of the phases could be realized by controlling
the magnetic fluxes, either by applying a small periodic modulation or by a
sudden quench of the flux. Eventually, the response of the system to these
perturbations can be used to measure the tensor Berry curvature in the space
of the superconducting phases and ultimately to determine the respective
topological invariant.
To conclude we emphasize that our proposal to realize tensor monopoles in
superconducting nanostructures could stimulate further research in the field
of tensor monopoles and the respective topology in superconducting systems and
Josephson matter systems. It is the first proposal of using superconducting
phases as synthetic dimensions to realize tensor monopoles in superconducting
systems. In principle this idea of the tensor Berry connection/curvature and
the respective tensor monopole could be expanded to higher-order tensors [21].
This seems experimentally accessibly in superconducting systems due to the
unlimited dimensions available in the space of superconducting phases
providing an alternative path to construct exotic topological states of
matter.

The authors acknowledge funding provided by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) Grant No. RA 2810/1
and SFB 1432 – Project-ID 425217212.

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##  Appendix A Derivation of the low energy Hamiltonian of the superconducting
circuit

The Hamiltonian for the circuit can be derived with the Lagrangian following
Ref. [41] by $\mathcal{L}=\mathcal{L}_{c}-U_{J}$ with the charging term of the
Lagrangian $\mathcal{L}_{c}$ and $U_{J}$ the Josephson energy of the
junctions.

$\displaystyle\mathcal{L}_{c}=$
$\displaystyle\frac{\hbar^{2}}{8e^{2}}\left(\sum_{i=1}^{3}C_{Li}\dot{\phi}_{L}^{2}+\sum_{i=1}^{3}C_{Ri}\dot{\phi}_{R}^{2}\right)=\frac{\hbar^{2}}{8e^{2}}\dot{\bm{\phi}}\bm{C}\dot{\bm{\phi}}$
(11) $\displaystyle U_{j}=$ $\displaystyle-
E_{J_{L1}}\cos(\phi_{L})-E_{J_{L2}}\cos(\phi_{L}-\varphi_{x})-E_{J_{L3}}\cos(\phi_{L}-\varphi_{x}-\varphi_{y})$
$\displaystyle-
E_{J_{R1}}\cos(-\phi_{R})-E_{J_{R2}}\cos(-\phi_{R}-\varphi_{z})-E_{J_{R1}}\cos(-\phi_{R}-\varphi_{z}-\varphi_{w})$
(12)

with the capacitance matrix $\bm{C}$, and $\bm{\phi}=(\phi_{L},\phi_{R})$.
Assuming gate voltages on each island which results in the offset charge
$(n_{gL},n_{gR})$ the Hamiltonian takes the form

$\displaystyle H_{circ}$
$\displaystyle=\frac{(2e)^{2}}{2}(\hat{\bm{n}}-\bm{n}_{g})^{T}\bm{C}^{-1}(\hat{\bm{n}}-\bm{n}_{g})+E_{J_{L\,1}}\cos\left(\hat{\phi}_{L}\right)+E_{J_{L\,2}}\cos\left(\hat{\phi}_{L}-\varphi_{x}\right)+E_{J_{L\,3}}\cos\left(\hat{\phi}_{L}-\varphi_{x}-\varphi_{y}\right)$
$\displaystyle+E_{J_{R\,1}}\cos\left(\hat{\phi}_{R}\right)+E_{J_{R\,2}}\cos\left(\hat{\phi}_{R}+\varphi_{z}\right)+E_{J_{R\,3}}\cos\left(\hat{\phi}_{R}+\varphi_{z}+\varphi_{w}\right)$
(13)

as presented in the main part of the manuscript. In the charging dominated
regime $E_{c}\gg E_{J}$ the charging states can be used as basis state with
the property of the phase operators [41]
$e^{i\hat{\phi}_{i}}\ket{n_{i}}=\ket{n_{i}-1}$ such that it introduces a
hopping between the charging states $\ket{L}$,$\ket{0}$, and $\ket{R}$ which
depends on the magnetic fluxes

$\displaystyle H_{0}$
$\displaystyle=\frac{E_{C}}{3}(1-2n_{gL})\ket{L}\bra{L}+\frac{E_{C}}{3}(1-2n_{gR})\ket{R}\bra{R}$
$\displaystyle-\left(E_{J_{L1}}+E_{J_{L2}}e^{i\varphi_{x}}+E_{J_{L3}}e^{i(\varphi_{x}+\varphi_{y})}\right)\ket{L}\bra{0}+h.c.$
$\displaystyle-\left(E_{J_{R1}}+E_{J_{R2}}e^{-i\varphi_{z}}+E_{J_{R3}}e^{-i(\varphi_{z}+\varphi_{w})}\right)\ket{0}\bra{R}+h.c.\,.$
(14)

##  Appendix B Integral: Hypercube enclosing the degenerate point

If the degenerate point is at position
$(\varphi_{x0},\varphi_{y0},\varphi_{z0},\varphi_{w0})$ the topological charge
can be expressed by the integral

$\displaystyle Q_{T}$
$\displaystyle=\frac{1}{2\pi^{2}}\int_{\varphi_{x0}-a}^{\varphi_{x0}+a}\int_{\varphi_{y0}-a}^{\varphi_{y0}+a}\int_{\varphi_{z0}-a}^{\varphi_{z0}+a}\left(H_{xyz}\big{|}_{\varphi_{w}=\varphi_{w0}+a}-H_{xyz}\big{|}_{\varphi_{w}=\varphi_{w0}-a}\right)d\varphi_{x}d\varphi_{y}d\varphi_{z}$
$\displaystyle-\frac{1}{2\pi^{2}}\int_{\varphi_{x0}-a}^{\varphi_{x0}+a}\int_{\varphi_{y0}-a}^{\varphi_{y0}+a}\int_{\varphi_{w0}-a}^{\varphi_{w0}+a}\left(H_{xyw}\big{|}_{\varphi_{z}=\varphi_{z0}+a}-H_{xyw}\big{|}_{\varphi_{z}=\varphi_{z0}-a}\right)d\varphi_{x}d\varphi_{y}d\varphi_{w}$
$\displaystyle+\frac{1}{2\pi^{2}}\int_{\varphi_{x0}-a}^{\varphi_{x0}+a}\int_{\varphi_{z0}-a}^{\varphi_{z0}+a}\int_{\varphi_{w0}-a}^{\varphi_{w0}+a}\left(H_{xzw}\big{|}_{\varphi_{y}=\varphi_{y0}+a}-H_{xzw}\big{|}_{\varphi_{y}=\phi_{y0}-a}\right)d\varphi_{x}d\varphi_{z}d\varphi_{w}$
$\displaystyle-\frac{1}{2\pi^{2}}\int_{\varphi_{y0}-a}^{\varphi_{y0}+a}\int_{\varphi_{z0}-a}^{\varphi_{z0}+a}\int_{\varphi_{w0}-a}^{\varphi_{w0}+a}\left(H_{yzw}\big{|}_{\varphi_{x}=\varphi_{x0}+a}-H_{yzw}\big{|}_{\varphi_{x}=\varphi_{x0}-a}\right)d\varphi_{y}d\varphi_{z}d\varphi_{w}$

with $2a$ the length of the hypercube in phase space enclosing the degenerate
point.

##  Appendix C Derivation of the low energy Hamiltonian of the triple dot
system

The starting Hamiltonian for the triple dot system with the superconducting
leads reads

$\displaystyle
H_{\mathrm{tot}}=H_{\mathrm{d}}+H_{\mathrm{c,d}}+\sum_{j}H_{\mathrm{s},j}+\sum_{\braket{\alpha,j}}H_{\mathrm{c},\mathrm{d}_{\alpha}-\mathrm{s}_{j}}+\sum_{\braket{i,j}}H_{\mathrm{c},\mathrm{s}_{i}-\mathrm{s}_{j}}\,,$
(16)

with the Hamiltonian of the dots

$\displaystyle
H_{\mathrm{d}}=\sum_{\alpha=L,M,R}\sum_{\sigma}\left(\epsilon_{\alpha}d_{\alpha\sigma}^{\dagger}d_{\alpha\sigma}+U_{C}n_{\alpha\downarrow}n_{\alpha\uparrow}\right)\,,$
(17)

the coupling between the dots

$\displaystyle
H_{\mathrm{c,d}}=\sum_{\sigma}v_{L}d_{L\sigma}^{\dagger}d_{M\sigma}+v_{R}d_{M\sigma}^{\dagger}d_{R\sigma}+h.c.\,,$
(18)

the BCS Hamiltonian of the superconducting leads

$\displaystyle
H_{\mathrm{s},j}=\sum_{\bm{k}\sigma}\epsilon_{\bm{k}}c_{j\bm{k}\sigma}^{\dagger}c_{j\bm{k}\sigma}+\sum_{\bm{k}}\left(\Delta
e^{i\phi_{j}}c_{j\bm{k}\uparrow}^{\dagger}c_{j-\bm{k}\downarrow}^{\dagger}+h.c.\right)\,,$
(19)

the coupling between neighboring dots and leads

$\displaystyle H_{\mathrm{c},\mathrm{d}_{\alpha}-\mathrm{s}_{j}}=\sum_{\sigma
k}v_{j}\left(d_{\alpha\sigma}^{\dagger}c_{jk\sigma}+h.c.\right)\,,$ (20)

and the coupling between neighboring leads

$\displaystyle H_{\mathrm{c},\mathrm{s}_{i}-\mathrm{s}_{j}}=\sum_{\sigma
k}w_{ij}\left(c_{ik\sigma}^{\dagger}c_{jk\sigma}+h.c.\right)\,.$ (21)

Since the couplings do not depend on the quasi-momentum $\bm{k}$ of the
electrons in the leads one can already integrate out $\bm{k}$ to obtain an
effective Green’s function for the leads in Spin-Nambu space

$\displaystyle g_{s,\phi_{j}}=-\frac{\pi
N_{0}}{\Delta^{2}-\epsilon^{2}}\sigma_{0}\otimes\left(\epsilon\tau_{0}+\Delta
e^{i\phi_{j}\tau_{3}}\tau_{1}\right)\,,$ (22)

where $\sigma_{j}$ are the Pauli matrices in spin space, $\tau_{i}$ are the
Pauli matrices in Nambu space, and $N_{0}$ is the density of states in the
normal state. In the low energy limit $\epsilon\ll\Delta$ the Green’s function
can be approximated to

$\displaystyle g_{s,\phi_{j}}\approx-\pi N_{0}\left(\sigma_{0}\otimes
e^{i\phi_{j}\tau_{3}}\tau_{1}\right)\,.$ (23)

For small coupling between the leads $w_{ij}\ll v_{j}$ we expand the dressed
Green’s function of the dots $G_{d}$ with the Dyson equation up to first order
in the coupling between the leads

$\displaystyle
G_{d}=g_{d}+g_{d}V_{ds}g_{s}\left(V_{sd}+V_{ss}g_{s}V_{sd}\right)G_{d}\,,$
(24)

with $g_{d}=\text{diag}(g_{L},g_{M},g_{R})$ the bare Green’s function of the
three dots,
$g_{s}=\text{diag}(g_{s,\phi_{x}},g_{s,\phi_{y}},g_{s,\phi_{0}},g_{s,\phi_{z}},g_{s,\phi_{w}})$
the bare Green’s function of the leads,

$\displaystyle V_{ss}=\begin{pmatrix}0&0&W_{x0}&0&0\\\ 0&0&W_{y0}&0&0\\\
W_{x0}^{\dagger}&W_{y0}^{\dagger}&0&W_{z0}&W_{w0}\\\
0&0&W_{z0}^{\dagger}&0&0\\\ 0&0&W_{w0}^{\dagger}&0&0\end{pmatrix}$ (25)

the interaction between the superconductors with
$W_{ij}=w_{ij}\left(\sigma_{0}\otimes\tau_{3}\right)$, and $V_{sd}$ the
interaction between dots and leads

$\displaystyle V_{ds}=\begin{pmatrix}V_{x}&V_{y}&0&0&0\\\ 0&0&V_{0}&0&0\\\
0&0&0&V_{z}&V_{w}\\\ \end{pmatrix}=V_{sd}^{\dagger}\,,$ (26)

with $V_{j}=v_{j}\left(\sigma_{0}\otimes\tau_{3}\right)$ . Finally we obtain
an effective Hamiltonian for the dots
$H_{\mathrm{eff}}=H_{\mathrm{d}}+H_{\mathrm{c,d}}+\Sigma_{0}+\Sigma_{1}$ with
the zeroth-order contribution

$\displaystyle\Sigma_{0}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\bm{d}_{L}^{\dagger}&\bm{d}_{M}^{\dagger}&\bm{d}_{R}^{\dagger}\end{pmatrix}V_{ds}g_{s}V_{sd}\begin{pmatrix}\bm{d}_{L}\\\
\bm{d}_{M}\\\ \bm{d}_{R}\end{pmatrix}$ $\displaystyle=\left(\pi
N_{0}\left(v_{x}^{2}e^{i\phi_{x}}+v_{y}^{2}e^{i\phi_{y}}\right)d_{L\uparrow}^{\dagger}d_{L\downarrow}^{\dagger}+\pi
N_{0}\left(v_{z}^{2}e^{i\phi_{z}}+v_{w}^{2}e^{i\phi_{w}}\right)d_{R\uparrow}^{\dagger}d_{R\downarrow}^{\dagger}+\pi
N_{0}v_{o}^{2}e^{i\phi_{0}}d_{M\uparrow}^{\dagger}d_{M\downarrow}^{\dagger}\right)+h.c.$
(27)

with the spinor
$\bm{d}_{\alpha}^{\dagger}=\left(d_{\alpha\uparrow}^{\dagger},d_{\alpha\downarrow},d_{\alpha\downarrow}^{\dagger},-d_{\alpha\uparrow}\right)$,
and the first order contribution

$\displaystyle\Sigma_{1}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}\bm{d}_{L}^{\dagger}&\bm{d}_{M}^{\dagger}&\bm{d}_{R}^{\dagger}\end{pmatrix}V_{ds}g_{s}V_{ss}g_{s}V_{sd}\begin{pmatrix}\bm{d}_{L}\\\
\bm{d}_{M}\\\ \bm{d}_{R}\end{pmatrix}$
$\displaystyle=-\sum_{\sigma}\left(\Gamma_{x}e^{i(\phi_{x}-\phi_{0})}+\Gamma_{y}e^{i(\phi_{y}-\phi_{0})}\right)d_{L\sigma}^{\dagger}d_{M\sigma}+h.c.-\sum_{\sigma}\left(\Gamma_{z}e^{i(\phi_{0}-\phi_{z})}+\Gamma_{w}e^{i(\phi_{0}-\phi_{w})}\right)d_{M\sigma}^{\dagger}d_{R\sigma}+h.c.\,,$
(28)

with $\Gamma_{j}=2\pi N_{0}^{2}v_{j}v_{0}w_{j0}$. Assuming a large Coulomb
energy $U_{c}\rightarrow\infty$ the Cooper pair injection terms appearing in
$\Sigma_{0}$ are suppressed as doubly occupied states have to pay the large
Coulomb energy. Hence the system conserves total electron number and the
effective single electron Hamiltonian can be determined to Eq. 8.


# Meromorphic mappings have the same inverse images of moving hyperplanes with
truncated multiplicities

Si Duc Quang

###### Abstract.

In 1999, H. Fujimoto proved that there exists an integer $l_{0}$ such that if
two meromorphic mappings $f$ and $g$ of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ have the same inverse images for $(2n+2)$
hyperplanes in general position with counting multiplicities to level $l_{0}$
then the map $f\times g$ is algebraically degenerate. The purpose of this
paper is to generalize the result of H. Fujimoto to the case of meromorphic
mappings and slowly moving hyperplanes. Also we improve this result by giving
an explicit estimate for the number $l_{0}$.

††2010 _Mathematics Subject Classification_ : Primary 32H04, 32A22; Secondary
32A35.††Key words and phrases: Degenerate, meromorphic mapping, truncated
multiplicity, hyperplane.

## Introduction

Let $f$ and $g$ be two meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$. Let $H_{1},...,H_{q}$ be $q$ hyperplanes of
${\mathbf{P}}^{n}({\mathbf{C}})$ in general position. Denote by
$\nu_{(f,H_{i})}$ the pull back divisor of $H_{i}$ by $f$. In 1975, H.
Fujimoto proved the following.

Theorem A (H. Fujimoto [2, Theorem II]). Assume that
$\nu_{(f,H_{i})}=\nu_{(g,H_{i})}\ (1\leq i\leq q)$. If $q=3n+2$ and either $f$
or $g$ is linearly non-degenerate over ${\mathbf{C}}$, i.e, the image does not
included in any hyperplane in ${\mathbf{P}}^{n}({\mathbf{C}})$. Then $f=g$.

We note that in this theorem, the condition $\nu_{(f,H_{i})}=\nu_{(g,H_{i})}\
(1\leq i\leq q)$ means that $f$ and $g$ have the same inverse images with
counting multiplicities for all hyperplanes. In 1999 H. Fujimoto [3]
considered the case where these inverse images are taken with multiplicities
truncated by a level $l_{0}$. He proved the following theorem, in which the
number $q$ of hyperplanes is also reduced.

Theorem B (H. Fujimoto [3, Theorem II]). Let $H_{1},...,H_{2n+2}$ be
hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position. Then
there exist an integer $l_{0}$ such that; for two algebraically nondegenerate
meromorphic mappings $f$ and $g$, if
$\min\\{\nu_{(f,H_{i})},l_{0}\\}=\min\\{\nu_{(g,H_{i})},l_{0}\\}\ (1\leq i\leq
2n+2)$ then the mapping $f\times g$ into ${\mathbf{P}}^{n}({\mathbf{C}})\times
P^{n}({\mathbf{C}})$ is algebraically degenerate.

Then the following questions arises naturally:

  * •

Are there any similar results to the above results of H. Fujimoto in the case
where fixed hyperplanes are replaced by moving hyperplanes?

  * •

Is there an explicit estimate for the integer $l_{0}$?

The purpose of the present paper is to give an answer for these questions. We
shall generalize and improve Theorem B to the case of moving hyperplanes, also
give an explicit estimate for the truncation level $l_{0}$. To state our
result, we first recall some following.

Let $f$, $a$ be two meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ with reduced representations
$f=(f_{0}:\dots:f_{n})$, $a=(a_{0}:\dots:a_{n})$ respectively. We say that $a$
is “small” with respect to $f$ if $||\ T_{a}(r)=o(T(r,f))$ as $r\to\infty$.
Put $(f,a)=\sum\limits_{i=0}^{n}a_{i}f_{i}$. We also call $a$ a slowly (with
respect to $f$) moving hyperplanes or moving target.

Let $a_{1},\dots,a_{q}$ $(q\geq n+1)$ be $q$ moving hyperplanes of
${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})$ with reduced
representations $a_{i}=(a_{i0}:\dots:a_{in})\ (1\leq j\leq q).$ We say that
$a_{1},\dots,a_{q}$ are located in general position if
$\det(a_{i_{k}l})\not\equiv 0$ for any $1\leq i_{0}<i_{1}<...<i_{n}\leq q.$ We
denote by; $\mathcal{M}$ the field of all meromorphic functions on
${\mathbf{C}}^{m}$, $\mathcal{R}\\{a_{i}\\}_{i=1}^{q}$ the smallest subfield
of $\mathcal{M}$ which contains ${\mathbf{C}}$ and all
$\dfrac{a_{jk}}{a_{jl}}\text{ with }a_{jl}\not\equiv 0,$ $\mathcal{R}_{f}$ the
field of all small (with respect to $f$) meromorphic function on
${\mathbf{C}}^{m}.$

Let $V$ be a projective subvariety of ${\mathbf{P}}^{N}({\mathbf{C}})$. Take a
homogeneous coordinates $(\omega_{0}:\cdots:\omega_{N})$ of
${\mathbf{P}}^{N}({\mathbf{C}})$. Let $F$ be a meromorphic mappings of
${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})$ with a reduced
representation $F=(f_{0}:\cdots:F_{N})$.

Definition C. The meromorphic mapping $F$ is said to be algebraically
degenerate over a subfield $\mathcal{R}$ of $\mathcal{M}$ if there exists a
homogeneous polynomial $Q\in\mathcal{R}[\omega_{0},...,\omega_{N}]$ with the
form:

$Q(z)(\omega_{0},...,\omega_{N})=\sum_{I\in\mathcal{I}_{d}}a_{I}(z)\omega^{I},$

where $d$ is an integer, $\mathcal{I}_{d}=\\{(i_{0},...,i_{N})\ ;\ 0\leq
i_{j}\leq d,\sum_{j=0}^{N}i_{j}=d\\},$ $a_{I}\in\mathcal{R}$ and
$\omega^{I}=\omega_{0}^{i_{0}}\cdots\omega_{N}^{i_{N}}$ for
$I=(i_{0},...,i_{N})$, such that

$\displaystyle\mathrm{(i)}$ $\displaystyle\ \
Q(z)(F_{0}(z),...,F_{N}(z))\equiv 0\ \text{ on }{\mathbf{C}}^{m},$
$\displaystyle\mathrm{(ii)}$ $\displaystyle\ \ \exists
z_{0}\in{\mathbf{C}}^{m},Q(z_{0})(\omega_{0},...,\omega_{N})\not\equiv 0\text{
on }V.$

Now let $f$ and $g$ be two meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ with representations

$f=(f_{0}:\cdots f_{n})\text{ and }g=(g_{0}:\cdots:g_{n}).$

We consider
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$ as a
projective subvariety of ${\mathbf{P}}^{(n+1)^{2}-1}({\mathbf{C}})$ by Segre
embedding. Then the map $f\times g$ into
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$ is
algebraically degenerate over a subfield $\mathcal{R}$ of $\mathcal{M}$ if
there exists a nontrivial polynomial

$Q(z)(\omega_{0},...,\omega_{n},\omega_{0}^{\prime},...,\omega_{n}^{\prime})=\sum_{\underset{i_{0}+\cdots+i_{n}=d}{I=(i_{0},...,i_{n})\in\mathbf{Z}^{n+1}_{+}}}\sum_{\underset{j_{0}+\cdots+j_{n}=d^{\prime}}{J=(j_{0},...,j_{n})\in\mathbf{Z}^{n+1}_{+}}}a_{IJ}(z)\omega^{I}\omega^{\prime
J},$

where $d,d^{\prime}$ are positive integers, $a_{IJ}\in\mathcal{R}$, such that

$Q(z)(f_{0}(z),...,f_{n}(z),g_{0}(z),...,g_{n}(z))\equiv 0.$

We now generalize and improve Theorem B to the following.

######  Main Theorem.

Let $f$ and $g$ be two meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$. Let $a_{1},...,a_{2n+2}$ be slowly (with
respect to $f$) moving hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ in
general position. Let $l_{0}$ be a positive integer. Assume that $f$ and $g$
are linearly non-degenerate over $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$ and

$\min\\{\nu_{(f,H_{i})},l_{0}\\}=\min\\{\nu_{(g,H_{i})},l_{0}\\}\ (1\leq i\leq
2n+2).$

If $l_{0}>2n^{3}q^{2}(q-1)(q-2),$ where $q=\binom{2n+2}{n+2}$, then the map
$f\times g$ into
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$ is
algebraically degenerate over $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$.

N.B. Concerning to finiteness or degeneracy problems of meromorphic mappings
with moving targets, there are many results given by M. Ru [6], Z. H. Tu [10],
D. D. Thai - S. D. Quang [8], G. Dethloff - T. V. Tan [1] and others. However
in all their results, they need an aditional assumption that $f$ and $g$ are
agree on inverse images of all targets. This is a strong condition and it is
very hard to examine.

Acknowledgements. The research of the author was supported in part by a
NAFOSTED grant of Vietnam.

##  1\. Basic notions and auxiliary results from Nevanlinna theory

(a). We set $||z||=\big{(}|z_{1}|^{2}+\dots+|z_{m}|^{2}\big{)}^{1/2}$ for
$z=(z_{1},\dots,z_{m})\in{\mathbf{C}}^{m}$ and define

$\displaystyle B(r):=\\{z\in{\mathbf{C}}^{m}:||z||<r\\},\quad
S(r):=\\{z\in{\mathbf{C}}^{m}:||z||=r\\}\ (0<r<\infty).$

Define

$v_{m-1}(z):=\big{(}dd^{c}||z||^{2}\big{)}^{m-1}\quad\quad\text{and}$
$\sigma_{m}(z):=d^{c}\text{log}||z||^{2}\land\big{(}dd^{c}\text{log}||z||^{2}\big{)}^{m-1}\text{on}\quad{\mathbf{C}}^{m}\setminus\\{0\\}.$

Let $F$ be a nonzero meromorphic function on a domain $\Omega$ in
${\mathbf{C}}^{m}$. For a set $\alpha=(\alpha_{1},...,\alpha_{m})$ of
nonnegative integers, we set $|\alpha|=\alpha_{1}+...+\alpha_{m}$ and
$\mathcal{D}^{\alpha}F=\dfrac{\partial^{|\alpha|}F}{\partial^{\alpha_{1}}z_{1}...\partial^{\alpha_{m}}z_{m}}.$
We denote by $\nu^{0}_{f}$ (resp. $\nu^{\infty}_{f}$) the zero divisor (resp.
pole divisor) of the function $f$.

For a divisor $\nu$ on ${\mathbf{C}}^{m}$, which is regarded as a function on
${\mathbf{C}}^{m}$ with values in $\mathbf{Z}$, and for positive integers
$k,M$ or $M=\infty$, we define the counting function of $\nu$ by

$\displaystyle\nu^{[M]}(z)$ $\displaystyle=\min\ \\{M,\nu(z)\\},$
$\displaystyle\nu_{>k}^{[M]}(z)$
$\displaystyle=\begin{cases}\nu^{[M]}(z)&\text{ if }\nu(z)>k,\\\ 0&\text{ if
}\nu(z)\leq k,\end{cases}$ $\displaystyle n(t)$
$\displaystyle=\begin{cases}\int\limits_{|\nu|\,\cap B(t)}\nu(z)v_{m-1}&\text{
if }m\geq 2,\\\ \sum\limits_{|z|\leq t}\nu(z)&\text{ if }m=1.\end{cases}$

Similarly, we define $n^{[M]}(t),\ n_{>k}^{[M]}(t).$

Define

$N(r,\nu)=\int\limits_{1}^{r}\dfrac{n(t)}{t^{2n-1}}dt\quad(1<r<\infty).$

Similarly, we define $N(r,\nu^{[M]}),\ N(r,\nu_{>k}^{[M]})$ and denote them by
$N^{[M]}(r,\nu),\ N_{>k}^{[M]}(r,\nu)$ respectively.

Let $\varphi:{\mathbf{C}}^{m}\longrightarrow{\mathbf{C}}$ be a meromorphic
function. Define

$N_{\varphi}(r)=N(r,\nu^{0}_{\varphi}),\
N_{\varphi}^{[M]}(r)=N^{[M]}(r,\nu^{0}_{\varphi}),\
N_{\varphi,>k}^{[M]}(r)=N_{>k}^{[M]}(r,\nu^{0}_{\varphi}).$

For brevity we will omit the character [M] if $M=\infty$.

(b). Let $f:{\mathbf{C}}^{m}\longrightarrow{\mathbf{P}}^{n}({\mathbf{C}})$ be
a meromorphic mapping. For arbitrarily fixed homogeneous coordinates
$(\omega_{0}:\cdots:\omega_{n})$ on ${\mathbf{P}}^{n}({\mathbf{C}})$, we take
a reduced representation $f=(f_{0}:\cdots:f_{n})$, which means that each
$f_{i}$ is a holomorphic function on ${\mathbf{C}}^{m}$ and
$f(z)=\big{(}f_{0}(z):\dots:f_{n}(z)\big{)}$ outside the analytic set
$\\{f_{0}=\cdots=f_{n}=0\\}$ of codimension $\geq 2$. Set
$\|f\|=\big{(}|f_{0}|^{2}+\dots+|f_{n}|^{2}\big{)}^{1/2}$.

The characteristic function of $f$ is defined by

$\displaystyle
T(r,f)=\int\limits_{S(r)}\text{log}\|f\|\sigma_{n}-\int\limits_{S(1)}\text{log}\|f\|\sigma_{n}.$

Let $a$ be a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ with reduced representation
$a=(a_{0}:\dots:a_{n})$. We define

$m_{f,a}(r)=\int\limits_{S(r)}\text{log}\dfrac{||f||\cdot||a||}{|(f,a)|}\sigma_{m}-\int\limits_{S(1)}\text{log}\dfrac{||f||\cdot||a||}{|(f,a)|}\sigma_{m},$

where $\|a\|=\big{(}|a_{0}|^{2}+\dots+|a_{n}|^{2}\big{)}^{1/2}$.

If $f,a:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ are meromorphic
mappings such that $(f,a)\not\equiv 0$, then the first main theorem for moving
targets in value distribution theory (see [5]) states

$T(r,f)+T(r,a)=m_{f,a}(r)+N_{(f,a)}(r).$

Let $\varphi$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}$, which
are occasionally regarded as a meromorphic map into
${\mathbf{P}}^{1}({\mathbf{C}})$. The proximity function of $\varphi$ is
defined by

$m(r,\varphi):=\int_{S(r)}\mathrm{log}\max\ (|\varphi|,1)\sigma_{m}.$

(c). As usual, by the notation $``||\ P"$ we mean the assertion $P$ holds for
all $r\in[0,\infty)$ excluding a Borel subset $E$ of the interval $[0,\infty)$
with $\int_{E}dr<\infty$.

The following plays essential roles in Nevanlinna theory.

######  Theorem 1.1 (see [7, Theorem 2.1],[9, Theorem 2]).

Let $f=(f_{0}:\cdots:f_{n})$ be a reduced representation of meromorphic
mapping $f$ of ${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})$.
Assume that $f_{n+1}$ is a holomorphic such that $f_{0}+\cdots
f_{n}+f_{n+1}=0$. If $\sum_{i\in I}f_{i}\neq 0$ for all
$I\subsetneq\\{0,...,n+1\\}$ then

$||\ T(r,f)\leq\sum_{i=0}^{n+1}N_{f_{i}}^{[n]}(r)+o(T(r,f)).$

######  Theorem 1.2 (see [5], [3, Theorem 5.5]).

Let $f$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}.$ Then

$\biggl{|}\biggl{|}\quad
m\biggl{(}r,\dfrac{\mathcal{D}^{\alpha}(f)}{f}\biggl{)}=O(\mathrm{log}^{+}T(r,f))\
(\alpha\in\mathbf{Z}^{m}_{+}).$

######  Theorem 1.3 (see [5, Theorem 5.2.29]).

Let $f$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}.$ with a
reduced representation $f=(f_{0}:\cdots:f_{n})$. Suppose that $f_{k}\not\equiv
0$. Then

$T(r,\frac{f_{j}}{f_{k}})\leq
T(r,f)\leq\sum_{j=0}^{n}T(r,\frac{f_{j}}{f_{j}})+O[1].$

(d). Let $h_{1},h_{2},...,h_{p}$ be finitely many nonzero meromorphic
functions on ${\mathbf{C}}^{m}$. By a rational function in logarithmic
derivatives of $h_{j}^{\prime}$s we mean a nonzero meromorphic function
$\varphi$ on ${\mathbf{C}}^{m}$ which is represented as

$\varphi=\dfrac{P(\cdots,\frac{\mathcal{D}^{\alpha}h_{j}}{h_{j}},\cdots)}{Q(\cdots,\frac{\mathcal{D}^{\alpha}h_{j}}{h_{j}},\cdots)}$

with polynomials $P(\cdots,X^{\alpha},\cdots)$ and
$Q(\cdots,X^{\alpha},\cdots)$

######  Proposition 1.4 (see [3, Proposition 3.4]).

Let $h_{1},h_{2},...,h_{p}\ (p\geq 2)$ be nonzero meromorphic functions on
${\mathbf{C}}^{m}$. Assume that

$h_{1}+h_{2}+\cdots+h_{p}=0$

Then, the set $\\{1,...,p\\}$ of indices has a partition

$\\{1,...,p\\}=J_{1}\cup J_{2}\cup\cdots\cup J_{k},\sharp J_{\alpha}\geq 2\
\forall\ \alpha,J_{\alpha}\cap J_{\beta}=\emptyset\text{ for }\alpha\neq\beta$

such that, for each $\alpha$,

$\displaystyle\mathrm{(i)}$ $\displaystyle\ \sum_{i\in J_{\alpha}}h_{i}=0,$
$\displaystyle\mathrm{(ii)}$ $\frac{h_{i}^{\prime}}{h_{i}}\ (i,i^{\prime}\in
J_{\alpha})$ are rational functions in logarithmic derivatives of
$h_{j}^{\prime}$s

##  2\. Proof of Main Theorem

In order to prove the main theorem, we need the following algebraic
propositions.

Let $H_{1},...,H_{2n+1}$ be $(2n+1)$ hyperplanes of
${\mathbf{P}}^{n}({\mathbf{C}})$ in general position given by

$H_{i}:\ \ x_{i0}\omega_{0}+x_{i1}\omega_{1}+\cdots+x_{in}\omega_{n}=0\ (1\leq
i\leq 2n+1).$

We consider the rational map
$\Phi:{\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})\longrightarrow{\mathbf{P}}^{2n}({\mathbf{C}})$
as follows:

For $v=(v_{0}:v_{1}\cdots:v_{n}),\
w=(w_{0}:w_{1}:\cdots:w_{n})\in{\mathbf{P}}^{n}({\mathbf{C}})$, we define the
value $\Phi(v,w)=(u_{0}:\cdots:u_{2n+1})\in{\mathbf{P}}^{2n}({\mathbf{C}})$ by

$u_{i}=\frac{x_{i0}v_{0}+x_{i1}v_{1}+\cdots+x_{in}v_{n}}{x_{i0}w_{0}+x_{i1}w_{1}+\cdots+x_{in}w_{n}}.$

######  Proposition 2.1 (see [3, Proposition 5.9]).

The map $\Phi$ is a birational map of
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$ onto
${\mathbf{P}}^{2n}({\mathbf{C}})$.

Now let $b_{1},...,b_{2n+1}$ be $(2n+1)$ moving hyperplanes of
${\mathbf{P}}^{n}({\mathbf{C}})$ in general position with reduced
representations

$b_{i}=(b_{i0}:b_{i1}:\cdots:b_{in})\ (1\leq i\leq 2n+1).$

Let $f$ and $g$ be two meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ with reduced representations

$f=(f_{0}:\cdots:f_{n})\ \text{ and }\ g=(g_{0}:\cdots:g_{n}).$

Define $h_{i}=\frac{(f,b_{i})}{(g,b_{i})}\ (1\leq i\leq 2n+1)$ and
$h_{I}=\prod_{i\in I}h_{i}$ for each subset $I$ of $\\{1,...,2n+1\\}.$ Set
$\mathcal{I}=\\{I=(i_{1},...,i_{n})\ ;\ 1\leq i_{1}<\cdots<i_{n}\leq 2n+1\\}$.
Let $\mathcal{R}$ be a subfield of $\mathcal{M}$ which contains
$\mathcal{R}\\{b_{i}\\}_{i=1}^{2n+1}$. We have the following proposition

######  Proposition 2.2.

If there exist functions $A_{I}\in\mathcal{R}\ (I\in\mathcal{I})$, not all
zero, such that

$\sum_{I\in\mathcal{I}}A_{I}h_{I}\equiv 0$

then the map $f\times g$ into
${\mathbf{P}}^{n}({\mathbf{C}})\times{\mathbf{P}}^{n}({\mathbf{C}})$ is
algebraically degenerate over $\mathcal{R}$.

###### Proof.

By changing the homogeneous coordinates of ${\mathbf{P}}^{n}({\mathbf{C}})$,
we ma assume that $b_{i0}\not\equiv 0\ (1\leq i\leq 2n+1)$. Since
$b_{1},...,b_{2n+1}$ are in general position, then for $1\leq i_{0}<\cdots
i_{n}\leq 2n+1$ we have $\det(b_{i_{j}k})_{0\leq j,k\leq n}\not\equiv 0$.
Therefor the set

$S=\bigl{(}\bigcap_{I\in\mathcal{I}}\\{z\in{\mathbf{C}}^{m}\ ;\
A_{I}(z)=0\\}\bigl{)}\cup\bigcup_{1\leq i_{0}<\cdots i_{n}\leq
2n+1}\\{z\in{\mathbf{C}}^{m}\ ;\ \det(b_{i_{j}k}(z))_{0\leq j,k\leq n}=0\\}$

is a proper analytic subset of ${\mathbf{C}}^{m}$. Take $z_{0}\not\in S$ and
set $x_{ij}=b_{ij}(z_{0})$.

For $v=(v_{0}:v_{1}\cdots:v_{n}),\
w=(w_{0}:w_{1}:\cdots:w_{n})\in{\mathbf{P}}^{n}({\mathbf{C}})$, we define the
map $\Phi(v,w)=(u_{0}:\cdots:u_{2n+1})\in{\mathbf{P}}^{2n}({\mathbf{C}})$ as
above. By Proposition 2.1, $\Phi$ is birational function. This implies that
the functions

$\sum_{I\in\mathcal{I}}A_{I}(z_{0})\prod_{i\in
I}\frac{\sum_{j=0}^{n}b_{ij}(z_{0})v_{j}}{\sum_{j=0}^{n}b_{ij}(z_{0})w_{j}}$

is a nontrivial rational function. It follows that

$Q(z)(v_{0},...,v_{n},w_{0},...,w_{n})=\dfrac{1}{\prod_{i=1}^{2n+1}b_{i0}}\sum_{I\in\mathcal{I}}A_{I}(z)\left(\prod_{i\in
I}\sum_{j=0}^{n}b_{ij}(z)v_{j}\right)\times\left(\prod_{i\in
I^{c}}\sum_{j=0}^{n}b_{ij}(z)w_{j}\right),$

where $I^{c}=\\{1,....,2n+1\\}\setminus I$, is a nontrivial polynomial with
coefficients in $\mathcal{R}$. Since the assumption of the proposition, it is
clear that

$Q(z)(f_{0}(z),...,f_{n}(z),g_{0}(z),...,g_{n}(z))\equiv 0.$

Hence $f\times g$ is algebraically degenerate over $\mathcal{R}$. ∎

######  Proposition 2.3.

Let $f$ be a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$ and let $b_{1},...,b_{n+1}$ be moving
hyperplanes of ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position with
reduced representations:

$f=(f_{0}:\cdots:f_{n}),\ b_{i}=(b_{i0}:\cdots:b_{in})\ (1\leq i\leq n+1).$

Then for each regular point $z_{0}$ of the analytic subset
$\bigcup_{i=1}^{n+1}\\{z\ ;\ (f,b_{i})(z)=0\\}$ with $z_{0}\not\in I(f)$, we
have

$\min_{1\leq i\leq
n+1}\nu^{0}_{(f,b_{i})}(z_{0})\leq\nu^{0}_{\det\Phi}(z_{0}),$

where $I(f)$ denotes the indeterminacy set of $f$ and $\Phi$ is the matrix
$(b_{ij};1\leq i\leq n+1,0\leq j\leq n).$

###### Proof.

Since $z_{0}\not\in I(f)$, we may assume that $f_{0}(z_{0})\neq 0.$ We
consider the following system of equations

$b_{i0}f_{0}+\cdots+b_{in}f_{n}=(f,b_{i})\ (1\leq i\leq n+1).$

Solving these equations, we obtain

$f_{0}=\frac{\det\Phi^{\prime}}{\det\Phi},$

where $\Phi^{\prime}$ is a matrix obtained from $\Phi$ by replacing the first
column of $\Phi$ by $\left(\begin{array}[]{cc}(f,b_{1})\\\ \vdots\\\
(f,b_{n})\end{array}\right)$. Therefore, we have

$\nu^{0}_{\det\Phi}(z_{0})=\nu^{0}_{\det\Phi^{\prime}}(z_{0})\geq\min_{1\leq
i\leq n+1}\nu^{0}_{(f,b_{i})}(z_{0}).$

The proposition is proved. ∎

Proof of Main Theorem. Assume that $f,g,a_{i}$ have reduced representations

$f=(f_{0}:\cdots:f_{n}),\ g=(g_{0}:\cdots:g_{n}),\
a_{i}=(a_{i0}:\cdots:a_{in}).$

By Theorem 1.1 we have

$\displaystyle\bigl{|}\bigl{|}\ \dfrac{2n+2}{n+2}T(r,f)$
$\displaystyle\leq\sum_{i=1}^{2n+2}N_{(f,a_{i})}^{[n]}(r)+o(T(r,f))$
$\displaystyle\leq n\cdot\sum_{i=1}^{2n+2}N_{(g,a_{i})}^{[1]}(r)+o(T(r,f))$
$\displaystyle\leq n(2n+2)(T(r,g))+o(T(r,f)).$

Then we have $||\ T(r,f)=O(T(r,g))$. Similarly we also have $||\
T(r,g)=O(T(r,f))$.

We suppose contrarily that the map $f\times g$ is algebraically non-degenerate
over $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}.$ Define
$h_{i}=\dfrac{(f,a_{i})}{(g,a_{i})}\ (1\leq i\leq 2n+2).$ Then
$\dfrac{h_{i}}{h_{j}}=\dfrac{(f,a_{i})\cdot(g,a_{j})}{(g,a_{i})\cdot(f,a_{j})}$
does not depend on the choice of representations of $f$ and $g$ . Since
$\sum_{k=0}^{n}a_{ik}f_{k}-h_{i}\cdot\sum_{k=0}^{n}a_{ik}g_{k}=0\ (1\leq i\leq
2n+2),$ it implies that

(2.1) $\displaystyle\Phi:=\det\
(a_{i0},...,a_{in},a_{i0}h_{i},...,a_{in}h_{i};1\leq i\leq 2n+2)\equiv 0.$

For each subset $I\subset\\{1,2,...,2n+2\\},$ put $h_{I}=\prod_{i\in I}h_{i}$.
Denote by $\mathcal{I}$ the set

$\mathcal{I}=\\{I=(i_{1},...,i_{n+1})\ ;\ 1\leq i_{1}<\cdots<i_{n+1}\leq
2n+2\\}.$

For each $I=(i_{1},...,i_{n+1})\in\mathcal{I}$, define

$\displaystyle A_{I}=(-1)^{\frac{(n+1)(n+2)}{2}+i_{1}+...+i_{n+1}}$
$\displaystyle\times\det(a_{i_{r}l};1\leq r\leq n+1,0\leq l\leq n)$
$\displaystyle\times\det(a_{j_{s}l};1\leq s\leq n+1,0\leq l\leq n),$

where $J=(j_{1},...,j_{n+1})\in\mathcal{I}$ such that $I\cup
J=\\{1,2,...,2n+2\\}.$

We define some following:

$\bullet$ $\mathcal{R}$: the field of rational functions in logarithmic
derivatives of functions in $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$,

$\bullet$ $G$: the group of all nonzero functions $\varphi$ so that there
exists an positive integer $m$, $\varphi^{m}$ is a rational function in
logarithmic derivatives of ${h_{i}}^{\prime}$s with coefficients in
$\mathcal{R}$,

$\bullet$ $\mathcal{H}$: the subgroup of the group $\mathcal{M}/G$ generated
by elements $[h_{1}],...,[h_{2n+2}]$.

Hence $\mathcal{H}$ is a finitely generated torsion-free abelian group. We
call $(x_{1},...,x_{p})$ a basis of $\mathcal{H}$. Then for each
$i\in\\{1,...,2n+2\\}$, we have

$[h_{i}]=x_{1}^{t_{i1}}\cdots x_{p}^{t_{ip}}.$

Put $t_{i}=(t_{i1},...,t_{ip})\in\mathbf{Z}^{p}$ and denote by $``\leqslant"$
the lexicographical order on $\mathbf{Z}^{p}$. Without loss of generality, we
may assume that

$t_{1}\leqslant t_{2}\leqslant\cdots\leqslant t_{2n+1}.$

Now the equality (2.1) implies that

$\sum_{I\in\mathcal{I}}A_{I}h_{I}=0.$

Applying Proposition 1.4 to meromorphic mappings $A_{I}h_{I}\
(I\in\mathcal{I})$, then we have a partition
$\mathcal{I}=\mathcal{I}_{1}\cup\cdots\cup\mathcal{I}_{k}$ with
$\mathcal{I}_{\alpha}\neq\emptyset$ and
$\mathcal{I}_{\alpha}\cap\mathcal{I}_{\beta}=\emptyset$ for $\alpha\neq\beta$
such that for each $\alpha$,

(2.2) $\displaystyle\sum_{I\in\mathcal{I}_{\alpha}}A_{I}h_{I}\equiv 0,$ (2.3)
$\displaystyle\frac{A_{I^{\prime}}h_{I^{\prime}}}{A_{I}h_{I}}\
(I,I^{\prime}\in\mathcal{I}_{\alpha})\text{ are rational functions in
logarithmic derivatives of ${A_{J}h_{J}}^{\prime}$s}.$

Moreover, we ma assume that $I_{\alpha}$ is minimal, i.e., there is no proper
subset $J_{\alpha}\subsetneq I_{\alpha}$ with
$\sum_{I\in\mathcal{J}_{\alpha}}A_{I}h_{I}\equiv 0$.

We distinguish the following two cases:

Case 1. Assume that there exists an index $i_{0}$ such that
$t_{i_{0}}<t_{i_{0}+1}$. We may assume that $i_{0}\leq n+1$ (otherwise we
consider the relation $``\geqslant"$ and change indices of
$\\{h_{1},...,h_{2n+2}\\}$).

Assume that $I=(1,2,...,n+1)\in\mathcal{I}_{1}$. By the assertion (2.3), for
each $J=(j_{1},...,j_{n+1})\in\mathcal{I}_{1}\ (1\leq j_{1}<\cdots<j_{n+1}\leq
2n+2)$, we have $[h_{I}]=[h_{J}]$. This implies that

$t_{1}+\cdots+t_{n+1}=t_{j_{1}}+\cdots+t_{j_{n+1}}.$

This yields that $t_{j_{i}}=t_{i}\ (1\leq i\leq n+1)$.

Suppose that $j_{i_{0}}>i_{0}$, then $t_{i_{0}}<t_{i_{0}+1}\leqslant
t_{j_{i_{0}}}$. This is a contradiction. Therefore $j_{i_{0}}=i_{0}$, and
hence $j_{1}=1,...,j_{i_{0}-1}=i_{0}-1.$ We conclude that
$J=(1,...,i_{0},j_{i_{0}+1},...,j_{n+1})$ and $i_{0}\leq n+1$ for each
$J\in\mathcal{I}_{1}.$

By (2.1), we have

$\sum_{I\in\mathcal{I}_{1}}A_{I}h_{I}=h_{i_{0}}\sum_{I\in\mathcal{I}_{1}}A_{I}h_{I\setminus\\{i_{0}\\}}\equiv
0.$

Thus

$\sum_{I\in\mathcal{I}_{1}}A_{I}h_{I\setminus\\{i_{0}\\}}\equiv 0.$

Then Proposition 2.2 shows that $f\times g$ is algebraically degenerate over
$\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$. It contradicts to the supposition.

Case 2. Assume that $t_{1}=\cdots t_{2n+2}$. It follows that
$\frac{h_{I}}{h_{J}}\in G$ for any $I,J\in\mathcal{I}$. Then we easily see
that $\frac{h_{i}}{h_{j}}\in G$ for all $1\leq i,j\leq 2n+2.$ Hence, there
exists a positive integer $m_{ij}$ such that
$\left(\frac{h_{i}}{h_{j}}\right)^{m_{ij}}$ is a rational funtion in
logarithmic derivatives of ${h_{s}}^{\prime}$s with coefficients in
$\mathcal{R}$. Therefore, by lemma on logarithmic derivatives, we have

(2.4) $\displaystyle\begin{split}\biggl{|}\biggl{|}\ \
m\bigl{(}r,\frac{h_{i}}{h_{j}}\bigl{)}&=\frac{1}{m_{ij}}m\bigl{(}r,\left(\frac{h_{i}}{h_{j}}\right)^{m_{ij}}\bigl{)}+O(1)\\\
&=O\bigl{(}\max
m\left(r,\frac{\mathcal{D}^{\alpha}(\frac{a_{sl}}{a_{sk}})}{\frac{a_{sl}}{a_{sk}}}\right)\bigl{)}+O\bigl{(}\max
m\left(r,\frac{\mathcal{D}^{\alpha}(h_{s})}{h_{s}}\right)\bigl{)}+O(1)\\\
&=O\bigl{(}\max
m\left(r,\frac{\mathcal{D}^{\alpha}(f,b_{s})}{(f,b_{s})}\right)\bigl{)}+O\bigl{(}\max
m\left(r,\frac{\mathcal{D}^{\alpha}(g,b_{s})}{(g,b_{s})}\right)\bigl{)}+o(T(r,f))\\\
&=o(T(r,f))+o(T(r,g))=o(T(r,f)).\end{split}$

######  Claim 2.4.

For each $i_{0}\in\\{1,...,2n+2\\},$ we have

$\displaystyle||\
N_{h_{i_{0}}}(r)\leq\dfrac{q^{2}(q-1)(q-2)}{2(l_{0}+1)}T(r,g)+o(T(r,g))$ and
$\displaystyle||\
N_{\frac{1}{h_{i_{0}}}}(r)\leq\dfrac{q^{2}(q-1)(q-2)}{2(l_{0}+1)}T(r,g)+o(T(r,g)).$

Indeed, fix an index $\alpha$. If $\sharp I_{\alpha}=2$, it is clear that we
have a nontrivial algebraic relation (over
$\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$) among
$f_{0},...,f_{n},g_{0},...,g_{n}$. It follows that $f\times g$ is
algebraically degenerate over $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$. this is a
contradiction. Then $\sharp I_{\alpha}>2$. Assume that
$\mathcal{I}_{\alpha}=\\{I_{0},...,I_{t+1}\\},\ t\geq 1$ and put
$J=I_{0}\cup\cdots\cup I_{t+1}.$ We consider the meromorphic mapping
$F_{\alpha}$ of ${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{t}({\mathbf{C}})$ with
the reduced representation

$F_{\alpha}=(dA_{I_{0}}h_{I_{0}}:\cdots:dA_{I_{t}}h_{I_{t}}),$

where $d$ is a meromorphic function. Then we see that each zero point of
$dA_{I_{i}}h_{I_{i}}\ (0\leq i\leq t+1)$ must be either zero or pole of some
$A_{I_{j}}h_{I_{j}}\ (0\leq j\leq t)$. Then by Theorem 1.1, we have

(2.5) $\displaystyle\begin{split}||\
T(r,F_{\alpha})&\leq\sum_{i=0}^{t+1}N^{[t]}_{dA_{I_{i}}h_{I_{i}}}(r)+o(T(r,F_{\alpha}))\leq\sum_{i=0}^{n+1}(N^{[t]}_{A_{I_{i}}h_{I_{i}}}(r)+N^{[t]}_{\frac{1}{A_{I_{i}}h_{I_{i}}}}(r))+o(T(r,F_{\alpha}))\\\
&\leq\sum_{i=0}^{t+1}(N^{[t]}_{h_{I_{i}}}(r)+N^{[t]}_{\frac{1}{h_{I_{i}}}}(r))+o(T(r,g))\leq
t\sum_{i=0}^{t+1}(N^{[1]}_{h_{I_{i}}}(r)+N^{[1]}_{\frac{1}{h_{I_{i}}}}(r))+o(T(r,g))\\\
&\leq
t(t+2)\sum_{i\in\mathcal{J}}(N^{[1]}_{h_{i}}(r)+N^{[1]}_{\frac{1}{h_{i}}}(r))+o(T(r,g))\\\
&\leq
t(t+2)\sum_{i\in\mathcal{J}}N^{[1]}_{(g,a_{i}),>l_{0}}(r)+o(T(r,g))\leq\frac{t(t+2)}{l_{0}+1}\sum_{i\in\mathcal{J}}N_{(g,a_{i}),>l_{0}}(r)+o(T(r,g))\\\
&\leq\frac{t(t+2)(2n+2)}{l_{0}+1}T(r,g)+o(T(r,g))\leq\frac{q(q+2)(2n+2)}{l_{0}+1}T(r,g)+o(T(r,g)),\end{split}$

where $q=\binom{2n+2}{n+1}.$

Take a regular point $z_{0}$ of the analytic subset

$\bigcup_{i=1}^{2n+2}\\{z\ ;\ (f,a_{i})=0\\}\text{ with }z_{0}\not\in I(f)\cup
I(g).$

We may assume that
$\nu^{0}_{h_{1}}(z_{0})\geq\nu^{0}_{h_{2}}(z_{0})\geq\cdots\geq\nu^{0}_{h_{2n+2}}(z_{0}).$
Set $I=(1,...,n+1)$. Then $I\in\mathcal{I}_{\alpha}$ with an index $\alpha$,
$1\leq\alpha\leq k.$

It also is easy to see that if $1\in I^{\prime}\ \forall
I^{\prime}\in\mathcal{I}_{\alpha}$, then

$\sum_{I^{\prime}\in\mathcal{I}_{\alpha}}A_{I^{\prime}}h_{I^{\prime}\setminus\\{1\\}}\equiv
0.$

Therefore, Proposition 2.2 shows that $f\times g$ is algebraically degenerate
(over $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$). This is a contradiction. Hence,
there exists $I^{\prime}\in\mathcal{I}_{\alpha}$ such that $1\not\in
I^{\prime}$. Assume that $I^{\prime}=(j_{1},...,j_{n+1})$ with
$1<j_{1}<\cdots<j_{n+1}$. By Proposition 2.3, it yields that
$\nu^{0}_{j_{n+1}}(z_{0})\leq\nu^{0}_{det(a_{jl};j\in I^{\prime},0\leq l\leq
n)}(z_{0})$. Then we have

$\displaystyle\nu^{0}_{h_{i_{0}}}(z_{0})$
$\displaystyle\leq\nu^{0}_{h_{1}}(z_{0})\leq\nu^{0}_{\frac{A_{I}h_{I}}{A_{I^{\prime}}h_{I^{\prime}}}}(z_{0})$
$\displaystyle+\nu^{0}_{\frac{A_{I^{\prime}}}{A_{i}}}(z_{0})+\nu^{0}_{j_{n+1}}(z_{0})$
$\displaystyle\leq\nu^{0}_{\frac{A_{I}h_{I}}{A_{I^{\prime}}h_{I^{\prime}}}}(z_{0})+\nu^{0}_{\frac{A_{I^{\prime}}}{A_{I}}}(z_{0})+\nu^{0}_{\det(a_{jl};j\in
I^{\prime},0\leq l\leq n)}(z_{0}).$

Thus we have

$\displaystyle\nu^{0}_{h_{i_{0}}}(z_{0})\leq$
$\displaystyle\sum_{\alpha=1}^{k}\sum_{I,I^{\prime}\in\mathcal{I}_{\alpha}}(\nu^{0}_{\frac{A_{I}h_{I}}{A_{I^{\prime}}h_{I^{\prime}}}}(z_{0})+\nu^{0}_{\frac{A_{I^{\prime}}}{A_{I}}}(z_{0}))$
$\displaystyle+\sum_{1\leq j_{1}<\cdots<j_{n+1}\leq
2n+2}\nu^{0}_{\det(a_{j_{i}l};1\leq i\leq n+1,0\leq l\leq n)}(z_{0}).$

The above inequality holds for all $z_{0}$ outside an analytic subset of
codimension at least two. Integrating both sides of this inequality, we obtain

$\displaystyle||\ N_{h_{i_{0}}}(r)$
$\displaystyle\leq\sum_{\alpha=1}^{k}\sum_{I,I^{\prime}\in\mathcal{I}_{\alpha}}(N_{\frac{A_{I}h_{I}}{A_{I^{\prime}}h_{I^{\prime}}}}(r)+N_{\frac{A_{I^{\prime}}}{A_{I}}}(r))+o(T(r,g))$
$\displaystyle\leq\sum_{\alpha=1}^{k}\sum_{I,I^{\prime}\in\mathcal{I}_{\alpha}}T(r,\frac{A_{I}h_{I}}{A_{I^{\prime}}h_{I^{\prime}}})+o(T(r,g))$
$\displaystyle\leq\sum_{\alpha=1}^{k}\sum_{I,I^{\prime}\in\mathcal{I}_{\alpha}}T(r,F_{\alpha})+o(T(r,g))$
$\displaystyle\leq\sum_{\alpha=1}^{k}\sum_{I,I^{\prime}\in\mathcal{I}_{\alpha}}\frac{q(q+2)(2n+2)}{l_{0}+1}T(r,g)+o(T(r,g))\
\ \ \text{ by (\ref{new3.2})},$
$\displaystyle\leq\sum_{I,I^{\prime}\in\mathcal{I}}\frac{q(q+2)(2n+2)}{l_{0}+1}T(r,g)+o(T(r,g))$
$\displaystyle=\frac{q^{2}(q-1)(q-2)}{2(l_{0}+1)}T(r,g)+o(T(r,g)).$

Similarly, we get

$||\
N_{\frac{1}{h_{i_{0}}}}(r)\leq\dfrac{q^{2}(q-1)(q-2)}{2(l_{0}+1)}T(r,g)+o(T(r,g)).$

We complete the proof of Claim 2.4

We now continue the proof of Main Theorem. By changing the homogeneous
coordinates of ${\mathbf{P}}^{n}({\mathbf{C}})$ if necessary, we may assume
that $a_{i0}\not\equiv 0$ for all $1\leq i\leq 2n+2$. Set

$\tilde{a}_{ij}=\frac{a_{ij}}{a_{i0}},\
\tilde{a}_{i}=(\tilde{a}_{i0},....,\tilde{a}_{in}),\
(f,\tilde{a}_{i})=\sum_{j=0}^{n}\tilde{a}_{ij}f_{j}\ \text{and
}(g,\tilde{a}_{i})=\sum_{j=0}^{n}\tilde{a}_{ij}g_{j}.$

Then there exist functions $b_{ij}\in\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}\
(n+2\leq i\leq 2n+2,1\leq j\leq n+1)$ such that

$\tilde{a}_{i}=\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j}.$

By the identity (2.1), we have

$\det\
(\tilde{a}_{i0},...,\tilde{a}_{in},\tilde{a}_{i0}h_{i},...,\tilde{a}_{in}h_{i};1\leq
i\leq 2n+2)\equiv 0.$

It easily implies that

$\det\
(\tilde{a}_{i0}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j0}h_{j},...,\tilde{a}_{in}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{jn}h_{j};n+2\leq
i\leq 2n+2,1\leq j\leq n+1)\equiv 0.$

Therefore, the matrix

$\Psi=\bigl{(}\tilde{a}_{i0}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j0}h_{j},...,\tilde{a}_{in}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{jn}h_{j};n+2\leq
i\leq 2n+2\bigl{)}$

has the rank at most $n$.

Suppose that ${\mathrm{rank}\,}\Psi<n$. Then, the determinant of the square
submatrix

$\bigl{(}\tilde{a}_{i1}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j0}h_{j},...,\tilde{a}_{in}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{jn}h_{j};n+2\leq
i\leq 2n+1\bigl{)}$

vanishes identically. By Proposition 2.2, it follows that $f\times g$ is
algebraically degenerate over $\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}$. This
contradicts to the supposition. Hence ${\mathrm{rank}\,}\Psi=n$.

Without loss of generality, we may assume that the determinant of the square
submatrix

$\bigl{(}\tilde{a}_{i1}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j0}h_{j},...,\tilde{a}_{in}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{jn}h_{j};n+2\leq
i\leq 2n+1\bigl{)}$

of $\Psi$ does not vanish identically. On the other hand, we have

$(\tilde{a}_{i0}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j0}h_{j})g_{0}+\cdots+(\tilde{a}_{in}h_{i}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{jn}h_{j})g_{n}=0\
(n+2\leq i\leq 2n+1).$

Thus

$\displaystyle(\tilde{a}_{i0}\frac{h_{i}}{h_{1}}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j0}\frac{h_{j}}{h_{1}})\frac{g_{0}}{g_{n}}$
$\displaystyle+\cdots+(\tilde{a}_{in}\frac{h_{i}}{h_{1}}-\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{j(n-1)}\frac{h_{j}}{h_{1}})\frac{g_{n-1}}{g_{n}}$
$\displaystyle=-\tilde{a}_{in}\frac{h_{i}}{h_{1}}+\sum_{j=1}^{n+1}b_{ij}\tilde{a}_{jn}\frac{h_{j}}{h_{1}}\
(n+2\leq i\leq 2n+1).$

We regard the above identities as a system of $n$ equations in unkown
variables $\dfrac{g_{0}}{g_{n}},...,\dfrac{g_{n-1}}{g_{n}}$ and solve these to
obtain that $\dfrac{g_{i}}{g_{n}}\ (0\leq i\leq n-1)$ has the form

$\frac{g_{i}}{g_{n}}=\frac{P_{i}}{Q_{i}},$

where $P_{i}$ and $Q_{i}$ are homogeneous polynomials in
$\dfrac{h_{j}}{h_{1}}\ (1\leq j\leq 2n+1)$ of degree $n$ with coefficients in
$\mathcal{R}\\{a_{j}\\}_{j=1}^{2n+2}$. Then by Theorem 1.2 we have

(2.6) $\displaystyle
T(r,g)\leq\sum_{i=0}^{n-1}T(r,\frac{g_{i}}{g_{n}})=\sum_{i=0}^{n-1}T(r,\frac{P_{i}}{Q_{i}})\leq
n^{2}\sum_{j=1}^{2n+1}T(r,\dfrac{h_{j}}{h_{1}})+o(T(r,g)).$

We note that $T(r,\dfrac{h_{1}}{h_{1}})=0$. By the equality (2.4) and Claim
2.4, for each $j\in\\{2,...,2n+1\\}$ we have

(2.7) $\displaystyle\begin{split}||\
T(r,\dfrac{h_{j}}{h_{1}})&=m(r,\dfrac{h_{j}}{h_{1}})+N(r,\nu^{\infty}_{\frac{h_{j}}{h_{1}}})+O(1)\leq
N_{h_{1}}(r)+N_{\frac{1}{h_{j}}}(r)+O(1)\\\
&\leq\dfrac{q^{2}(q-1)(q-2)}{(l_{0}+1)}T(r,g)+o(T(r,g)).\end{split}$

Combining (2.6) and (2.7), we obtain

$T(r,g)\leq 2n^{3}\dfrac{q^{2}(q-1)(q-2)}{(l_{0}+1)}T(r,g)+o(T(r,g)).$

Letting $r\longrightarrow+\infty$, we get

$1\leq 2n^{3}\dfrac{q^{2}(q-1)(q-2)}{(l_{0}+1)}\Leftrightarrow l_{0}+1\leq
2n^{3}q^{2}(q-1)(q-2).$

This is a contradiction.

Then from Case 1 and Case 2, we see that the supposition is impossible. Hence,
$f\times g$ is algebraically degenerate over
$\mathcal{R}\\{a_{i}\\}_{i=1}^{2n+2}.$ The theorem is proved. $\square$

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Department of Mathematics
Hanoi University of Education
136-Xuan Thuy, Cau Giay, Hanoi
Vietnam

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