# Extension of Fujimoto’s uniqueness theorems

Kai Zhou

###### Abstract

Hirotaka Fujimoto considered two meromorphic maps $f$ and $g$ of
$\mathbb{C}^{m}$ into $\mathbb{P}^{n}$ such that $f^{*}(H_{j})=g^{*}(H_{j})$
($1\leq j\leq q$) for $q$ hyperplanes $H_{j}$ in $\mathbb{P}^{n}$ in general
position and proved $f=g$ under suitable conditions. This paper considers the
case where $f$ is into $\mathbb{P}^{n}$ and $g$ is into $\mathbb{P}^{N}$ and
gives extensions of some of Fujimoto’s uniqueness theorems. The dimensions $N$
and $n$ are proved to be equal under suitable conditions. New and interesting
phenomena also occur. ††2010 Mathematics Subject Classification. 32H30, 32H25.
††Key words and phrases. complex projective spaces, uniqueness, meromorphic
maps.

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries
  3. 3 Extension of Theorem 1.1
  4. 4 Extension of Theorem 1.3
  5. 5 The Zariski closure of the image of $f\times g$ in $\mathbb{P}^{n}\times\mathbb{P}^{N}$
  6. 6 The equality of dimensions of target spaces when $g$ is algebraically non-degenerate
  7. 7 Open questions and further remarks
  8. Appendix A Proof of Lemmas 2.7 and 2.9
  9. Appendix B The algebraic non-degeneracy of $g$ in Example 6.4

##  1 Introduction and main results

In the 1920s, George Pólya [12] and Rolf Nevanlinna [10] proved some
uniqueness theorems for meromorphic functions. For instance, Nevanlinna [10]
obtained the famous five-value theorem which says that two nonconstant
meromorphic functions on the complex plane $\mathbb{C}$ are necessarily equal
if they have the same inverse image sets for five distinct points in the
Riemann sphere.

In the 1970s, Hirotaka Fujimoto [6, 7, 8] gave some generalizations of the
uniqueness results of Pólya [12] and Nevanlinna [10] to the case of
meromorphic maps into the $n$-dimensional complex projective space
$\mathbb{P}^{n}.$ For instance, Fujimoto proved in [6] the following famous
uniqueness theorem.

######  Theorem 1.1.

Let $f$ and $g$ be meromorphic maps of $\mathbb{C}^{m}$ into $\mathbb{P}^{n}$
such that $f(\mathbb{C}^{m})\not\subseteq H_{j}$,
$g(\mathbb{C}^{m})\not\subseteq H_{j}$ and $f^{*}(H_{j})=g^{*}(H_{j})$ $(1\leq
j\leq q)$ for $q$ hyperplanes $H_{1},\dots,H_{q}$ in $\mathbb{P}^{n}$ in
general position. If $g$ is linearly non-degenerate and $q\geq 3n+2,$ then
$f=g.$

Since Fujimoto’s research, there have been lots of papers which deal with the
uniqueness problem in higher dimensions, see for example [5, 15, 9, 1, 3, 13]
and the references therein. In the uniqueness results of these papers, the
target spaces of different holomorphic maps are same. However the uniqueness
problem where the identity of target spaces is not a prior assumption is
seldom studied. Katsutoshi Yamanoi considered in [16, subsection 3.2] the
uniqueness problem for holomorphic curves into possibly different abelian
varieties. In [4], Pietro Corvaja and Junjiro Noguchi extended Yamanoi’s
result to the case of semi-abelian varieties. A corollary of the main theorem
of [4] is the following.

######  Theorem 1.2 (see Corollary 1.14 in [4]).

Let $f:\mathbb{C}\to(\mathbb{C}^{*})^{n}$ and
$g:\mathbb{C}\to(\mathbb{C}^{*})^{N}$ be algebraically non-degenerate
holomorphic maps such that $f^{-1}(D_{n})=g^{-1}(D_{N}).$ Then $N=n$ and there
is an isomorphism $\varphi:(\mathbb{C}^{*})^{n}\to(\mathbb{C}^{*})^{n}$ such
that $g=\varphi\circ f$ and $\varphi(D_{n})=D_{N}.$

Here, for an integer $n\geq 1$, $D_{n}$ denotes the following hyperplane in
$(\mathbb{C}^{*})^{n}$:

$D_{n}=\\{(x_{1},\dots,x_{n})\in(\mathbb{C}^{*})^{n}\,|\,x_{1}+\dots+x_{n}=1\\}.$

The aim of this paper is to extend the uniqueness theorems of Fujimoto to the
case where the target spaces are possibly different dimensional complex
projective spaces.

One of our main results is the following which extends Fujimoto’s Theorem 1.1.

######  Theorem I.

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$ be hyperplanes in
$\mathbb{P}^{n}$ in general position and $\\{H^{\prime}_{j}\\}_{j=1}^{q}$ be
hyperplanes in $\mathbb{P}^{N}$ in general position. Let
$f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ and $g:\mathbb{C}^{m}\to\mathbb{P}^{N}$ be
meromorphic maps such that $f(\mathbb{C}^{m})\not\subseteq H_{j}$,
$g(\mathbb{C}^{m})\not\subseteq H^{\prime}_{j}$ and
$f^{*}(H_{j})=g^{*}(H^{\prime}_{j})$ for any $1\leq j\leq q.$ If $g$ is
linearly non-degenerate and $q\geq N+2n+2,$ then $N=n$ and there are a
projective linear transformation $L$ of $\mathbb{P}^{n}$ and $(n+2)$ distinct
indices $k_{1},\dots,k_{n+2}\in\\{1,\dots,q\\}$ such that $f=L(g)$ and

$L(H^{\prime}_{k_{i}})=H_{k_{i}},\quad 1\leq i\leq n+2.$

As a matter of fact, we shall prove in Section 3 an improved version of
Theorem I, namely Theorem 3.1.

In [6], Fujimoto proved also the following theorem.

######  Theorem 1.3.

Let $f$, $g$ and $H_{1},\dots,H_{q}$ be as in Theorem 1.1. If $n\geq 2,$ and
$g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree $\leq 2$ in
$\mathbb{P}^{n},$ and $q\geq 3n+1,$ then $f=g.$

We shall extend the above theorem of Fujimoto to the following.

######  Theorem II.

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$,
$\\{H^{\prime}_{j}\\}_{j=1}^{q}$, $f$ and $g$ be as in Theorem I. If
$g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree
$\leq\min\\{n+1,N-n+2\\}$ in $\mathbb{P}^{N}$ and

$q\geq\max\big{\\{}3n+1,N+\frac{3}{2}n+\frac{3}{2}\big{\\}},$

then $N=n$ and there is a projective linear transformation $L$ of
$\mathbb{P}^{n}$ such that $f=L(g)$ and $L$ satisfies the following condition:
either

$L(H^{\prime}_{k_{i}})=H_{k_{i}},\quad 1\leq i\leq n+2,$

for $(n+2)$ distinct indices $k_{1},\dots,k_{n+2}\in\\{1,\dots,q\\},$ or

$L(H^{\prime}_{k_{i}})=H_{k_{i}},\,\,1\leq i\leq n+1,\quad\mbox{and}\quad
L(H^{\prime}_{k_{n+2}})=H_{k_{n+3}},\,\,L(H^{\prime}_{k_{n+3}})=H_{k_{n+2}},$

for $(n+3)$ distinct indices $k_{1},\dots,k_{n+3}\in\\{1,\dots,q\\}.$

The above Theorem II will be proved in Section 4. Indeed, we shall prove
Theorem 4.1 which slightly improves Theorem II.

######  Remark 1.4.

For any $(n+3)$ hyperplanes $H_{1},\dots,H_{n+3}$ in $\mathbb{P}^{n}$ _in
general position_ where $n\geq 2,$ one can easily prove that there _does not
exist_ a projective linear transformation $L$ of $\mathbb{P}^{n}$ such that

$L(H_{i})=H_{i},\,\,1\leq i\leq n+1,\quad\mbox{and}\quad
L(H_{n+2})=H_{n+3},\,\,L(H_{n+3})=H_{n+2}.$

So Theorem II recovers Fujimoto’s Theorem 1.3.

######  Remark 1.5.

Let $g:\mathbb{C}^{m}\to\mathbb{P}^{n}$ be any linearly non-degenerate
meromorphic map. Let $\\{H^{\prime}_{j}\\}_{j=1}^{q}$ be any hyperplanes in
$\mathbb{P}^{n}$ in general position. Let $L$ be a projective linear
transformation of $\mathbb{P}^{n}.$ Then $f:=L(g)$ is a linearly non-
degenerate meromorphic map and $H_{j}:=L(H^{\prime}_{j})$, $1\leq j\leq q,$
are hyperplanes in general position. The meromorphic maps $f$ and $g$ satisfy
the condition that

$f^{*}(H_{j})=g^{*}(H^{\prime}_{j}),\quad 1\leq j\leq q.$

The following is a direct corollary of Theorems 3.1 and 4.1.

######  Corollary 1.6.

Let $N$ and $n$ be integers such that $1\leq n\leq N\leq\frac{3}{2}n.$ Let
$\\{H_{j}\\}_{j=1}^{q}$, $\\{H^{\prime}_{j}\\}_{j=1}^{q}$, $f$ and $g$ be as
in Theorem I.

  1. (i)

If $g$ is linearly non-degenerate and $q\geq 3n+2,$ then the conclusion of
Theorem I holds.

  2. (ii)

If $g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree
$\leq(N-n+2)$ in $\mathbb{P}^{N}$ and $q\geq 3n+1,$ then the conclusion of
Theorem II holds.

Moreover, Fujimoto proved in [7] and [8] the following deep theorem.

######  Theorem 1.7.

Let $f$ and $g$ be meromorphic maps of $\mathbb{C}^{m}$ into $\mathbb{P}^{n}$
such that $f(\mathbb{C}^{m})\not\subseteq H_{j}$,
$g(\mathbb{C}^{m})\not\subseteq H_{j}$ and $f^{*}(H_{j})=g^{*}(H_{j})$ $(1\leq
j\leq 2n+3)$ for $(2n+3)$ hyperplanes $H_{j}$ in $\mathbb{P}^{n}$ in general
position. If $g$ is algebraically non-degenerate, then $f=g.$

We refer the reader to [17] for a note on Fujimoto’s proof of the above
theorem.

Fujimoto obtained the above Theorem 1.7 after a very deep analysis of the case
when the number of hyperplanes is $(2n+2)$ and $g$ is algebraically non-
degenerate.

When trying to extend Theorem 1.7, we get a partial result, namely, Theorem
6.1 which is also one of the main results of this paper. Let $N$, $n$,
$\\{H_{j}\\}_{j=1}^{q}$, $\\{H^{\prime}_{j}\\}_{j=1}^{q}$, $f$ and $g$ be as
in Theorem I. Theorem 6.1 says that the dimensions $N$ and $n$ are necessarily
equal if $g$ is algebraically non-degenerate and $q\geq N+n+2.$ So the
extension of Theorem 1.7 becomes studying the case when there are two families
$\\{H_{j}\\}_{j=1}^{2n+3}$ and $\\{H^{\prime}_{j}\\}_{j=1}^{2n+3}$ of
hyperplanes in a complex projective space $\mathbb{P}^{n}.$ We cannot yet get
a satisfactory result and a discussion about this will be given in Section 7.

This paper is organized as follows. In Section 2, we recall some definitions
and include some auxiliary results. In particular, we shall give some
combinatorial lemmas which are essentially due to Fujimoto. In Section 3, we
consider the extension of Theorem 1.1 and two theorems shall be proved. In
Section 4, we give an extension of Theorem 1.3. In Section 5, we consider the
dimension of the Zariski closure $V_{f\times g}$ of the image of $f\times g$
in $\mathbb{P}^{n}\times\mathbb{P}^{N}.$ The main theorem of Section 5 will be
essentially used in Section 6. We shall also show that $V_{f\times g}$ is an
irreducible algebraic set of dimension $\leq N.$ In Section 6, we shall show
the equality of dimensions when $g$ is algebraically non-degenerate and $q\geq
N+n+2.$ In Section 7, we shall give further remarks about the extension of
Theorem 1.7 and also ask a question concerning the main theorem of Section 6.
After Section 7, there are two appendices in which we give proofs of some
lemmas or conclusions in previous sections.

### Acknowledgements.

The author thanks Professor Lu Jin and Professor Qiming Yan for their constant
encouragements. The author also thanks for Professor Qiming Yan’s constant
help.

##  2 Preliminaries

### 2.1 Meromorphic maps and hyperplanes in complex projective spaces.

Let $f$ be a meromorphic map of $\mathbb{C}^{m}$ into $\mathbb{P}^{n}.$ Denote
by $I(f)$ the indeterminacy locus of $f,$ which is an analytic subset of
$\mathbb{C}^{m}$ of codimension $\geq 2.$ A _reduced representation_ of $f$ is
a $(n+1)$-tuple $(f_{0},\dots,f_{n})$ of holomorphic functions on
$\mathbb{C}^{m}$ such that

$\\{f_{0}=f_{1}=\dots=f_{n}=0\\}=I(f)$

and

$f(z)=[f_{0}(z):\dots:f_{n}(z)]\quad\forall z\in\mathbb{C}^{m}\setminus I(f).$

Any meromorphic map of $\mathbb{C}^{m}$ into $\mathbb{P}^{n}$ has a reduced
representation.

We also recall the following definition.

######  Definition 2.1.

A meromorphic map of $\mathbb{C}^{m}$ into $\mathbb{P}^{n}$ is said to be
_linearly_ (resp. _algebraically_) _non-degenerate_ , if its image is not
contained in any hyperplane (resp. hypersurface) in $\mathbb{P}^{n}.$

Let

$H=\big{\\{}[x_{0}:\dots:x_{n}]\in\mathbb{P}^{n}\,|\,a_{0}x_{0}+\dots+a_{n}x_{n}=0\big{\\}}$

be a hyperplane in $\mathbb{P}^{n}$ where
$(a_{0},\dots,a_{n})\in\mathbb{C}^{n+1}\setminus\\{(0,\dots,0)\\}.$ We shall
say that the linear form $a_{0}X_{0}+\dots+a_{n}X_{n}$ defines the hyperplane
$H.$

######  Definition 2.2.

Let $\\{H_{j}\\}_{j=1}^{q}$ be a family of hyperplanes in $\mathbb{P}^{n}.$
Take a linear form $L_{j}=a^{j}_{0}X_{0}+\dots+a^{j}_{n}X_{n}$ that defines
$H_{j}$ for each $1\leq j\leq q.$ The family $\\{H_{j}\\}_{j=1}^{q}$ of
hyperplanes is said to be _in general position_ , if, for any distinct indices
$i_{1},\dots,i_{k}\in\\{1,\dots,q\\}$ with $1\leq k\leq n+1,$ the linear forms
$L_{i_{1}},\dots,L_{i_{k}}$ are linearly independent over $\mathbb{C}.$

Let $\\{H_{j}\\}_{j=1}^{q}$ and $\\{L_{j}\\}_{j=1}^{q}$ be as above. If $q\geq
n+1,$ then $\\{H_{j}\\}_{j=1}^{q}$ is in general position if and only if

$\det\big{(}a^{i_{k}}_{0},\dots,a^{i_{k}}_{n};\,1\leq k\leq n+1\big{)}\neq 0$

for any $(n+1)$ distinct indices $i_{1},\dots,i_{n+1}\in\\{1,\dots,q\\}.$

######  Definition 2.3.

Let $f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ be a meromorphic map and let $H$ be a
hyperplane in $\mathbb{P}^{n}.$ By taking a reduced representation
$(f_{0},\dots,f_{n})$ of $f$ and taking a linear form
$a_{0}X_{0}+\dots+a_{n}X_{n}$ that defines $H,$ we define

$(f,H):=a_{0}f_{0}+\dots+a_{n}f_{n},$

which is a holomorphic function on $\mathbb{C}^{m}.$ When
$f(\mathbb{C}^{m})\not\subseteq H,$ the pullback divisor $f^{*}H$ is defined
to be the zero divisor of the holomorphic function $(f,H),$ which is
independent of the different choices of reduced representations of $f$ and
linear forms that define $H.$

### 2.2 The Borel Lemma.

Following Fujimoto’s method, we shall essentially use the following result
which is referred to as the Borel Lemma.

######  Theorem 2.4 (Borel Lemma, see [2] and Corollary 4.2 in [6]).

Let $h_{1},\dots,h_{q}$ be nowhere zero holomorphic functions on
$\mathbb{C}^{m}$ which satisfy the following equation:

$c_{1}h_{1}+\dots+c_{q}h_{q}\equiv 0,$

where $c_{1},\dots,c_{q}$ are nonzero complex numbers. Then, for each $h_{i},$
there is some $h_{j}$ with $j\neq i$ such that $h_{j}/h_{i}$ is constant.

We now give some notations.

######  Notation.

Denote by $\mathcal{H}^{*}=\mathcal{H}^{*}_{m}$ the multiplicative group of
all nowhere zero holomorphic functions on $\mathbb{C}^{m}.$ We regard the set
$\mathbb{C}^{*}$ of all nonzero complex numbers as a subgroup of
$\mathcal{H}^{*}.$ For any $h\in\mathcal{H}^{*},$ we denote by $[h]$ the
equivalence class in the quotient group $\mathcal{H}^{*}/\mathbb{C}^{*}$ that
contains $h.$

We emphasize that the quotient group $\mathcal{H}^{*}/\mathbb{C}^{*}$ is a
_torsion-free_ abelian group.

A corollary of the Borel Lemma is the following.

######  Proposition 2.5 (see Proposition 4.5 in [6]).

Let $\eta_{1},\dots,\eta_{t}$ be nowhere zero holomorphic functions on
$\mathbb{C}^{m}$ and assume they are multiplicatively independent, namely,

$\eta_{1}^{n_{1}}\cdot\eta_{2}^{n_{2}}\cdots\eta_{t}^{n_{t}}\not\in\mathbb{C}^{*}$

for any $(n_{1},\dots,n_{t})\in\mathbb{Z}^{t}\setminus\\{(0,\dots,0)\\}.$ If a
polynomial $P(X_{1},\dots,X_{t})\in\mathbb{C}[X_{1},\dots,X_{t}]$ satisfies

$P(\eta_{1},\dots,\eta_{t})\equiv 0,$

then $P(X_{1},\dots,X_{t})$ is the zero polynomial.

### 2.3 Combinatorial lemmas.

Let $G$ be a torsion-free abelian group. Let $A=(\alpha_{1},\dots,\alpha_{q})$
be a $q$-tuple of elements in $G.$ We denote by
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ the subgroup of $G$ generated by
$\alpha_{1},\dots,\alpha_{q}.$ Because $G$ is torsion-free,
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ is free and is of finite rank. We
denote by rank$\\{\alpha_{1},\dots,\alpha_{q}\\}$ the rank of the free abelian
group $\langle\alpha_{1},\dots,\alpha_{q}\rangle.$

The following definition is due to Fujimoto [6].

######  Definition 2.6.

Let $(G,\cdot)$ be a torsion-free abelian group. Let $q\geq r>s\geq 1$ be
integers. A $q$-tuple $A=(\alpha_{1},\dots,\alpha_{q})$ of elements in $G$ is
said to _have the property_ $(P_{r,s})$ if arbitrarily chosen $r$ elements
$\alpha_{l(1)},\dots,\alpha_{l(r)}$ in $A$ ($1\leq l(1)<\dots<l(r)\leq q$)
satisfy the condition that, for any $s$ distinct indices
$i_{1},\dots,i_{s}\in\\{1,\dots,r\\},$ there exist distinct indices
$j_{1},\dots,j_{s}\in\\{1,\dots,r\\}$ with
$\\{j_{1},\dots,j_{s}\\}\neq\\{i_{1},\dots,i_{s}\\}$ such that

$\alpha_{l(i_{1})}\cdot\alpha_{l(i_{2})}\cdots\alpha_{l(i_{s})}=\alpha_{l(j_{1})}\cdot\alpha_{l(j_{2})}\cdots\alpha_{l(j_{s})}.$

The following lemma is a refinement of Fujimoto’s Lemma 2.6 in [6].

######  Lemma 2.7.

Let $(G,\cdot)$ be a torsion-free abelian group. Let
$A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple of elements in $G$ that has
the property $(P_{r,s}),$ where $q\geq r>s\geq 1.$ Then there exist $(q-r+2)$
distinct indices $i_{1},\dots,i_{q-r+2}\in\\{1,\dots,q\\}$ such that

$\alpha_{i_{1}}=\alpha_{i_{2}}=\dots=\alpha_{i_{q-r+2}}.$

Furthermore, when $r\leq 2s,$ the conclusion can be refined as follows:

  1. (i)

if $q\geq 2s-1,$ then there exist $q-2(r-s)+2=:l$ distinct indices
$i_{1},\dots,i_{l}\in\\{1,\dots,q\\}$ such that

$\alpha_{i_{1}}=\alpha_{i_{2}}=\dots=\alpha_{i_{l}};$

  2. (ii)

if $q<2s-1,$ then there exist $2(q-r+2)$ distinct indices
$i_{1},\dots,i_{q-r+2},$ $j_{1},\dots,j_{q-r+2}$ in $\\{1,\dots,q\\}$ such
that

$\alpha_{i_{1}}=\alpha_{i_{2}}=\dots=\alpha_{i_{q-r+2}}\quad\mbox{and}\quad\alpha_{j_{1}}=\alpha_{j_{2}}=\dots=\alpha_{j_{q-r+2}}.$

######  Remark 2.8.

A $q$-tuple $A=(\alpha_{1},\dots,\alpha_{q})$ has the property $(P_{r,s})$ if
and only if it has the property $(P_{r,r-s}).$ So the condition $r\leq 2s$ is
not a real restriction.

The proof of Lemma 2.7 is given in Appendix A. Here we give a remark.
Following Fujimoto’s proof, we arrange the $\alpha_{i}$’s in a suitable order:

$\alpha_{i_{1}}\prec\alpha_{i_{2}}\prec\dots\prec\alpha_{i_{q}}.$

We prove that

$\alpha_{i_{s}}=\alpha_{i_{s+1}}=\dots=\alpha_{i_{q-(r-s)+1}}.$

Also we have

$\alpha_{i_{r-s}}=\alpha_{i_{r-s+1}}=\dots=\alpha_{i_{q-s+1}}.$

Then, when $r-s\leq s,$ by comparing the two numbers $(q-s+1)$ and $s,$ the
conclusion is naturally divided into two cases. Thus we get the conclusions of
Lemma 2.7.

The next lemma is a refinement of Fujimoto’s Lemma 2.7 in [6].

######  Lemma 2.9.

Let $(G,\cdot)$ be a torsion-free abelian group. Let $q\geq r>s\geq 1$ be
integers such that $2\leq r-s\leq s$ and $q\geq 2s-1.$ Let
$A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple of elements in $G$ that has
the property $(P_{r,s}).$ Set $l:=q-2(r-s)+2.$ Then, when $r-s\geq 3,$ at
least one of the following three cases occurs:

  1. $(\alpha)$

there exist $(l+1)$ distinct indices $i_{1},\dots,i_{l+1}\in\\{1,\dots,q\\}$
such that

$\alpha_{i_{1}}=\alpha_{i_{2}}=\dots=\alpha_{i_{l+1}};$

  2. $(\beta)$

there exist $(l+2)$ distinct indices $i_{1},\dots,i_{l+2}\in\\{1,\dots,q\\}$
such that

$\alpha_{i_{1}}=\alpha_{i_{2}}=\dots=\alpha_{i_{l}}\quad\mbox{and}\quad\alpha_{i_{l+1}}=\alpha_{i_{l+2}};$

  3. $(\gamma)$

there exist an integer $k$ with $2\leq k\leq\min\\{r-s,2s-r+2\\}$ and $(l+k)$
distinct indices $i_{1},\dots,i_{l+k}\in\\{1,\dots,q\\}$ such that

$\alpha_{i_{1}}=\alpha_{i_{2}}=\dots=\alpha_{i_{l}}\quad\mbox{and}\quad\alpha_{i_{l+1}}\cdot\alpha_{i_{l+2}}\cdots\alpha_{i_{l+k}}=\alpha_{i_{1}}^{k}.$

When $r-s=2,$ one of the cases $(\alpha)$ and $(\gamma)$ occurs.

The basic idea of the proof of Lemmas 2.7 and 2.9 is essentially same as the
idea of Fujimoto’s proof in [6, Section 2], though some new discussions are
also needed. We shall give a proof of Lemmas 2.7 and 2.9 in Appendix A.

To state another lemma, we need the following notation.

######  Notation.

For elements $\alpha_{1},\alpha_{2},\dots,\alpha_{q}$,
$\tilde{\alpha}_{1},\tilde{\alpha}_{2},\dots,\tilde{\alpha}_{q}$ in an abelian
group $(G,\cdot),$ by the notation

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{q}=\tilde{\alpha}_{1}:\tilde{\alpha}_{2}:\dots:\tilde{\alpha}_{q},$

we mean that $\alpha_{i}=\beta\tilde{\alpha}_{i}$ ($1\leq i\leq q$) for some
element $\beta\in G.$

Now we recall the following lemma.

######  Lemma 2.10 (see Lemma 3.6 in [7] and [17]).

Let $s$ and $q$ be integers with $2\leq s<q\leq 2s.$ Let $(G,\cdot)$ be a
torsion-free abelian group. Let $A=(\alpha_{1},\dots,\alpha_{q})$ be a
$q$-tuple of elements in $G$ that has the property $(P_{q,s}),$ and assume
that at least one $\alpha_{i}$ equals the unit element $1$ of $G.$ Then

  1. (i)

rank$\\{\alpha_{1},\dots,\alpha_{q}\\}=:t\leq s-1;$

  2. (ii)

if $t=s-1,$ then $q=2s$ and there is a basis
$\\{\beta_{1},\dots,\beta_{s-1}\\}$ of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ such that the $\alpha_{i}$ are
represented, after a suitable change of indices, as one of the following two
types:

     * (A)

$s$ is odd and

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}=1:1:\beta_{1}:\beta_{1}:\beta_{2}:\beta_{2}:\dots:\beta_{s-1}:\beta_{s-1};$

     * (B)

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}=1:1:\dots:1:\beta_{1}:\dots:\beta_{s-1}:(\beta_{1}\cdots\beta_{a_{1}})^{-1}:(\beta_{a_{1}+1}\cdots\beta_{a_{2}})^{-1}:\dots:(\beta_{a_{k-1}+1}\cdots\beta_{a_{k}})^{-1},$

where $0\leq k\leq s-1$, $1\leq a_{1}<a_{2}<\dots<a_{k}\leq s-1,$ and the unit
element $1$ appears $(s+1-k)$ times in the right hand side.

We have the following easy observation.

######  Observation 2.11.

Let $(\alpha_{1},\dots,\alpha_{q})$ and
$(\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{q})$ be two $q$-tuples of elements
in a torsion-free abelian group $(G,\cdot).$ Assume that there exist indices
$i_{0}$ and $j_{0}$ such that $\alpha_{i_{0}}=\tilde{\alpha}_{j_{0}}=1,$ and
there is an element $\beta\in G$ such that
$\alpha_{i}=\beta\tilde{\alpha}_{i}$ for any $1\leq i\leq q.$ Then

$\langle\alpha_{1},\dots,\alpha_{q}\rangle=\langle\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{q}\rangle.$

Now let $A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple that has the property
$(P_{r,s})$ where $q\geq r>s\geq 1$ and $r\leq 2s,$ and assume that at least
one $\alpha_{i}$ equals the unit element $1.$ Set $t:={\rm
rank}\\{\alpha_{1},\dots,\alpha_{q}\\}.$ By Lemma 2.7, we may assume that
$\alpha_{r-1}=\alpha_{r}=\dots=\alpha_{q}.$ Put
$\alpha^{\prime}_{i}:=\alpha_{r-1}^{-1}\cdot\alpha_{i}$ for each $1\leq i\leq
q.$ Consider the new $r$-tuple
$(\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{r})$ which also has the property
$(P_{r,s}).$ By Observation 2.11, we know that ${\rm
rank}\\{\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{r}\\}=t.$ If $s\geq 2,$
then Lemma 2.10 tells us that $t\leq s-1,$ and $t=s-1$ implies $r=2s.$ If
$s=1,$ then $r=2$ and $t=0.$ Whatever the case, we have $t\leq s-1,$ and
$t=s-1$ implies $r=2s.$

The above discussion shows the following result.

######  Lemma 2.12.

Let $(G,\cdot)$ be a torsion-free abelian group. Let $q\geq r>s\geq 1$ be
integers such that $r\leq 2s.$ Let $A=(\alpha_{1},\dots,\alpha_{q})$ be a
$q$-tuple of elements in $G$ that has the property $(P_{r,s}),$ and assume at
least one $\alpha_{i}$ equals the unit element $1$ of $G.$ Set $t:={\rm
rank}\\{\alpha_{1},\dots,\alpha_{q}\\}.$ Then $t\leq s-1,$ and $t=s-1$ implies
$r=2s.$

##  3 Extension of Theorem 1.1

We give first the following extension of Fujimoto’s Theorem 1.1.

######  Theorem 3.1.

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$ be hyperplanes in
$\mathbb{P}^{n}$ in general position and $\\{H^{\prime}_{j}\\}_{j=1}^{q}$ be
hyperplanes in $\mathbb{P}^{N}$ in general position. Let
$f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ and $g:\mathbb{C}^{m}\to\mathbb{P}^{N}$ be
meromorphic maps such that $f(\mathbb{C}^{m})\not\subseteq H_{j}$,
$g(\mathbb{C}^{m})\not\subseteq H^{\prime}_{j}$ and
$f^{*}(H_{j})=g^{*}(H^{\prime}_{j})$ for any $1\leq j\leq q.$ If $g$ is
linearly non-degenerate and

$q\geq\min\big{\\{}N+2n+2,\max\big{\\{}2N+1,3n+2\big{\\}}\big{\\}},$

then $N=n$ and there are a projective linear transformation $L$ of
$\mathbb{P}^{n}$ and $(n+2)$ distinct indices
$k_{1},\dots,k_{n+2}\in\\{1,\dots,q\\}$ such that $f=L(g)$ and

$L(H^{\prime}_{k_{i}})=H_{k_{i}},\quad 1\leq i\leq n+2.$

###### Proof.

Take reduced representations $(f_{0},\dots,f_{n})$ and $(g_{0},\dots,g_{N})$
of $f$ and $g,$ respectively. For each $1\leq j\leq q,$ take linear forms
$a^{j}_{0}X_{0}+\dots+a^{j}_{n}X_{n}$ and
$b^{j}_{0}Y_{0}+\dots+b^{j}_{N}Y_{N}$ that define $H_{j}$ and
$H^{\prime}_{j},$ respectively.

Define, for each $1\leq i\leq q,$

$h_{i}:=\frac{(f,H_{i})}{(g,H^{\prime}_{i})},$ (3.2)

where

$(f,H_{i})=a^{i}_{0}f_{0}+\dots+a^{i}_{n}f_{n}\quad\mbox{and}\quad(g,H^{\prime}_{i})=b^{i}_{0}g_{0}+\dots+b^{i}_{N}g_{N}.$

By the assumption, we know that each $h_{i}$ is a nowhere zero holomorphic
function on $\mathbb{C}^{m}.$

We note that the condition on $q$ implies $q\geq N+n+2.$ Choose $(N+n+2)$
functions among the $h_{i}$’s arbitrarily, say $h_{1},\dots,h_{N+n+2}.$

From definition, we get

$a^{i}_{0}f_{0}+\dots+a^{i}_{n}f_{n}-b^{i}_{0}h_{i}g_{0}-\dots-b^{i}_{N}h_{i}g_{N}=0,\quad
1\leq i\leq N+n+2.$

Eliminating $f_{0},\dots,f_{n},g_{0}\dots,g_{N}$ from these equations, we
conclude that

$\det\big{(}a^{i}_{0},\dots,a^{i}_{n},b^{i}_{0}h_{i},\dots,b^{i}_{N}h_{i};\,1\leq
i\leq N+n+2\big{)}\equiv 0.$

By the Laplace expansion formula, we deduce from the above equation the
following identity:

$\sum_{1\leq i_{1}<i_{2}<\dots<i_{N+1}\leq N+n+2}A_{i_{1},\dots,i_{N+1}}\cdot
h_{i_{1}}\cdot h_{i_{2}}\cdots h_{i_{N+1}}\equiv 0,$ (3.3)

where

$\displaystyle A_{i_{1},\dots,i_{N+1}}=\pm$
$\displaystyle\det\big{(}a^{i}_{0},\dots,a^{i}_{n};\,i\in\\{1,\dots,N+n+2\\}\setminus\\{i_{1},\dots,i_{N+1}\\}\big{)}$
$\displaystyle\cdot\det\big{(}b^{i}_{0},\dots,b^{i}_{N};\,i\in\\{i_{1},\dots,i_{N+1}\\}\big{)}.$

Because $\\{H_{j}\\}_{j=1}^{q}$ and $\\{H^{\prime}_{j}\\}_{j=1}^{q}$ are both
in general position, the constants $A_{i_{1},\dots,i_{N+1}}$ are all nonzero.
Thus by making use of the Borel Lemma (Theorem 2.4), we deduce from the above
identity (3.3) that, for any $(N+1)$ distinct indices
$i_{1},\dots,i_{N+1}\in\\{1,\dots,N+n+2\\},$ there exist distinct
$j_{1},\dots,j_{N+1}\in\\{1,\dots,N+n+2\\}$ with
$\\{j_{1},\dots,j_{N+1}\\}\neq\\{i_{1},\dots,i_{N+1}\\}$ such that

$(h_{j_{1}}\cdot h_{j_{2}}\cdots h_{j_{N+1}})/(h_{i_{1}}\cdot h_{i_{2}}\cdots
h_{i_{N+1}})\in\mathbb{C}^{*},$

which gives (see Section 2 for notations)

$[h_{j_{1}}]\cdot[h_{j_{2}}]\cdots[h_{j_{N+1}}]=[h_{i_{1}}]\cdot[h_{i_{2}}]\cdots[h_{i_{N+1}}](\in\mathcal{H}^{*}/\mathbb{C}^{*}).$

Using the Definition 2.6, the above discussion in fact shows that the
$q$-tuple $([h_{1}],\dots,[h_{q}])$ whose elements are in the quotient group
$\mathcal{H}^{*}/\mathbb{C}^{*}$ has the property $(P_{N+n+2,N+1}).$

Then by Lemma 2.7 and using the condition on $q,$ we get that there are
$(n+2)$ distinct indices $k_{1},\dots,k_{n+2}\in\\{1,\dots,q\\}$ such that

$[h_{k_{1}}]=[h_{k_{2}}]=\dots=[h_{k_{n+2}}].$

In fact, if $q\geq N+2n+2$ then

$q-(N+n+2)+2\geq n+2$

and the above conclusion follows from Lemma 2.7. If
$q\geq\max\\{2N+1,3n+2\\},$ then

$q-2(n+1)+2\geq n+2$

and the above conclusion follows from the refined conclusion (i) of Lemma 2.7.

By choosing a new reduced representation of $f$ if necessary, we may assume
that $h_{k_{1}}=:c_{1}$, $h_{k_{2}}=:c_{2}$, $\dots$, $h_{k_{n+2}}=:c_{n+2}$
are all constants.

In the following, if there is no confusion, we shall denote by $f$, $g$,
$H_{i}$ and $H^{\prime}_{i}$ ($1\leq i\leq q$) the row vectors
$(f_{0},\dots,f_{n})$, $(g_{0},\dots,g_{N})$, $(a^{i}_{0},\dots,a^{i}_{n})$
and $(b^{i}_{0},\dots,b^{i}_{N})$ ($1\leq i\leq q$), respectively. Then, for
any $1\leq i\leq q,$ $(f,H_{i})$ equals the inner product $\langle
f,H_{i}\rangle,$ and $(g,H^{\prime}_{i})$ equals $\langle
g,H^{\prime}_{i}\rangle.$

Since $H_{k_{1}},H_{k_{2}},\dots,H_{k_{n+2}}$ are hyperplanes in
$\mathbb{P}^{n}$ in general position, we can write

$H_{k_{n+2}}=x_{1}H_{k_{1}}+x_{2}H_{k_{2}}+\dots+x_{n+1}H_{k_{n+1}},$

where $x_{1},x_{2},\dots,x_{n+1}$ are nonzero constants.

We note that $\langle f,H_{k_{i}}\rangle=c_{i}\langle
g,H^{\prime}_{k_{i}}\rangle$ for any $1\leq i\leq n+2.$ So

$\langle f,H_{k_{n+2}}\rangle=\sum_{i=1}^{n+1}x_{i}\langle
f,H_{k_{i}}\rangle=\sum_{i=1}^{n+1}x_{i}c_{i}\langle
g,H^{\prime}_{k_{i}}\rangle=\langle
g,\sum_{i=1}^{n+1}x_{i}c_{i}H^{\prime}_{k_{i}}\rangle.$

Thus

$c_{n+2}\langle g,H^{\prime}_{k_{n+2}}\rangle=\langle
g,\sum_{i=1}^{n+1}x_{i}c_{i}H^{\prime}_{k_{i}}\rangle.$

It then follows that

$\langle
g,c_{n+2}H^{\prime}_{k_{n+2}}-\sum_{i=1}^{n+1}x_{i}c_{i}H^{\prime}_{k_{i}}\rangle=0.$

By the linear non-degeneracy of $g,$ we conclude that

$c_{n+2}H^{\prime}_{k_{n+2}}-\sum_{i=1}^{n+1}x_{i}c_{i}H^{\prime}_{k_{i}}=0.$
(3.4)

The equation (3.4) shows that the $(n+2)$ vectors $H^{\prime}_{k_{1}}$,
$H^{\prime}_{k_{2}}$, $\dots$, $H^{\prime}_{k_{n+2}}$ in $\mathbb{C}^{N+1}$
are linearly dependent over $\mathbb{C}.$ If $N>n,$ then $n+2\leq N+1$ and
(3.4) gives a contradiction because $\\{H^{\prime}_{j}\\}$ are hyperplanes in
$\mathbb{P}^{N}$ in general position. So it necessarily holds that $N=n.$

Now the matrices

$\left(\begin{array}[]{c}H_{k_{1}}\\\ \vdots\\\
H_{k_{n+1}}\end{array}\right)=:A\quad\mbox{and}\quad\left(\begin{array}[]{c}H^{\prime}_{k_{1}}\\\
\vdots\\\ H^{\prime}_{k_{n+1}}\end{array}\right)=:B$

are both invertible square matrices of order $(n+1).$

Denote by $C$ the diagonal matrix ${\rm diag}(c_{1},\dots,c_{n+1}).$ Then we
have

$\left(\begin{array}[]{c}f_{0}\\\ \vdots\\\
f_{n}\end{array}\right)=A^{-1}CB\left(\begin{array}[]{c}g_{0}\\\ \vdots\\\
g_{n}\end{array}\right).$

Put $\tilde{L}:=A^{-1}CB.$ We write the homogeneous coordinate of a point in
$\mathbb{P}^{n}$ as a column vector. Then the correspondence

$\left(\begin{array}[]{c}w_{0}\\\ \vdots\\\
w_{n}\end{array}\right)\mapsto\tilde{L}\left(\begin{array}[]{c}w_{0}\\\
\vdots\\\ w_{n}\end{array}\right)$

defines a projective linear transformation of $\mathbb{P}^{n}$ which is
denoted by $L.$ We see that $f=L(g).$

For any $1\leq i\leq n+1,$ by

$H^{\prime}_{k_{i}}B^{-1}=(0,\dots,0,\underset{\underset{i-{\rm
th}}{\uparrow}}{1},0,\dots,0),$

we deduce easily that $L(H^{\prime}_{k_{i}})=H_{k_{i}}.$

Because the $H^{\prime}_{j}$’s are hyperplanes in $\mathbb{P}^{n}$ in general
position, we can write

$H^{\prime}_{k_{n+2}}=y_{1}H^{\prime}_{k_{1}}+y_{2}H^{\prime}_{k_{2}}+\dots+y_{n+1}H^{\prime}_{k_{n+1}},$

where $y_{1},y_{2},\dots,y_{n+1}$ are nonzero constants. Substituting this
into (3.4), we get

$\sum_{i=1}^{n+1}(c_{n+2}y_{i}-x_{i}c_{i})H^{\prime}_{k_{i}}=0,$

which implies that

$c_{i}=\frac{y_{i}}{x_{i}}\cdot c_{n+2},\quad 1\leq i\leq n+1.$

Then easy computation tells us that $L(H^{\prime}_{k_{n+2}})=H_{k_{n+2}}.$
This finishes the proof of Theorem 3.1. ∎

Next we shall reduce the number of hyperplanes in the condition of Theorem 3.1
under a stronger non-degeneracy condition of $g.$

######  Theorem 3.5.

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$,
$\\{H^{\prime}_{j}\\}_{j=1}^{q}$, $f$ and $g$ be as in Theorem 3.1. If
$g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree $\leq 2$ in
$\mathbb{P}^{N}$ and

$q\geq\max\big{\\{}3n+2,N+\frac{3}{2}n+\frac{3}{2}\big{\\}},$

then the conclusion of Theorem 3.1 holds.

###### Proof.

Let $(f_{0},\dots,f_{n})$, $(g_{0},\dots,g_{N})$,
$a^{j}_{0}X_{0}+\dots+a^{j}_{n}X_{n}$ and
$b^{j}_{0}Y_{0}+\dots+b^{j}_{N}Y_{N}$ ($1\leq j\leq q$) be as in the proof of
Theorem 3.1. For each $1\leq i\leq q,$ let $h_{i}$ be defined as (3.2). Then
each $h_{i}$ is a nowhere zero holomorphic function on $\mathbb{C}^{m}.$

Since $N+3n/2+3/2\geq N+n+2,$ the assumption on $q$ implies that $q\geq
N+n+2.$ As in the proof of Theorem 3.1, we know that the $q$-tuple
$([h_{1}],\dots,[h_{q}])$ has the property $(P_{N+n+2,N+1}).$

Consider first the case when $q\geq 2N+1.$ Since $q-2(n+1)+2\geq n+2,$ the
refined conclusion (i) of Lemma 2.7 tells us that there are $(n+2)$ distinct
indices $k_{1},\dots,k_{n+2}\in\\{1,\dots,q\\}$ such that

$[h_{k_{1}}]=[h_{k_{2}}]=\dots=[h_{k_{n+2}}].$

Then the same discussion as in the proof of Theorem 3.1 shows the conclusion
of Theorem 3.1 holds.

So, it suffices to prove next that the case of $q<2N+1$ cannot happen. Assume
$q<2N+1.$ We shall show a contradiction.

By the refined conclusion (ii) of Lemma 2.7, we get there are $2(q-N-n)$
distinct indices
$i_{0},\dots,i_{q-N-n-1},j_{0},\dots,j_{q-N-n-1}\in\\{1,\dots,q\\}$ such that

$[h_{i_{0}}]=[h_{i_{1}}]=\dots=[h_{i_{q-N-n-1}}]\quad\mbox{and}\quad[h_{j_{0}}]=[h_{j_{1}}]=\dots=[h_{j_{q-N-n-1}}].$
(3.6)

By $q\geq N+3n/2+3/2,$ we have

$2(q-N-n-1)\geq n+1.$

Let $l$ be the largest integer that does not exceed $(n+1)/2,$ and set
$\tilde{l}:=n+1-l.$ One checks easily that

$1\leq l\leq\tilde{l}\leq q-N-n-1\quad\mbox{and}\quad l+\tilde{l}=n+1.$

From (3.6), we know that $h_{i_{k}}/h_{i_{0}}=:c_{k}\,(1\leq k\leq l)$ and
$h_{j_{k}}/h_{j_{0}}=:\tilde{c}_{k}\,(1\leq k\leq\tilde{l})$ are all nonzero
constants.

As in the proof of Theorem 3.1, sometimes we shall use the symbols $f,g$,
$H_{j},H^{\prime}_{j}\,(1\leq j\leq q)$ to denote the corresponding row
vectors.

Since $\\{H_{j}\\}$ are hyperplanes in $\mathbb{P}^{n}$ in general position,
we can write

$H_{i_{0}}=x_{1}H_{i_{1}}+\dots+x_{l}H_{i_{l}}+x_{l+1}H_{j_{1}}+\dots+x_{n+1}H_{j_{\tilde{l}}}$

and

$H_{j_{0}}=\tilde{x}_{1}H_{i_{1}}+\dots+\tilde{x}_{l}H_{i_{l}}+\tilde{x}_{l+1}H_{j_{1}}+\dots+\tilde{x}_{n+1}H_{j_{\tilde{l}}},$

where $x_{1},\dots,x_{n+1},\tilde{x}_{1},\dots,\tilde{x}_{n+1}$ are nonzero
constants.

We then have

$\langle f,H_{i_{0}}\rangle=\sum_{k=1}^{l}x_{k}\langle
f,H_{i_{k}}\rangle+\sum_{k=1}^{\tilde{l}}x_{l+k}\langle f,H_{j_{k}}\rangle.$

So

$\displaystyle\langle g,H^{\prime}_{i_{0}}\rangle h_{i_{0}}=\langle
f,H_{i_{0}}\rangle$ $\displaystyle=\sum_{k=1}^{l}x_{k}c_{k}h_{i_{0}}\langle
g,H^{\prime}_{i_{k}}\rangle+\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}h_{j_{0}}\langle
g,H^{\prime}_{j_{k}}\rangle$ $\displaystyle=\langle
g,\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}}\rangle h_{i_{0}}+\langle
g,\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\rangle
h_{j_{0}},$

which implies

$\langle
g,H^{\prime}_{i_{0}}-\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}}\rangle
h_{i_{0}}=\langle
g,\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\rangle
h_{j_{0}}.$

Similarly, we have

$\langle
g,H^{\prime}_{j_{0}}-\sum_{k=1}^{\tilde{l}}\tilde{x}_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\rangle
h_{j_{0}}=\langle g,\sum_{k=1}^{l}\tilde{x}_{k}c_{k}H^{\prime}_{i_{k}}\rangle
h_{i_{0}}.$

From the above two equations, we conclude

$\displaystyle\langle
g,H^{\prime}_{i_{0}}-\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}}\rangle$
$\displaystyle\cdot\langle
g,H^{\prime}_{j_{0}}-\sum_{k=1}^{\tilde{l}}\tilde{x}_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\rangle$
$\displaystyle=$ $\displaystyle\langle
g,\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\rangle\cdot\langle
g,\sum_{k=1}^{l}\tilde{x}_{k}c_{k}H^{\prime}_{i_{k}}\rangle.$ (3.7)

We note that the vectors

$H^{\prime}_{i_{0}}-\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}},\quad
H^{\prime}_{j_{0}}-\sum_{k=1}^{\tilde{l}}\tilde{x}_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}},\quad\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\quad\mbox{and}\quad\sum_{k=1}^{l}\tilde{x}_{k}c_{k}H^{\prime}_{i_{k}}$

are all nonzero vectors in $\mathbb{C}^{N+1}.$

Because both the left and right hand sides of (3) can be written as a
homogeneous polynomial of degree 2 in $g_{0},\dots,g_{N},$ we conclude by the
non-degeneracy condition of $g$ that (3) holds as an identity of polynomials
in $g_{0},\dots,g_{N}.$ This tells us that the linear polynomial $\langle
g,H^{\prime}_{i_{0}}-\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}}\rangle$ is a
constant multiple of $\langle
g,\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}\rangle,$ or is
a constant multiple of $\langle
g,\sum_{k=1}^{l}\tilde{x}_{k}c_{k}H^{\prime}_{i_{k}}\rangle.$ The latter
implies that $H^{\prime}_{i_{0}},H^{\prime}_{i_{1}},\dots,H^{\prime}_{i_{l}}$
are linearly dependent over $\mathbb{C},$ which contradicts the “in general
position” assumption of $\\{H^{\prime}_{j}\\}.$ So it is necessary that
$H^{\prime}_{i_{0}}-\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}}$ is a constant
multiple of $\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}},$
namely, there is a nonzero constant $c$ such that

$H^{\prime}_{i_{0}}-\sum_{k=1}^{l}x_{k}c_{k}H^{\prime}_{i_{k}}-c\sum_{k=1}^{\tilde{l}}x_{l+k}\tilde{c}_{k}H^{\prime}_{j_{k}}=0.$

If $N>n,$ then $n+2\leq N+1$ and the $(n+2)$ vectors
$H^{\prime}_{i_{0}},H^{\prime}_{i_{1}},\dots,H^{\prime}_{i_{l}}$,
$H^{\prime}_{j_{1}},\dots,H^{\prime}_{j_{\tilde{l}}}$ in $\mathbb{C}^{N+1}$
are linearly independent over $\mathbb{C},$ which contradicts the above
equation. Hence $N=n.$

Thus we get

$q\geq N+n+2=2N+2,$

which contradicts our assumption that $q<2N+1.$ This shows the case of
$q<2N+1$ cannot happen and the proof is thus finished. ∎

##  4 Extension of Theorem 1.3

We shall prove in this section the following theorem which extends Fujimoto’s
Theorem 1.3.

######  Theorem 4.1.

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$,
$\\{H^{\prime}_{j}\\}_{j=1}^{q}$, $f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ and
$g:\mathbb{C}^{m}\to\mathbb{P}^{N}$ be as in Theorem 3.1. If
$g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree
$\leq\min\\{n+1,N-n+2\\}$ in $\mathbb{P}^{N}$ and

$q\geq\max\big{\\{}3n+1,\min\big{\\{}2N+1,N+\frac{3}{2}n+\frac{3}{2}\big{\\}}\big{\\}},$

then $N=n$ and there is a projective linear transformation $L$ of
$\mathbb{P}^{n}$ such that $f=L(g)$ and $L$ satisfies the following condition:
either

$L(H^{\prime}_{k_{i}})=H_{k_{i}},\quad 1\leq i\leq n+2,$

for $(n+2)$ distinct indices $k_{1},\dots,k_{n+2}\in\\{1,\dots,q\\},$ or

$L(H^{\prime}_{k_{i}})=H_{k_{i}},\,\,1\leq i\leq n+1,\quad\mbox{and}\quad
L(H^{\prime}_{k_{n+2}})=H_{k_{n+3}},\,\,L(H^{\prime}_{k_{n+3}})=H_{k_{n+2}},$

for $(n+3)$ distinct indices $k_{1},\dots,k_{n+3}\in\\{1,\dots,q\\}.$

######  Remark 4.2.

For positive integers $N$ and $n$ with $N\geq n,$ we set

$\alpha(n,N):=\max\big{\\{}3n+1,N+\frac{3}{2}n+\frac{3}{2}\big{\\}}$

and

$\beta(n,N):=\max\big{\\{}3n+1,\min\big{\\{}2N+1,N+\frac{3}{2}n+\frac{3}{2}\big{\\}}\big{\\}}.$

To compare $\alpha(n,N)$ and $\beta(n,N),$ we consider the following three
cases.

  * •

$N\geq\frac{3}{2}n+\frac{1}{2}.$ In this case, $\beta(n,N)=\alpha(n,N).$

  * •

$N\leq\frac{3}{2}n-\frac{1}{2}.$ In this case, $\beta(n,N)=\alpha(n,N)=3n+1.$

  * •

$\frac{3}{2}n-\frac{1}{2}<N<\frac{3}{2}n+\frac{1}{2}.$ In this case, it is
necessary that $n$ is even and $N=\frac{3}{2}n.$ Thus $\beta(n,N)=3n+1$ and
$\alpha(n,N)=3n+\frac{3}{2}.$

###### Proof of Theorem 4.1.

Let $(f_{0},\dots,f_{n})$, $(g_{0},\dots,g_{N})$,
$a^{j}_{0}X_{0}+\dots+a^{j}_{n}X_{n}$ and
$b^{j}_{0}Y_{0}+\dots+b^{j}_{N}Y_{N}$ ($1\leq j\leq q$) be as in the proof of
Theorem 3.1. For each $1\leq i\leq q,$ define $h_{i}$ as (3.2). Then each
$h_{i}$ is a nowhere zero holomorphic function on $\mathbb{C}^{m}.$

One checks easily that the assumption on $q$ implies $q\geq N+n+2.$ As in the
proof of Theorem 3.1, we know that the $q$-tuple $([h_{1}],\dots,[h_{q}])$ has
the property $(P_{N+n+2,N+1}).$

By assumption, we have

$q\geq\min\big{\\{}2N+1,N+\frac{3}{2}n+\frac{3}{2}\big{\\}}.$

If $q\geq N+3n/2+3/2,$ then the same argument as in the proof of Theorem 3.5
tells us that $q\geq 2N+1,$ because $g(\mathbb{C}^{m})$ is not contained in
any hypersurface of degree $\leq 2$ in $\mathbb{P}^{N}.$ So we always have

$q\geq 2N+1.$

Then by making use of Lemma 2.9 and using the condition that $q\geq 3n+1,$ we
conclude that at least one of the following three cases occurs: (we renumber
the indices for simplicity)

  1. $(\alpha)$

$[h_{1}]=[h_{2}]=\dots=[h_{n+2}];$

  2. $(\beta)$

$[h_{1}]=[h_{2}]=\dots=[h_{n+1}]$ and $[h_{n+2}]=[h_{n+3}];$

  3. $(\gamma)$

$[h_{1}]=[h_{2}]=\dots=[h_{n+1}]$ and
$[h_{n+2}]\cdot[h_{n+3}]\cdots[h_{n+1+k}]=[h_{1}]^{k},$ where $k$ is a
positive integer such that $2\leq k\leq\min\\{n+1,N-n+2\\}.$

If case ($\alpha$) occurs, then the same argument as in the proof of Theorem
3.1 tells us that $N=n$ and there is a projective linear transformation $L$ of
$\mathbb{P}^{n}$ such that $f=L(g)$ and $L(H^{\prime}_{i})=H_{i}$ for any
$1\leq i\leq n+2.$

Now assume case ($\beta$) occurs. By choosing a new reduced representation of
$f,$ we assume that $h_{1}=:c_{1}$, $h_{2}=:c_{2}$, $\dots$,
$h_{n+1}=:c_{n+1}$ are all constants. We also know that
$h_{n+2}=c_{n+2}h_{n+3}$ for a nonzero constant $c_{n+2}.$

In the following, we sometimes use the symbols $f,g$,
$H_{j},H^{\prime}_{j}\,(1\leq j\leq q)$ to denote the corresponding row
vectors as in the proof of Theorem 3.1.

Since the $\\{H_{j}\\}$ are hyperplanes in $\mathbb{P}^{n}$ in general
position, we can write

$H_{n+2}=x_{n+2,1}H_{1}+x_{n+2,2}H_{2}+\dots+x_{n+2,n+1}H_{n+1}$

and

$H_{n+3}=x_{n+3,1}H_{1}+x_{n+3,2}H_{2}+\dots+x_{n+3,n+1}H_{n+1},$

where $x_{n+2,i}$ and $x_{n+3,i}$ ($1\leq i\leq n+1$) are nonzero constants.

Since $\langle f,H_{i}\rangle=c_{i}\langle g,H^{\prime}_{i}\rangle$ ($1\leq
i\leq n+1$), we have

$\langle f,H_{n+2}\rangle=\sum_{i=1}^{n+1}x_{n+2,i}\langle
f,H_{i}\rangle=\sum_{i=1}^{n+1}x_{n+2,i}c_{i}\langle
g,H^{\prime}_{i}\rangle=\langle
g,\sum_{i=1}^{n+1}x_{n+2,i}c_{i}H^{\prime}_{i}\rangle.$

Similarly, we have

$\langle f,H_{n+3}\rangle=\langle
g,\sum_{i=1}^{n+1}x_{n+3,i}c_{i}H^{\prime}_{i}\rangle.$

We note that both $\sum_{i=1}^{n+1}x_{n+2,i}c_{i}H^{\prime}_{i}$ and
$\sum_{i=1}^{n+1}x_{n+3,i}c_{i}H^{\prime}_{i}$ are nonzero vectors in
$\mathbb{C}^{N+1}.$

Substituting the above two equalities into the equation
$h_{n+2}=c_{n+2}h_{n+3},$ we deduce that

$\langle g,\sum_{i=1}^{n+1}x_{n+2,i}c_{i}H^{\prime}_{i}\rangle\cdot\langle
g,H^{\prime}_{n+3}\rangle=c_{n+2}\cdot\langle
g,\sum_{i=1}^{n+1}x_{n+3,i}c_{i}H^{\prime}_{i}\rangle\cdot\langle
g,H^{\prime}_{n+2}\rangle.$

Because $g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree
$\leq 2,$ the above equation holds as an identity of polynomials in
$g_{0},\dots,g_{N}.$ Since the linear polynomial $\langle
g,H^{\prime}_{n+2}\rangle$ cannot be a constant multiple of $\langle
g,H^{\prime}_{n+3}\rangle,$ we conclude that $\langle
g,H^{\prime}_{n+2}\rangle$ is a constant multiple of $\langle
g,\sum_{i=1}^{n+1}x_{n+2,i}c_{i}H^{\prime}_{i}\rangle.$ This means that there
is a nonzero constant $c$ such that

$H^{\prime}_{n+2}-c\sum_{i=1}^{n+1}x_{n+2,i}c_{i}H^{\prime}_{i}=0,$

which implies that the $(n+2)$ vectors $H^{\prime}_{1},\dots,H^{\prime}_{n+2}$
are linearly dependent over $\mathbb{C}.$ The same reasoning as in the proof
of Theorem 3.1 shows $N=n.$

Now using the same method as in the proof of Theorem 3.1, we conclude that
there is a projective linear transformation $L$ of $\mathbb{P}^{n}$ such that
$f=L(g)$ and $L(H^{\prime}_{i})=H_{i}$ for any $1\leq i\leq n+2.$

Finally, we assume case ($\gamma$) occurs. By choosing a new reduced
representation of $f,$ we assume that $h_{1}=:c_{1}$, $h_{2}=:c_{2}$, $\dots$,
$h_{n+1}=:c_{n+1}$ are all constants and

$h_{n+2}\cdot h_{n+3}\cdots h_{n+1+k}=:c_{n+2}$

is also a constant.

As before, we can write

$H_{j}=x_{j,1}H_{1}+x_{j,2}H_{2}+\dots+x_{j,n+1}H_{n+1},\quad n+2\leq j\leq
n+1+k,$

where $x_{j,1},\dots,x_{j,n+1}$ are nonzero constants.

Since $\langle f,H_{i}\rangle=c_{i}\langle g,H^{\prime}_{i}\rangle$ ($1\leq
i\leq n+1$), we have

$\langle f,H_{j}\rangle=\langle
g,\sum_{i=1}^{n+1}x_{j,i}c_{i}H^{\prime}_{i}\rangle,\quad n+2\leq j\leq
n+1+k.$

We note that $\sum_{i=1}^{n+1}x_{j,i}c_{i}H^{\prime}_{i}$ ($n+2\leq j\leq
n+1+k$) are all nonzero vectors in $\mathbb{C}^{N+1}.$

Substituting these equalities into the equation $h_{n+2}\cdot h_{n+3}\cdots
h_{n+1+k}=c_{n+2},$ we conclude that

$\prod_{j=n+2}^{n+1+k}\langle
g,\sum_{i=1}^{n+1}x_{j,i}c_{i}H^{\prime}_{i}\rangle=c_{n+2}\cdot\prod_{j=n+2}^{n+1+k}\langle
g,H^{\prime}_{j}\rangle.$

Because $g(\mathbb{C}^{m})$ is not contained in any hypersurface of degree
$\leq\min\\{n+1,N-n+2\\},$ the above equation holds as an identity of
polynomials in $g_{0},\dots,g_{N}.$ Then we see that there is a bijection
$\sigma:\\{n+2,\dots,n+1+k\\}\to\\{n+2,\dots,n+1+k\\}$ such that the linear
polynomial $\langle g,\sum_{i=1}^{n+1}x_{j,i}c_{i}H^{\prime}_{i}\rangle$ is a
constant multiple of $\langle g,H^{\prime}_{\sigma(j)}\rangle$ for any
$n+2\leq j\leq n+1+k.$ This means that there are nonzero constants
$\tilde{c}_{j}\,(n+2\leq j\leq n+1+k)$ such that

$\sum_{i=1}^{n+1}x_{j,i}c_{i}H^{\prime}_{i}-\tilde{c}_{j}H^{\prime}_{\sigma(j)}=0,\quad
n+2\leq j\leq n+1+k.$ (4.3)

The case of $j=n+2$ implies that the $(n+2)$ vectors
$H^{\prime}_{1},\dots,H^{\prime}_{n+1},H^{\prime}_{\sigma(n+2)}$ are linearly
dependent over $\mathbb{C}.$ The same reasoning as in the proof of Theorem 3.1
shows $N=n.$

Then we get $k=2$ and the above $\sigma$ is a bijection from $\\{n+2,n+3\\}$
to $\\{n+2,n+3\\}.$

As in the proof of Theorem 3.1, we write the homogeneous coordinate of a point
in $\mathbb{P}^{n}$ as a column vector and let $\tilde{L}=A^{-1}CB,$ where

$A=\left(\begin{array}[]{c}H_{1}\\\ \vdots\\\ H_{n+1}\end{array}\right),\quad
B=\left(\begin{array}[]{c}H^{\prime}_{1}\\\ \vdots\\\
H^{\prime}_{n+1}\end{array}\right)\quad\mbox{and}\quad C={\rm
diag}(c_{1},\dots,c_{n+1}).$

The matrix $\tilde{L}$ defines a projective linear transformation $L$ of
$\mathbb{P}^{n}$ and this $L$ satisfies the condition that $f=L(g)$ and
$L(H^{\prime}_{i})=H_{i}$ for $1\leq i\leq n+1.$

We write

$H^{\prime}_{j}=y_{j,1}H^{\prime}_{1}+y_{j,2}H^{\prime}_{2}+\dots+y_{j,n+1}H^{\prime}_{n+1},\quad
n+2\leq j\leq n+3,$

where $y_{j,1},\dots,y_{j,n+1}$ are nonzero constants.

We conclude from (4.3) that, for any $j\in\\{n+2,n+3\\},$

$c_{i}=\tilde{c}_{j}\cdot\frac{y_{\sigma(j),i}}{x_{j,i}},\quad 1\leq i\leq
n+1.$

Then direct computation shows $L(H^{\prime}_{\sigma(j)})=H_{j}$ for any
$j\in\\{n+2,n+3\\}.$

If $\sigma$ is the identity map, then we see that $L(H^{\prime}_{i})=H_{i}$
for any $1\leq i\leq n+3.$ If $\sigma$ is not the identity map, then we see
that

$L(H^{\prime}_{i})=H_{i},\,\,1\leq i\leq n+1,\quad\mbox{and}\quad
L(H^{\prime}_{n+2})=H_{n+3},\,\,L(H^{\prime}_{n+3})=H_{n+2}.$

This completes the proof of Theorem 4.1. ∎

By putting $N=n=m=1$ and $q=4,$ we get from Theorem 4.1 the following result
concerning meromorphic functions.

######  Corollary 4.4.

Let $\\{a_{j}\\}_{j=1}^{4}$ and $\\{b_{j}\\}_{j=1}^{4}$ be two families of
distinct points in the Riemann sphere. If two nonconstant meromorphic
functions $f$ and $g$ on $\mathbb{C}$ satisfy the condition that
$f^{*}(a_{j})=g^{*}(b_{j})(1\leq j\leq 4),$ then there is a Möbius
transformation $L$ such that $f=L(g)$ and $L$ satisfies the following
condition: either

$L(b_{k_{1}})=a_{k_{1}},L(b_{k_{2}})=a_{k_{2}},L(b_{k_{3}})=a_{k_{3}}$

for distinct $k_{1},k_{2},k_{3}\in\\{1,2,3,4\\},$ or

$L(b_{k_{1}})=a_{k_{1}},L(b_{k_{2}})=a_{k_{2}},L(b_{k_{3}})=a_{k_{4}},L(b_{k_{4}})=a_{k_{3}}$

for a permutation $(k_{1},k_{2},k_{3},k_{4})$ of $\\{1,2,3,4\\}.$

##  5 The Zariski closure of the image of $f\times g$ in
$\mathbb{P}^{n}\times\mathbb{P}^{N}$

This section is an extension of [7, Section 5].

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$ be hyperplanes in
$\mathbb{P}^{n}$ in general position and $\\{H^{\prime}_{j}\\}_{j=1}^{q}$ be
hyperplanes in $\mathbb{P}^{N}$ in general position with

$q\geq N+n+2.$

Let $f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ and
$g:\mathbb{C}^{m}\to\mathbb{P}^{N}$ be meromorphic maps such that
$f(\mathbb{C}^{m})\not\subseteq H_{j}$, $g(\mathbb{C}^{m})\not\subseteq
H^{\prime}_{j}$ and $f^{*}(H_{j})=g^{*}(H^{\prime}_{j})$ for any $1\leq j\leq
q.$

Let $f\times g$ be the holomorphic map of
$\mathbb{C}^{m}\setminus\big{(}I(f)\cup I(g)\big{)}$ into
$\mathbb{P}^{n}\times\mathbb{P}^{N}$ that is given by $(f\times
g)(z)=(f(z),g(z))$ for $z\in\mathbb{C}^{m}\setminus\big{(}I(f)\cup
I(g)\big{)},$ where $I(f)$ and $I(g)$ are the indeterminacy loci of $f$ and
$g,$ respectively.

######  Definition 5.1.

We define $V_{f\times g}$ to be the Zariski closure of the image of $f\times
g$ in $\mathbb{P}^{n}\times\mathbb{P}^{N},$ namely, the intersection of all
algebraic sets in $\mathbb{P}^{n}\times\mathbb{P}^{N}$ that contain the image
of $f\times g.$

One sees easily the following.

######  Proposition 5.2.

The algebraic set $V_{f\times g}$ is irreducible.

###### Proof.

Assume $V_{f\times g}$ is reducible. This means that there are algebraic sets
$V_{1}$ and $V_{2}$ in $\mathbb{P}^{n}\times\mathbb{P}^{N}$ with
$V_{1}\subsetneqq V_{f\times g}$ and $V_{2}\subsetneqq V_{f\times g}$ such
that $V_{f\times g}=V_{1}\cup V_{2}.$ Then $(f\times g)^{-1}(V_{1})=:A_{1}$
and $(f\times g)^{-1}(V_{2})=:A_{2}$ are both analytic subsets of
$\mathbb{C}^{m}\setminus\big{(}I(f)\cup I(g)\big{)}$ and
$\mathbb{C}^{m}\setminus\big{(}I(f)\cup I(g)\big{)}=A_{1}\cup A_{2}.$ The
definition of $V_{f\times g}$ implies that $A_{1}$ and $A_{2}$ are both proper
analytic subsets of $\mathbb{C}^{m}\setminus\big{(}I(f)\cup I(g)\big{)}.$ This
gives a contradiction because $\mathbb{C}^{m}\setminus\big{(}I(f)\cup
I(g)\big{)}$ is irreducible. Thus $V_{f\times g}$ is necessarily irreducible.
∎

Let $(f_{0},\dots,f_{n})$, $(g_{0},\dots,g_{N})$, and
$a^{j}_{0}X_{0}+\dots+a^{j}_{n}X_{n}$, $b^{j}_{0}Y_{0}+\dots+b^{j}_{N}Y_{N}$
($1\leq j\leq q$) be as in the proof of Theorem 3.1. For each $1\leq i\leq q,$
let $h_{i}$ be defined as (3.2). Then each $h_{i}$ is a nowhere zero
holomorphic function on $\mathbb{C}^{m}.$ By choosing a new reduced
representation of $f$ if necessary, we may assume at least one $h_{i}$ is
constant.

Consider the $q$-tuple $([h_{1}],\dots,[h_{q}])$ of elements in
$\mathcal{H}^{*}/\mathbb{C}^{*}.$ As in the proof of Theorem 3.1, the
$q$-tuple $([h_{1}],\dots,[h_{q}])$ has the property $(P_{N+n+2,N+1}).$ We
define

$t:={\rm rank}\big{\\{}[h_{1}],\dots,[h_{q}]\big{\\}}.$

######  Remark 5.3.

The number $t$ is independent of the different choices of reduced
representations of $f$ and $g$ and linear forms that define hyperplanes
$H_{j}$ and $H^{\prime}_{j},$ as long as at least one $h_{i}$ is constant.

By Lemma 2.10 (i) (or Lemma 2.12), we have the following.

######  Proposition 5.4.

$0\leq t\leq N.$

Now we give the following main theorem of this section. Let notations be as
above.

######  Theorem 5.5.

Suppose the $q$-tuple $([h_{1}],\dots,[h_{q}])$ does not have the property
$(P_{2s-(N-n),s})$ for some positive integer $s$ with $N-n+1\leq s\leq N+1.$
Then

$\max\\{t,\dim V_{f\times g}\\}\leq N-s+t.$

###### Proof.

Clearly we have $N-n+1\leq s\leq N$ and $t\geq 1.$

Take $\eta_{1},\dots,\eta_{t}$ in $\mathcal{H}^{*}$ such that
$\\{[\eta_{1}],\dots,[\eta_{t}]\\}$ is a basis of the free abelian group
$\langle[h_{1}],\dots,[h_{q}]\rangle.$ Then the functions $h_{i}$ are
represented as follows:

$h_{i}=c_{i}\cdot\eta_{1}^{l_{i,1}}\cdots\eta_{t}^{l_{i,t}},\quad 1\leq i\leq
q,$ (5.6)

where $l_{i,1},\dots,l_{i,t}$ are integers and $c_{i}$ are nonzero constants.

By assumption, there are $h_{i_{1}},\dots,h_{i_{2s-(N-n)}}$ such that the
$(2s-(N-n))$-tuple $([h_{i_{1}}],\dots,[h_{i_{2s-(N-n)}}])$ does not have the
property $(P_{2s-(N-n),s}),$ where $i_{1},\dots,i_{2s-(N-n)}$ are distinct
indices in $\\{1,\dots,q\\}.$ Without loss of generality, we assume the
$(2s-(N-n))$-tuple $([h_{1}],\dots,[h_{2s-(N-n)}])$ does not have the property
$(P_{2s-(N-n),s}).$

Put $s_{1}:=s-(N-n).$ We list the following equalities and inequalities for
readers’ convenience:

$s_{1}+s=2s-(N-n),\quad s_{1}+(N+1-s)=n+1,$ $1\leq s_{1}\leq n,\quad 1\leq
N+1-s\leq n,\quad(2s-(N-n))+(N+1-s)=s+n+1.$

By making a transformation of homogenous coordinates on $\mathbb{P}^{n},$ we
may assume that

$(a^{i}_{0},\dots,a^{i}_{n})=(0,\dots,0,\underset{i-{\rm
th}}{\underset{\uparrow}{1}},0,\dots,0),\quad 1\leq i\leq s_{1},$

and

$(a^{i}_{0},\dots,a^{i}_{n})=(0,\dots,0,\underset{(i-s)-{\rm
th}}{\underset{\uparrow}{1}},0,\dots,0),\quad 2s-(N-n)+1\leq i\leq s+n+1.$

By making a transformation of homogenous coordinates on $\mathbb{P}^{N},$ we
may assume that

$(b^{i}_{0},\dots,b^{i}_{N})=(0,\dots,0,\underset{i-{\rm
th}}{\underset{\uparrow}{1}},0,\dots,0),\quad 1\leq i\leq s,$

and

$(b^{i}_{0},\dots,b^{i}_{N})=(0,\dots,0,\underset{(i-s_{1})-{\rm
th}}{\underset{\uparrow}{1}},0,\dots,0),\quad 2s-(N-n)+1\leq i\leq s+n+1.$

After these two transformations, we know that

$\det\big{(}a^{i}_{0},\dots,a^{i}_{s_{1}-1};\,i=i_{1},\dots,i_{s_{1}}\big{)}\neq
0$

for any $s_{1}$ distinct indices
$i_{1},\dots,i_{s_{1}}\in\\{1,\dots,2s-(N-n)\\},$ and

$\det\big{(}b^{i}_{0},\dots,b^{i}_{s-1};\,i=j_{1},\dots,j_{s}\big{)}\neq 0$

for any $s$ distinct indices $j_{1},\dots,j_{s}\in\\{1,\dots,2s-(N-n)\\}.$ We
also know that the functions $f_{0},\dots,f_{n}$, $g_{0},\dots,g_{N}$ are all
not identically zero.

Now we claim that

$\det\big{(}a^{i}_{0},\dots,a^{i}_{s_{1}-1},b^{i}_{0}h_{i},\dots,b^{i}_{s-1}h_{i};\,i=1,\dots,2s-(N-n)\big{)}\not\equiv
0.$ (5.7)

In fact, if the left hand side of (5.7) is identically zero, then using the
Laplace expansion formula and the Borel Lemma as in the proof of Theorem 3.1
we conclude $([h_{1}],\dots,[h_{2s-(N-n)}])$ has the property
$(P_{2s-(N-n),s}),$ which contradicts the assumption. Thus we have (5.7).

From the definition of $h_{i},$ we get the following equations:

$a^{i}_{0}f_{0}+\dots+a^{i}_{n}f_{n}-b^{i}_{0}h_{i}g_{0}-\dots-b^{i}_{N}h_{i}g_{N}=0,\quad
1\leq i\leq 2s-(N-n),$

and

$f_{i-s-1}=h_{i}g_{i-s_{1}-1},\quad 2s-(N-n)+1\leq i\leq s+n+1.$ (5.8)

Substituting (5.8) into the first $(2s-(N-n))$ equations, we get

$a^{i}_{0}f_{0}+\dots+a^{i}_{s_{1}-1}f_{s_{1}-1}-b^{i}_{0}h_{i}g_{0}-\dots-b^{i}_{s-1}h_{i}g_{s-1}=B_{i},\quad
1\leq i\leq 2s-(N-n),$ (5.9)

where

$B_{i}=(b^{i}_{s}h_{i}-a^{i}_{s_{1}}h_{2s-(N-n)+1})g_{s}+\dots+(b^{i}_{N}h_{i}-a^{i}_{n}h_{s+n+1})g_{N}.$

By (5.7), we can apply the Cramer’s rule to obtain the following expressions
of $f_{j}\,(0\leq j\leq s_{1}-1)$ and $g_{j}\,(0\leq j\leq s-1)$:

$f_{j}=\frac{\tilde{P}_{j,s}(h_{1},\dots,h_{s+n+1})g_{s}+\dots+\tilde{P}_{j,N}(h_{1},\dots,h_{s+n+1})g_{N}}{\tilde{P}(h_{1},\dots,h_{2s-(N-n)})},\quad
0\leq j\leq s_{1}-1,$
$g_{j}=\frac{\tilde{Q}_{j,s}(h_{1},\dots,h_{s+n+1})g_{s}+\dots+\tilde{Q}_{j,N}(h_{1},\dots,h_{s+n+1})g_{N}}{\tilde{P}(h_{1},\dots,h_{2s-(N-n)})},\quad
0\leq j\leq s-1,$

where $\tilde{P}$, $\tilde{P}_{j,s},\dots,\tilde{P}_{j,N}$,
$\tilde{Q}_{j,s},\dots,\tilde{Q}_{j,N}$ are polynomials with constant
coefficients and $\tilde{P}$ is not zero.

Combining the above expressions with (5.8), we get expressions of
$f_{j}/f_{n}.$ Then by substituting (5.6) into these expressions and also the
above expressions of $g_{j},$ we get the following:

$\frac{f_{j}}{f_{n}}=\frac{P_{j,s}(1,\eta_{1},\dots,\eta_{t})g_{s}+\dots+P_{j,N}(1,\eta_{1},\dots,\eta_{t})g_{N}}{P_{0}(1,\eta_{1},\dots,\eta_{t})g_{N}},\quad
0\leq j\leq s_{1}-1,$ (5.10)
$g_{j}=\frac{Q_{j,s}(1,\eta_{1},\dots,\eta_{t})g_{s}+\dots+Q_{j,N}(1,\eta_{1},\dots,\eta_{t})g_{N}}{P(1,\eta_{1},\dots,\eta_{t})},\quad
0\leq j\leq s-1,$ (5.11)
$\frac{f_{j}}{f_{n}}=\frac{R_{j}^{+}(1,\eta_{1},\dots,\eta_{t})g_{j+N-n}}{R_{j}^{-}(1,\eta_{1},\dots,\eta_{t})g_{N}},\quad
s_{1}\leq j\leq n-1,$ (5.12)

where $P_{0},P\in\mathbb{C}[z_{0},z_{1},\dots,z_{t}]$ are homogeneous
polynomials of degree $d\geq 1,$ and each of the polynomials
$P_{j,s},\dots,P_{j,N}$, $Q_{j,s},\dots,Q_{j,N}$ is either zero or a
homogeneous polynomial of degree $d,$ and $R_{j}^{+}$ and $R_{j}^{-}$ are
monomials with $\deg R_{j}^{+}=\deg R_{j}^{-}.$

Now let $\eta$ be the holomorphic map of $\mathbb{C}^{m}$ into
$\mathbb{P}^{t}$ that is given by $\eta(z)=[1:\eta_{1}(z):\dots:\eta_{t}(z)]$
for $z\in\mathbb{C}^{m},$ and let $f\times g\times\eta$ be the holomorphic map
of $\mathbb{C}^{m}\setminus\big{(}I(f)\cup I(g)\big{)}$ into
$\mathbb{P}^{n}\times\mathbb{P}^{N}\times\mathbb{P}^{t}$ that is given by
$(f\times g\times\eta)(z)=(f(z),g(z),\eta(z))$ for
$z\in\mathbb{C}^{m}\setminus\big{(}I(f)\cup I(g)\big{)}.$ Let $V_{f,g,\eta}$
be the Zariski closure of the image of $f\times g\times\eta$ in
$\mathbb{P}^{n}\times\mathbb{P}^{N}\times\mathbb{P}^{t}.$ The same argument as
in the proof of Proposition 5.2 shows that the algebraic set $V_{f,g,\eta}$ is
irreducible.

Let $V$ be the algebraic set in
$\mathbb{P}^{n}\times\mathbb{P}^{N}\times\mathbb{P}^{t}$ which is defined as
follows:

$\displaystyle
V:=\big{\\{}([x_{0}:\dots:x_{n}],[y_{0}:\dots:y_{N}],[z_{0}:\dots:z_{t}])\in\mathbb{P}^{n}\times\mathbb{P}^{N}\times\mathbb{P}^{t}\,|\hskip
40.00006pt$ $\displaystyle
x_{j}P_{0}(z_{0},\dots,z_{t})y_{N}-x_{n}P_{j,s}(z_{0},\dots,z_{t})y_{s}-\dots-
x_{n}P_{j,N}(z_{0},\dots,z_{t})y_{N}=0,$ $\displaystyle 0\leq j\leq s_{1}-1,$
$\displaystyle
y_{j}P(z_{0},\dots,z_{t})-Q_{j,s}(z_{0},\dots,z_{t})y_{s}-\dots-
Q_{j,N}(z_{0},\dots,z_{t})y_{N}=0,$ $\displaystyle 0\leq j\leq s-1,$
$\displaystyle
x_{j}R_{j}^{-}(z_{0},\dots,z_{t})y_{N}-x_{n}R_{j}^{+}(z_{0},\dots,z_{t})y_{j+N-n}=0,\quad
s_{1}\leq j\leq n-1\big{\\}}.$

Then by (5.10), (5.11) and (5.12), we see that

$V\supseteq V_{f,g,\eta}.$

Let $U$ be the subset of $V$ that contains all points
$([x_{0}:\dots:x_{n}],[y_{0}:\dots:y_{N}],[z_{0}:\dots:z_{t}])$ in $V$ such
that

$x_{n}\neq 0,\quad y_{N}\neq 0,\quad z_{0}\neq 0,\quad
P_{0}(z_{0},\dots,z_{t})\neq 0,\quad P(z_{0},\dots,z_{t})\neq 0,$
$R_{j}^{-}(z_{0},\dots,z_{t})\neq 0,\quad s_{1}\leq j\leq n-1.$

Then $U$ is a Zariski-open subset of $V.$ Because $f_{n}$, $g_{N}$,
$P_{0}(1,\eta_{1},\dots,\eta_{t})$, $P(1,\eta_{1},\dots,\eta_{t})$ and
$R_{j}^{-}(1,\eta_{1},\dots,\eta_{t})\,(s_{1}\leq j\leq n-1)$ are nonzero
holomorphic functions on $\mathbb{C}^{m},$ we see that

$V_{f,g,\eta}\cap U\neq\emptyset.$

Let $\phi:U\to\mathbb{C}^{N-s+t}$ be the map

$([x_{0}:\dots:x_{n}],[y_{0}:\dots:y_{N}],[z_{0}:\dots:z_{t}])\mapsto\Big{(}\frac{y_{s}}{y_{N}},\dots,\frac{y_{N-1}}{y_{N}},\frac{z_{1}}{z_{0}},\dots,\frac{z_{t}}{z_{0}}\Big{)}.$

From the definition of $V$ and $U,$ we easily check that $\phi$ is injective.
Now since $V_{f,g,\eta}\cap U$ is a nonempty Zariski-open subset of
$V_{f,g,\eta}$ and $V_{f,g,\eta}$ is irreducible, we conclude from the
injectivity of $\phi|_{V_{f,g,\eta}\cap U}:V_{f,g,\eta}\cap
U\to\mathbb{C}^{N-s+t}$ that

$\dim V_{f,g,\eta}\leq N-s+t.$

Let
$\pi_{12}:\mathbb{P}^{n}\times\mathbb{P}^{N}\times\mathbb{P}^{t}\to\mathbb{P}^{n}\times\mathbb{P}^{N}$
be the projection onto the first two components and let
$\pi_{3}:\mathbb{P}^{n}\times\mathbb{P}^{N}\times\mathbb{P}^{t}\to\mathbb{P}^{t}$
be the projection onto the third component. Since $\pi_{12}(V_{f,g,\eta})$
contains the image of $f\times g,$ one sees that
$\pi_{12}(V_{f,g,\eta})\supseteq V_{f\times g}$ and thus

$\dim V_{f\times g}\leq\dim\pi_{12}(V_{f,g,\eta})\leq\dim V_{f,g,\eta}.$

Since $\eta_{1},\dots,\eta_{t}$ are multiplicatively independent, we conclude
from Proposition 2.5 that $\eta:\mathbb{C}^{m}\to\mathbb{P}^{t}$ is
algebraically non-degenerate. It then follows from
$\pi_{3}(V_{f,g,\eta})\supseteq\eta(\mathbb{C}^{m})$ that
$\pi_{3}(V_{f,g,\eta})=\mathbb{P}^{t}.$ Thus

$t=\dim\pi_{3}(V_{f,g,\eta})\leq\dim V_{f,g,\eta}.$

Now it follows from the above inequalities that

$\max\\{t,\dim V_{f\times g}\\}\leq N-s+t.$

This proves Theorem 5.5. ∎

With the help of Lemma 2.10 (i), we shall give the following corollary of
Theorem 5.5.

######  Corollary 5.13.

$V_{f\times g}$ is always of dimension $\leq N.$

###### Proof.

We consider the following two cases separately.

$\bullet$ $t\geq N-n+1.$

In this case, we have $N-n+1\leq t\leq N.$ If the $q$-tuple
$([h_{1}],\dots,[h_{q}])$ has the property $(P_{2t-(N-n),t}),$ then Lemma 2.10
(i) (or Lemma 2.12) implies that $t\leq t-1,$ which is a contradiction. Thus
the $q$-tuple $([h_{1}],\dots,[h_{q}])$ does not have the property
$(P_{2t-(N-n),t}).$ Then by putting $s=t$ in Theorem 5.5, we get

$\dim V_{f\times g}\leq N.$

Corollary 5.13 is thus proved in this case.

$\bullet$ $t\leq N-n.$

Assume $\dim V_{f\times g}>N.$

We claim the $q$-tuple $([h_{1}],\dots,[h_{q}])$ has the property
$(P_{N-n+2,N-n+1}).$ Indeed, if $([h_{1}],\dots,[h_{q}])$ does not have the
property $(P_{N-n+2,N-n+1}),$ noticing that $N-n+2=2(N-n+1)-(N-n),$ we can put
$s=N-n+1$ in Theorem 5.5 and conclude

$\dim V_{f\times g}\leq N-(N-n+1)+t\leq(n-1)+(N-n)=N-1,$

which contradicts our assumption.

Now applying Lemma 2.7 and noticing that $q-(N-n+2)+2\geq 2n+2,$ we know that
there are $(2n+2)$ distinct indices $i_{1},\dots,i_{2n+2}\in\\{1,\dots,q\\}$
such that

$[h_{i_{1}}]=[h_{i_{2}}]=\dots=[h_{i_{2n+2}}].$

By choosing a new reduced representation of $f,$ we assume that
$h_{i_{1}}=:c_{1}$, $h_{i_{2}}=:c_{2}$, $\dots$, $h_{i_{n+1}}=:c_{n+1}$ are
all constants. Then we see easily that there exist a projective linear
transformation $A$ of $\mathbb{P}^{n}$ and a projective linear transformation
$B$ of $\mathbb{P}^{N}$ such that the image of $A(f)\times B(g)$ is contained
in

$\big{\\{}([x_{0}:\dots:x_{n}],[y_{0}:\dots:y_{N}])\in\mathbb{P}^{n}\times\mathbb{P}^{N}\,|\,x_{k}y_{0}-x_{0}y_{k}=0,\,k=1,\dots,n\big{\\}}$

which is an algebraic set in $\mathbb{P}^{n}\times\mathbb{P}^{N}$ of dimension
$N.$ This implies

$\dim V_{f\times g}\leq N,$

which contradicts our assumption.

Hence we also get $\dim V_{f\times g}\leq N$ in this case. ∎

##  6 The equality of dimensions of target spaces when $g$ is algebraically
non-degenerate

We shall prove in this section the following theorem.

######  Theorem 6.1.

Let $N\geq n\geq 1$ be integers. Let $\\{H_{j}\\}_{j=1}^{q}$ be hyperplanes in
$\mathbb{P}^{n}$ in general position and $\\{H^{\prime}_{j}\\}_{j=1}^{q}$ be
hyperplanes in $\mathbb{P}^{N}$ in general position with $q\geq N+n+2.$ Let
$f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ and $g:\mathbb{C}^{m}\to\mathbb{P}^{N}$ be
meromorphic maps such that $f(\mathbb{C}^{m})\not\subseteq H_{j}$,
$g(\mathbb{C}^{m})\not\subseteq H^{\prime}_{j}$ and
$f^{*}(H_{j})=g^{*}(H^{\prime}_{j})$ for any $1\leq j\leq q.$ If $g$ is
algebraically non-degenerate, then $N=n.$

As a matter of fact, we shall prove the following stronger result.

######  Proposition 6.2.

Let $N$, $n$, $\\{H_{j}\\}_{j=1}^{q}$, $\\{H^{\prime}_{j}\\}_{j=1}^{q}$, $f$
and $g$ be as in Theorem 6.1. Let $V_{f\times g}$ be as in Definition 5.1. If
$g$ is linearly non-degenerate and $\dim V_{f\times g}\geq N,$ then $N=n.$

The following proposition shows Proposition 6.2 implies Theorem 6.1.

######  Proposition 6.3.

Let $f:\mathbb{C}^{m}\to\mathbb{P}^{n}$ and
$g:\mathbb{C}^{m}\to\mathbb{P}^{N}$ be meromorphic maps and let $V_{f\times
g}$ be as in Definition 5.1. If $g$ is algebraically non-degenerate, then
$\dim V_{f\times g}\geq N.$

###### Proof.

Let $\pi_{2}:\mathbb{P}^{n}\times\mathbb{P}^{N}\to\mathbb{P}^{N}$ be the
projection onto the second component. Since $\pi_{2}(V_{f\times g})$ contains
the image of $g,$ we conclude from the algebraic non-degeneracy of $g$ that
$\pi_{2}(V_{f\times g})=\mathbb{P}^{N}.$ Thus

$\dim V_{f\times g}\geq\dim\pi_{2}(V_{f\times g})=N,$

which proves Proposition 6.3. ∎

###### Proof of Proposition 6.2.

Let $(f_{0},\dots,f_{n})$, $(g_{0},\dots,g_{N})$ and
$a^{j}_{0}X_{0}+\dots+a^{j}_{n}X_{n}$, $b^{j}_{0}Y_{0}+\dots+b^{j}_{N}Y_{N}$
($1\leq j\leq q$) be as in Section 5. Define $h_{i}$ as (3.2) for each $1\leq
i\leq q.$ Then each $h_{i}$ is a nowhere zero holomorphic function on
$\mathbb{C}^{m}.$ By choosing a new reduced representation of $f,$ we assume
at least one $h_{i}$ is constant. As in the proof of Theorem 3.1, the
$q$-tuple $([h_{1}],\dots,[h_{q}])$ has the property $(P_{N+n+2,N+1}).$ It
then follows from Lemma 2.10 (i) (or Lemma 2.12) that

${\rm rank}\big{\\{}[h_{1}],\dots,[h_{q}]\big{\\}}=:t\leq N.$

To show $N=n,$ we consider the following two cases separately.

$\bullet$ $t<N-n.$

We claim the $q$-tuple $([h_{1}],\dots,[h_{q}])$ has the property
$(P_{N-n+2,N-n+1}).$ Indeed, if $([h_{1}],\dots,[h_{q}])$ does not have the
property $(P_{N-n+2,N-n+1}),$ noticing that $N-n+2=2(N-n+1)-(N-n),$ we can put
$s=N-n+1$ in Theorem 5.5 and conclude

$\dim V_{f\times g}\leq N-(N-n+1)+t<(n-1)+(N-n)=N-1,$

which contradicts the assumption that $\dim V_{f\times g}\geq N.$

Now applying Lemma 2.7 and noticing that $q-(N-n+2)+2\geq 2n+2,$ we know that
there are $(2n+2)$ distinct indices $i_{1},\dots,i_{2n+2}\in\\{1,\dots,q\\}$
such that

$[h_{i_{1}}]=[h_{i_{2}}]=\dots=[h_{i_{2n+2}}].$

Since $g$ is assumed to be linearly non-degenerate, we conclude by using the
same method as in the proof of Theorem 3.1 that $N=n.$

$\bullet$ $t\geq N-n.$

In this case, we have $N-n+1\leq t+1\leq N+1.$ If the $q$-tuple
$([h_{1}],\dots,[h_{q}])$ does not have the property $(P_{2(t+1)-(N-n),t+1}),$
then by putting $s=t+1$ in Theorem 5.5 we get

$\dim V_{f\times g}\leq N-(t+1)+t=N-1,$

which contradicts the assumption that $\dim V_{f\times g}\geq N.$ Therefore
the $q$-tuple $([h_{1}],\dots,[h_{q}])$ necessarily has the property
$(P_{2(t+1)-(N-n),t+1}).$

Now applying the conclusion (ii) of Lemma 2.10 (or Lemma 2.12), we get

$2(t+1)-(N-n)=2(t+1),$

which implies $N=n.$

The proof of Proposition 6.2 is now completed. ∎

We guess Theorem 6.1 is optimal in the sense that the number “$N+n+2$” cannot
be replaced by “$N+n+1$”. When $n=1,$ this is true as is showed in the
following example.

######  Example 6.4.

Let $N\geq 2$ be any integer. Let $G_{1},\dots,G_{N}$ be holomorphic functions
on $\mathbb{C}$ which are given by $G_{j}(z)=z^{j}$, $1\leq j\leq N,$
respectively. Let $a_{1},\dots,a_{N}$ be distinct nonzero complex numbers.
Using the Borel Lemma, we know that $1+e^{G_{1}}+\dots+e^{G_{N}}$ and
$1+a_{1}e^{G_{1}}+\dots+a_{N}e^{G_{N}}$ are both nonzero holomorphic functions
on $\mathbb{C}.$ Take a nonzero holomorphic function $F$ on $\mathbb{C}$ such
that

$f_{0}:=\frac{1+e^{G_{1}}+\dots+e^{G_{N}}}{F}\quad\mbox{and}\quad
f_{1}:=\frac{1+a_{1}e^{G_{1}}+\dots+a_{N}e^{G_{N}}}{F}$

are holomorphic functions on $\mathbb{C}$ without common zeros. Using the
Borel Lemma, we see the meromorphic function $f:=f_{1}/f_{0}$ is nonconstant.
Let $g:\mathbb{C}\to\mathbb{P}^{N}$ be the holomorphic map that has a reduced
representation $(g_{0},\dots,g_{N}),$ where

$g_{0}=-f_{0}\quad\mbox{and}\quad g_{j}=e^{G_{j}}(f_{1}-a_{j}f_{0}),\quad
1\leq j\leq N.$

Then $g$ is algebraically non-degenerate (see Appendix B). We also note that
$g_{0}+\dots+g_{N}=-f_{1}.$ Let $a_{N+1}=\infty$ and $a_{N+2}=0$ be points in
the Riemann sphere $\mathbb{C}\cup\\{\infty\\}.$ Let
$H^{\prime}_{1}=\\{Y_{1}=0\\}$, $H^{\prime}_{2}=\\{Y_{2}=0\\}$, $\dots$,
$H^{\prime}_{N}=\\{Y_{N}=0\\}$, $H^{\prime}_{N+1}=\\{Y_{0}=0\\}$,
$H^{\prime}_{N+2}=\\{Y_{0}+\cdots+Y_{N}=0\\}$ be hyperplanes in
$\mathbb{P}^{N}.$ Clearly, $H^{\prime}_{1},\dots,H^{\prime}_{N+2}$ are in
general position. The holomorphic map $g:\mathbb{C}\to\mathbb{P}^{N}$ and the
meromorphic function $f$ on $\mathbb{C}$ satisfy the condition that

$f^{*}(a_{j})=g^{*}(H^{\prime}_{j}),\quad 1\leq j\leq N+2.$

##  7 Open questions and further remarks

In view of Theorem 6.1, the task of extending Fujimoto’s Theorem 1.7 becomes
answering the following question.

######  Question 7.1.

Let $\\{H_{j}\\}_{j=1}^{2n+3}$ and $\\{H^{\prime}_{j}\\}_{j=1}^{2n+3}$ be two
families of hyperplanes in $\mathbb{P}^{n}$ and each of them is in general
position. Let $f$ and $g$ be meromorphic maps of $\mathbb{C}^{m}$ into
$\mathbb{P}^{n}$ such that $f(\mathbb{C}^{m})\not\subseteq H_{j}$,
$g(\mathbb{C}^{m})\not\subseteq H^{\prime}_{j}$ and
$f^{*}(H_{j})=g^{*}(H^{\prime}_{j})$ for any $1\leq j\leq 2n+3.$ Suppose that
$g$ is algebraically non-degenerate. Is there a projective linear
transformation $L$ of $\mathbb{P}^{n}$ such that $f=L(g)$ and $L$ satisfies
certain condition which forces $L$ to be the identity map when
$\\{H_{j}\\}_{j=1}^{2n+3}=\\{H^{\prime}_{j}\\}_{j=1}^{2n+3}$?

Consider the case where the number of hyperplanes is $(2n+2)$ in each family
and assume $g$ is algebraically non-degenerate. Let notations be as in Section
5. By making use of Theorem 5.5, we see that the $(2n+2)$-tuple
$([h_{1}],\dots,[h_{2n+2}])$ has the property $(P_{2t+2,t+1}).$ Then using
Lemma 2.10, we conclude that the $[h_{i}]$ are represented, after a suitable
change of indices, as one of the following two cases:

  * (a)

$[h_{1}]:[h_{2}]:\dots:[h_{2n+2}]=1:1:\dots:1:1:\beta_{1}:\beta_{1}:\dots:\beta_{t}:\beta_{t};$

  * (b)

$[h_{1}]:[h_{2}]:\dots:[h_{2n+2}]=1:1:\dots:1:\beta_{1}:\dots:\beta_{t}:(\beta_{1}\cdots\beta_{a_{1}})^{-1}:(\beta_{a_{1}+1}\cdots\beta_{a_{2}})^{-1}:\dots:(\beta_{a_{k-1}+1}\cdots\beta_{a_{k}})^{-1},$

where $0\leq k\leq t$, $1\leq a_{1}<a_{2}<\dots<a_{k}\leq t.$

When $\\{H_{j}\\}=\\{H^{\prime}_{j}\\},$ Fujimoto shows that if case (a)
occurs then $f=g$ (see [7]), and if case (b) occurs then $a_{k}=t$ (see [7,
8]). Using the property $a_{k}=t$ in case (b), Fujimoto proves Theorem 1.7.
However, when $\\{H_{j}\\}\neq\\{H^{\prime}_{j}\\},$ it seems that essential
difficulties exist in the study of both case (a) and case (b).

We note that Question 7.1 has an affirmative answer when $n=1$ or $n=2.$ This
can be seen by putting $N=n=1$ and $q=5$ in Theorem I, and putting $N=n=2$ and
$q=7$ in Theorem II.

We guess the number “$N+n+2$” in Theorem 6.1 cannot be replaced by “$N+n+1$”.
So we ask the following question.

######  Question 7.2.

For any positive integers $N$ and $n$ with $N>n,$ are there an algebraically
non-degenerate holomorphic map $g:\mathbb{C}\to\mathbb{P}^{N},$ a linearly
non-degenerate holomorphic map $f:\mathbb{C}\to\mathbb{P}^{n},$ hyperplanes
$\\{H^{\prime}_{j}\\}_{j=1}^{N+n+1}$ in $\mathbb{P}^{N}$ in general position,
and hyperplanes $\\{H_{j}\\}_{j=1}^{N+n+1}$ in $\mathbb{P}^{n}$ in general
position such that $f^{*}(H_{j})=g^{*}(H^{\prime}_{j})$ for any $1\leq j\leq
N+n+1$?

We note that Example 6.4 gives an affirmative answer for Question 7.2 in the
case of $n=1.$

## Appendix A Proof of Lemmas 2.7 and 2.9

Let $(G,\cdot)$ be a torsion-free abelian group and let
$A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple of elements in $G$ that has
the property $(P_{r,s}),$ where $q\geq r>s\geq 1.$ Since $G$ is torsion-free,
the subgroup $\langle\alpha_{1},\dots,\alpha_{q}\rangle$ is free. We take a
basis $\\{\eta_{1},\dots,\eta_{t}\\}$ of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle.$ Then every $\alpha_{i}$ ($1\leq
i\leq q$) is uniquely represented as follows:

$\alpha_{i}=\eta_{1}^{l(i,1)}\cdot\eta_{2}^{l(i,2)}\cdots\eta_{t}^{l(i,t)},$
(A.1)

where $l(i,1),\dots,l(i,t)$ are integers.

Take integers $p_{1},\dots,p_{t}$ such that the integers $l_{i}$ ($1\leq i\leq
q$), which are defined as follows:

$l_{i}:=l(i,1)\cdot p_{1}+\dots+l(i,t)\cdot p_{t},$

satisfy the condition that, for any indices
$i_{1},\dots,i_{k},j_{1},\dots,j_{k}\in\\{1,\dots,q\\}$ with $1\leq k\leq s,$

$l_{i_{1}}+\dots+l_{i_{k}}=l_{j_{1}}+\dots+l_{j_{k}}\quad\mbox{if and only
if}\quad\alpha_{i_{1}}\cdot\cdots\cdot\alpha_{i_{k}}=\alpha_{j_{1}}\cdot\cdots\cdot\alpha_{j_{k}}.$
(A.2)

In fact, we may put $p_{1}=1,$ and once $p_{1},\dots,p_{u-1}$ are chosen, we
then put

$p_{u}=2s\sum_{\tau=1}^{u-1}\Big{(}\max_{1\leq i\leq q}|l(i,\tau)|\cdot
p_{\tau}\Big{)}+1.$

For such chosen $p_{1},\dots,p_{t},$ one checks easily that the integers
$l_{i}$ defined as above satisfy the condition that, for any
$i_{1},\dots,i_{k},j_{1},\dots,j_{k}\in\\{1,\dots,q\\}$ with $1\leq k\leq s,$

$l_{i_{1}}+\dots+l_{i_{k}}=l_{j_{1}}+\dots+l_{j_{k}}$

if and only if

$l(i_{1},\tau)+\dots+l(i_{k},\tau)=l(j_{1},\tau)+\dots+l(j_{k},\tau)\quad\forall\,1\leq\tau\leq
t.$

By (A.1), the latter statement is equivalent to

$\alpha_{i_{1}}\cdot\cdots\cdot\alpha_{i_{k}}=\alpha_{j_{1}}\cdot\cdots\cdot\alpha_{j_{k}}.$

So the integers $l_{i}$ ($1\leq i\leq q$) satisfy the condition (A.2).

In particular, we know that $l_{i}=l_{j}$ if and only if
$\alpha_{i}=\alpha_{j}$ for any $i,j\in\\{1,\dots,q\\}.$ We also know that,
for any $i_{1},\dots,i_{s},j_{1},\dots,j_{s}\in\\{1,\dots,q\\},$

$l_{i_{1}}+\dots+l_{i_{s}}=l_{j_{1}}+\dots+l_{j_{s}}\quad\mbox{if and only
if}\quad\alpha_{i_{1}}\cdot\cdots\cdot\alpha_{i_{s}}=\alpha_{j_{1}}\cdot\cdots\cdot\alpha_{j_{s}}.$

Changing the indices if necessary, we may assume that

$l_{1}\leq l_{2}\leq\cdots\leq l_{q}.$

Choose the following $r$ elements among the $\alpha_{i}$’s:

$\alpha_{1},\alpha_{2},\dots,\alpha_{s},\alpha_{q-(r-s)+1},\alpha_{q-(r-s)+2},\dots,\alpha_{q}.$

Assume that $l_{s}<l_{q-(r-s)+1}.$ Consider the $s$ elements
$\alpha_{1},\alpha_{2},\dots,\alpha_{s}.$ By assumption, there are indices
$j_{1},\dots,j_{s}\in\\{1,\dots,s,q-(r-s)+1,\dots,q\\}$ with
$j_{1}<j_{2}<\dots<j_{s}$ and $\\{j_{1},\dots,j_{s}\\}\neq\\{1,\dots,s\\},$
such that

$\alpha_{1}\cdot\alpha_{2}\cdots\alpha_{s}=\alpha_{j_{1}}\cdot\alpha_{j_{2}}\cdots\alpha_{j_{s}},$

which implies

$l_{1}+l_{2}+\dots+l_{s}=l_{j_{1}}+l_{j_{2}}+\dots+l_{j_{s}}.$ (A.3)

Since $j_{s}\geq q-(r-s)+1,$ we know that

$l_{j_{s}}\geq l_{q-(r-s)+1}>l_{s}.$

We also see that $l_{j_{1}}\geq l_{1}$, $l_{j_{2}}\geq l_{2}$, $\dots$,
$l_{j_{s-1}}\geq l_{s-1}.$ Thus we have

$l_{j_{1}}+\dots+l_{j_{s-1}}+l_{j_{s}}>l_{1}+\dots+l_{s-1}+l_{s},$

which contradicts the equality (A.3). Hence $l_{s}=l_{q-(r-s)+1}.$ Then we get

$l_{s}=l_{s+1}=\dots=l_{q-(r-s)+1},$

which implies

$\alpha_{s}=\alpha_{s+1}=\dots=\alpha_{q-(r-s)+1}.$

Similarly, by considering the following $r$ elements:

$\alpha_{1},\alpha_{2},\dots,\alpha_{r-s},\alpha_{q-s+1},\alpha_{q-s+2},\dots,\alpha_{q},$

we conclude that

$l_{r-s}=l_{r-s+1}=\dots=l_{q-s+1}.$

Now we assume that $r\leq 2s.$

If $q\geq 2s-1,$ which means $q-s+1\geq s,$ then

$l_{r-s}=l_{r-s+1}=\dots=l_{q-(r-s)+1}$

which implies

$\alpha_{r-s}=\alpha_{r-s+1}=\dots=\alpha_{q-(r-s)+1}.$

If $q<2s-1,$ which means $q-s+1<s,$ then

$\alpha_{r-s}=\alpha_{r-s+1}=\dots=\alpha_{q-s+1}\quad\mbox{and}\quad\alpha_{s}=\alpha_{s+1}=\dots=\alpha_{q-(r-s)+1}.$

This completes the proof of Lemma 2.7.

Assume further that $q\geq 2s-1$ and $r-s\geq 2.$ We now know that

$l_{1}\leq\dots\leq l_{r-s-1}\leq l_{r-s}=l_{r-s+1}=\dots=l_{q-(r-s)+1}\leq
l_{q-(r-s)+2}\leq\dots\leq l_{q}.$

If $l_{r-s-1}=l_{r-s}$ or $l_{q-(r-s)+1}=l_{q-(r-s)+2},$ which implies

$\alpha_{r-s-1}=\alpha_{r-s}\quad\mbox{or}\quad\alpha_{q-(r-s)+1}=\alpha_{q-(r-s)+2},$

then the case ($\alpha$) of Lemma 2.9 occurs.

So we assume $l_{r-s-1}<l_{r-s}$ and $l_{q-(r-s)+1}<l_{q-(r-s)+2}$ in the
following. We now have

$l_{1}\leq\dots\leq
l_{r-s-1}<l_{r-s}=l_{r-s+1}=\dots=l_{q-(r-s)+1}<l_{q-(r-s)+2}\leq\dots\leq
l_{q}.$

Choose the following $r$ elements among $\alpha_{i}$’s:

$\alpha_{1},\dots,\alpha_{r-s},\alpha_{r-s+1},\dots,\alpha_{s},\alpha_{q-(r-s)+1},\dots,\alpha_{q}.$

We note that $s-(r-s)=2s-r.$

Consider the $s$ elements
$\alpha_{i_{1}},\alpha_{i_{2}},\dots,\alpha_{i_{s}},$ where

$i_{1}=1,\dots,i_{r-s-2}=r-s-2,i_{r-s-1}=r-s,\dots,i_{s-1}=s,i_{s}=q-(r-s)+1.$

By assumption, there are indices
$j_{1},\dots,j_{s}\in\\{1,\dots,s,q-(r-s)+1,\dots,q\\}$ with
$j_{1}<j_{2}<\dots<j_{s}$ and
$\\{j_{1},\dots,j_{s}\\}=:J\neq\\{i_{1},\dots,i_{s}\\},$ such that

$\alpha_{i_{1}}\cdot\alpha_{i_{2}}\cdots\alpha_{i_{s}}=\alpha_{j_{1}}\cdot\alpha_{j_{2}}\cdots\alpha_{j_{s}},$

which implies

$l_{i_{1}}+l_{i_{2}}+\dots+l_{i_{s}}=l_{j_{1}}+l_{j_{2}}+\dots+l_{j_{s}}.$
(A.4)

First of all, we see that $r-s-1\in J.$ For, if $r-s-1\not\in J,$ then by
$l_{q-(r-s)+1}<l_{q-(r-s)+2}$ we see the right hand side of (A.4) is strictly
greater than the left hand side of (A.4), which gives a contradiction.

We assume $r-s-1=j_{\kappa},$ where $1\leq\kappa\leq s.$ It is obvious that
$1\leq\kappa\leq r-s-1.$ Next, we consider the following two cases.

$\bullet$ $\kappa<r-s-1.$ (This is possible only when $r-s\geq 3,$ so we
exclude this case when $r-s=2.$)

Since $l_{j_{u}}\geq l_{r-s}\geq l_{i_{u}}$ for any $\kappa+1\leq u\leq s,$
and $l_{j_{u}}\geq l_{i_{u}}$ for any $1\leq u\leq\kappa,$ we conclude from
(A.4) that $l_{j_{u}}=l_{i_{u}}$ for any $1\leq u\leq s.$ In particular,
putting $u=\kappa,$ we get $l_{r-s-1}=l_{\kappa}.$ It follows that
$l_{r-s-1}=l_{r-s-2},$ which implies

$\alpha_{r-s-2}=\alpha_{r-s-1}.$

This shows the case ($\beta$) of Lemma 2.9 occurs.

$\bullet$ $\kappa=r-s-1.$

In this case, we necessarily have $j_{u}=u$ for any $1\leq u\leq r-s-2.$ By

$l_{j_{r-s-1}}(=l_{r-s-1})<l_{i_{r-s-1}}(=l_{r-s}),$

we conclude from (A.4) that

$\\{j_{r-s},\dots,j_{s}\\}\cap\\{q-(r-s)+2,\dots,q\\}\neq\emptyset.$

We assume that
$\\{j_{r-s},\dots,j_{s}\\}\cap\\{q-(r-s)+2,\dots,q\\}=\\{u_{1},\dots,u_{d}\\}.$
One sees easily that

$1\leq d\leq\min\\{2s-r+1,r-s-1\\}.$

By cancelling some terms in the both sides of (A.4) suitably, we get

$l_{r-s}+d\cdot l_{r-s}=l_{r-s-1}+l_{u_{1}}+\dots+l_{u_{d}}.$

Since the $l_{i}$ have the property (A.2), we obtain

$\alpha_{r-s-1}\cdot\alpha_{u_{1}}\cdots\alpha_{u_{d}}=\alpha_{r-s}^{d+1},$

which shows the case ($\gamma$) of Lemma 2.9 occurs.

The proof of Lemma 2.9 is now completed.

## Appendix B The algebraic non-degeneracy of $g$ in Example 6.4

We shall freely use the standard notations and results from Nevanlinna theory.
For details of Nevanlinna theory, the reader may refer to books [14] and [11].

We need the following lemma.

######  Lemma B.1.

Let $f$ and $a_{0},a_{1},\dots,a_{n}$ be meromorphic functions on
$\mathbb{C}.$ Then

$T\big{(}r,a_{n}f^{n}+a_{n-1}f^{n-1}+\dots+a_{1}f+a_{0}\big{)}\leq
nT(r,f)+O\Big{(}\sum_{i=0}^{n}T(r,a_{i})\Big{)}+O(1).$

###### Proof.

The proof is easy by using the induction on $n.$ The details are omitted here.
∎

We have the following result.

######  Proposition B.2.

For any integer $n\geq 1,$ the holomorphic map $F_{n}$ of $\mathbb{C}$ into
$\mathbb{P}^{n}$ which is defined as

$F_{n}(z)=[1:e^{z}:e^{z^{2}}:\dots:e^{z^{n}}],$

is algebraically non-degenerate.

###### Proof.

With the help of Lemma B.1, this proposition can be easily proved by the
induction on $n.$ The details are also omitted here. ∎

Now we are going to show that the holomorphic map $g$ in Example 6.4 is
algebraically non-degenerate. Let notations be as in Example 6.4.

Put

$\hat{g}_{j}:=g_{j},\quad 0\leq j\leq
N-1,\quad\mbox{and}\quad\hat{g}_{N}:=(1-a_{N})g_{0}+g_{1}+\dots+g_{N}.$

By $g_{0}+\dots+g_{N}=-f_{1},$ we get $\hat{g}_{N}=a_{N}f_{0}-f_{1}.$ From the
expressions of $f_{0}$ and $f_{1},$ we get the following expressions of the
functions $F\hat{g}_{j}\,(0\leq j\leq N)$:

$\displaystyle F\hat{g}_{0}=-\big{(}1+e^{G_{1}}+\dots+e^{G_{N}}\big{)},$
$\displaystyle F\hat{g}_{j}=e^{G_{j}}\Big{[}(1-a_{j})+\sum_{i=1,i\neq
j}^{N-1}(a_{i}-a_{j})e^{G_{i}}+(a_{N}-a_{j})e^{G_{N}}\Big{]},\quad 1\leq j\leq
N-1,$ $\displaystyle
F\hat{g}_{N}=(a_{N}-1)+\sum_{i=1}^{N-1}(a_{N}-a_{i})e^{G_{i}}.$

Assume $g$ is not algebraically non-degenerate. Then we see easily that there
is a homogeneous polynomial $Q(X_{0},\dots,X_{N})$ of degree $d\geq 1$ such
that

$Q(F\hat{g}_{0},F\hat{g}_{1},\dots,F\hat{g}_{N})\equiv 0.$

Noticing that $F\hat{g}_{N}\not\equiv 0,$ we conclude that the above equation
can be written in the following form:

$\displaystyle P(1,e^{G_{1}}$
$\displaystyle,\dots,e^{G_{N-1}})\cdot(e^{G_{N}})^{d-k}\cdot(F\hat{g}_{N})^{k}$
$\displaystyle+P_{d-k-1}(e^{G_{1}},\dots,e^{G_{N-1}})\cdot(e^{G_{N}})^{d-k-1}+\cdots$
$\displaystyle\hskip 20.00003pt+P_{1}(e^{G_{1}},\dots,e^{G_{N-1}})\cdot
e^{G_{N}}+P_{0}(e^{G_{1}},\dots,e^{G_{N-1}})\equiv 0,$ (B.3)

where $k$ is an integer with $0\leq k\leq d-1,$ and
$P(X_{0},X_{1},\dots,X_{N-1})$ is a nonzero homogeneous polynomial of degree
$(d-k),$ and $P_{d-k-1},\dots,P_{0}$ are polynomials.

By Proposition B.2, we know $P(1,e^{G_{1}},\dots,e^{G_{N-1}})\not\equiv 0.$ We
note that $T(r,e^{G_{i}})=o\big{(}T(r,e^{G_{N}})\big{)}$ for any $1\leq i\leq
N-1.$ So applying Lemma B.1, we get from (Appendix B The algebraic non-
degeneracy of $g$ in Example 6.4) that

$(d-k)T(r,e^{G_{N}})+O(1)\leq(d-k-1)T(r,e^{G_{N}})+o\big{(}T(r,e^{G_{N}})\big{)}.$

This implies

$d-k\leq d-k-1,$

which is a contradiction.

Thus $g$ is algebraically non-degenerate.

## References

  * [1] Yoshihiro Aihara.  Unicity theorems for meromorphic mappings with deficiencies.  Complex Variables, 42(3):259–268, 2000.
  * [2] Emile Borel.  Sur les zéros des fonctions entières.  Acta Math., 20:357–396, 1897.
  * [3] Zhihua Chen and Qiming Yan.  Uniqueness theorem of meromorphic mappings into $\mathbb{P}^{N}(\mathbb{C})$ sharing $2N+3$ hyperplanes regardless of multiplicities.  Internat. J. Math., 20(6):717–726, 2009.
  * [4] Pietro Corvaja and Junjiro Noguchi.  A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties.  Math. Ann., 353(2):439–464, 2012.
  * [5] S. J. Drouilhet.  A unicity theorem for meromorphic mappings between algebraic varieties.  Trans. Amer. Math. Soc., 265(2):349–358, 1981.
  * [6] Hirotaka Fujimoto.  The uniqueness problem of meromorphic maps into the complex projective space.  Nagoya Math. J., 58:1–23, 1975.
  * [7] Hirotaka Fujimoto.  A uniqueness theorem of algebraically non-degenerate meromorphic maps into $P^{N}(\mathbf{C})$.  Nagoya Math. J., 64:117–147, 1976.
  * [8] Hirotaka Fujimoto.  Remarks to the uniqueness problem of meromorphic maps into $P^{N}(\mathbf{C})$, I.  Nagoya Math. J., 71:13–24, 1978.
  * [9] Hirotaka Fujimoto.  Uniqueness problem with truncated multiplicities in value distribution theory, II.  Nagoya Math. J., 155:161–188, 1999.
  * [10] Rolf Nevanlinna.  Einige Eindeutigkeitssätze in der Theorie der meromorphen Funktionen.  Acta Math., 48(3-4):367–391, 1926.
  * [11] Junjiro Noguchi and Jörg Winkelmann.  Nevanlinna Theory in Several Complex Variables and Diophantine Approximation.  Springer, Tokyo, 2014.
  * [12] George Pólya.  Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen.  Mat. Tidsskrift B, pages 16–21, 1921.  https://zbmath.org/?q=an%3A48.0354.03.
  * [13] Si Duc Quang.  Unicity of meromorphic mappings sharing few hyperplanes.  Ann. Polon. Math., 102(3):255–270, 2011.
  * [14] Min Ru.  Nevanlinna Theory and Its Relation to Diophantine Approximation.  World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  * [15] Leonard M. Smiley.  Geometric conditions for unicity of holomorphic curves.  Contemp. Math., 25:149–154, 1983.
  * [16] Katsutoshi Yamanoi.  Holomorphic curves in Abelian varieties and intersections with higher codimensional subvarieties.  Forum Math., 16(5):749–788, 2004.
  * [17] Kai Zhou.  A note on Fujimoto’s uniqueness theorem with $(2n+3)$ hyperplanes.  2023\.  arXiv:2307.16595 [math.CV].

School of Mathematical Sciences, Tongji University, Shanghai 200092, People’s
Republic of China

E-mail address<EMAIL_ADDRESS>