2. Some auxiliary results

Let $X\subset{\mathbb{P}}^n({\mathbb{C}})$ be a projective variety of dimension k and degree δ. For $\textbf{a} = (a_0,\ldots,a_n)\in\mathbb Z^{n+1}_{\ge 0}$ we write $\textbf{x}^\textbf{a}$ for the monomial $x^{a_0}_0\cdots x^{a_n}_n$. Let I_X be the prime ideal in ${\mathbb{C}}[x_0,\ldots,x_n]$ defining X. Let ${\mathbb{C}}[x_0,\ldots,x_n]_u$ be the vector space of homogeneous polynomials in ${\mathbb{C}}[x_0,\ldots,x_n]$ of degree u (including 0). For $u= 1, 2,\ldots,$ put $(I_X)_u:={\mathbb{C}}[x_0,\ldots,x_n]_u\cap I_X$ and define the Hilbert function H_X of X by

\begin{align*} H_X(u):=\dim{\mathbb{C}}[x_0,\ldots,x_n]_u/(I_X)_u. \end{align*}

Let $\textbf{c}=(c_0,\ldots,c_n)$ be a tuple in $\mathbb R^{n+1}_{\ge 0}$ and let $e_X(\textbf{c})$ be the Chow weight of X with respect to c. The u-th Hilbert weight $S_X(u,\textbf{c})$ of X with respect to c is defined by

\begin{align*} S_X(u,\textbf{c}):=\max\sum_{i=1}^{H_X(u)}\textbf{a}_i\cdot\textbf{c}, \end{align*}

where the maximum is taken over all sets of monomials $\textbf{x}^{\textbf{a}_1},\ldots,\textbf{x}^{\textbf{a}_{H_X(u)}}$ whose residue classes modulo I_X form a basis of ${\mathbb{C}}[x_0,\ldots,x_n]_u/(I_X)_u.$

The following theorem is due to Evertse and Ferretti [Reference Evertse and Ferretti2].

Theorem 2.1. (see [Reference Evertse and Ferretti2, Theorem 4.1]) Let $X\subset{\mathbb{P}}^n({\mathbb{C}})$ be an algebraic variety of dimension k and degree δ. Let $u \gt \delta$ be an integer and let $\textbf{c}=(c_0,\ldots,c_n)\in\mathbb R^{n+1}_{\geqslant 0}$. Then

\begin{equation*} \dfrac{1}{uH_X(u)}S_X(u,\textbf{c})\ge\dfrac{1}{(k+1)\delta}e_X(\textbf{c})-\dfrac{(2k+1)\delta}{u}\cdot\left (\max_{i=0,\ldots,n}c_i\right).\end{equation*}

The following lemma is due to the Quang [Reference Quang5].

Lemma 2.2. (see [Reference Quang5, Lemma 3.2]) Let Y be a projective subvariety of ${\mathbb{P}}^R({\mathbb{C}})$ of dimension $k\ge 1$ and degree δ_Y. Let $\ell\ (\ell\ge k+1)$ be an integer and let $\textbf{c}=(c_0,\ldots,c_R)$ be a tuple of non-negative reals. Let $\mathcal H=\{H_0,\ldots,H_R\}$ be a set of hyperplanes in ${\mathbb{P}}^R({\mathbb{C}})$ defined by $H_{i}=\{y_{i}=0\}\ (0\le i\le R)$. Let $\{i_1,\ldots, i_\ell\}$ be a subset of $\{0,\ldots,R\}$ such that:

1. (1) $c_{i_\ell}=\min\{c_{i_1},\ldots,c_{i_\ell}\}$,

2. (2) $Y\cap\,\bigcap_{j=1}^{\ell-1}H_{i_j}\ne \emptyset$,

3. (3) and $Y\not\subset H_{i_j}$ for all $j=1,\ldots,\ell$.

Let $\Delta_{\mathcal H,Y}$ be the distributive constant of the family $\mathcal H=\{H_{i_j}\}_{j=1}^\ell$ with respect to Y. Then

\begin{equation*}e_Y(\textbf{c})\ge \frac{\delta_Y}{\Delta_{\mathcal H,Y}}(c_{i_1}+\cdots+c_{i_\ell}).\end{equation*}

The following theorem is due to Ru and Sibony [Reference Ru and Sibony8].

Theorem 2.3. (reformulation of [Reference Ru and Sibony8, Theorem 4.8]) Let f be a linearly non-degenerate holomorphic map from $\Delta (R)\ (0 \lt R\le +\infty)$ into $\mathbb{P}^n(\mathbb{C})$. Let $H_1,\ldots,H_q$ be q arbitrary hyperplanes in ${\mathbb{P}}^n({\mathbb{C}})$. Then, for every ɛ > 0, we have

\begin{align*} \biggl\|\ &\int_0^{2\pi}\max_K\log\sum_{j\in K}\frac{\|{\bf{f}}\|}{|H_j({\bf{f}})|}\frac{d\theta}{2\pi}+N_W(r)\\ &\le (n+1)T_f(r)+\frac{n(n+1)}{2}(c_f+\epsilon)T_f(r), \end{align*}

where $W=\det (f_i^{(k)}; 0\le i,k\le n)$ for a reduced representation $\mathbf f=(f_0,\dots,f_n)$ of f.

Note that, in the original theorem [Reference Ru and Sibony8, Theorem 4.8], the last term of the right hand side of the above inequality is more complex and the inequality holds for all r outside an exceptional set $S\subset(0,R)$ such that $\int_S{\rm exp}((c_f+\epsilon)T_f(r))dr \lt +\infty$. In this reformulation, we just simplify that term but the exceptional set is a subset $S'\subset(0,R)$ such that $\int_{S'}{\rm exp}((c_f+\epsilon')T_f(r))dr \lt +\infty$ for some $\epsilon' \gt 0$.

Let $\mathcal Q=\{Q_1,\ldots,Q_q\}$ be a family moving hypersurfaces in ${\mathbb{P}}^N({\mathbb{C}})$ given by

\begin{equation*} Q_i(z)(\textbf{x})=\sum_{I\in\mathcal T_{d_i}}a_{iI}(z)\textbf{x}^I,\end{equation*}

where $\textbf{x}=(x_0,\ldots,x_N)$, $\textbf{x}^I=x_0^{i_0}\cdots x_N^{i_N}$ for $I=(i_0,\ldots,i_N)$. Denote by $\mathcal C_{\mathcal Q}$ the set of all non-negative functions $h : \mathbf{C}^m\setminus A\longrightarrow [0,+\infty]$, which are of the form:

\begin{equation*} h=\dfrac{|g_1|+\cdots +|g_l|}{|g_{l+1}|+\cdots +|g_{l+k}|}, \end{equation*}

where $k,l\in\mathbf{N},\ g_1,...., g_{l+k}\in\mathcal K_{\mathcal Q}\setminus\{0\}$ and A is a discrete subset of $\Delta(R)$, which may depend on $g_1,....,g_{l+k}$. Then, for $h\in\mathcal C_{\mathcal Q}$ we have

\begin{equation*}\int_{0}^{2\pi}\log^+(h)\frac{d\theta}{2\pi}=O(\max T_{a_{iI}/a_{iJ}}(r)).\end{equation*}

Also, for every moving hypersurface Q in $\mathcal K_{\mathcal Q}[x_0,\ldots,x_N]$ of degree d, we have

\begin{equation*} Q(z)(\textbf{x})\le c(z)\|\textbf{x}\|^d, \end{equation*}

for some $c\in\mathcal C_{\mathcal Q}$.

Lemma 2.4. (see [Reference Quang6, Lemma 3.2]) Let V be a projective variety of ${\mathbb{P}}^N({\mathbb{C}})$. With the above notation, let $1\le j_1\le\cdots\le j_k\le q$. Suppose that there exists $z_0\in\Delta(R)$ such that $V\cap\,\bigcap_{s=1}^kQ_{j_s}(z_0)^*=\emptyset.$ Then we have $V\cap\bigcap_{s=1}^kQ_{j_s}(z)^*=\emptyset$ for every $z\in\Delta(R)$ outside a discrete subset, and there exists a function $c\in\mathcal C_{\mathcal K}$ such that

\begin{equation*} \|\textbf{f}(z)\|\le c(z)\max_{1\le s\le k}\{Q_{j_s}(\textbf{f})(z)\}.\end{equation*}

3. Proof of theorem 1.1

Replacing Q_j by $Q^{\frac{d}{d_{j}}}$ if necessary, we may assume that $Q_{1},\ldots,Q_{q}$ have the same degree d and $Q_i=\sum_{I\in\mathcal T_d}a_{iI}x^I\ (i=1,\ldots,q)$. Take a point z ₀ such that $a_{iI_0}(z_0)\ne 0$ for all i, and

\begin{equation*}\Delta_V=\max_{\emptyset\ne\Gamma\subset\{1,\dots,q\}}\frac{\sharp\Gamma}{\dim V-\dim V\cap\bigcap_{j\in\Gamma}{\widetilde Q}_i(z_0)^\ast}.\end{equation*}

It is suffice for us to consider the case where $q \gt \Delta_V(n+1)$. Denote by $\sigma_1,\ldots,\sigma_{n_0}$ all bijections from $\{0,\ldots,q-1\}$ into $\{1,\ldots,q\}$, where $n_0=q!$. For each σ_i, it is easy to see that $\bigcap_{j=0}^{q-2}\tilde Q_{\sigma_i(j)}(z_0)^*\cap V=\emptyset$. Then there exists a smallest index $\ell_i\le q-2$ such that $\bigcap_{j=0}^{\ell_i}\tilde Q_{\sigma_i(j)}(z_0)^*\cap V=\emptyset$. Hence $\bigcap_{j=0}^{\ell_i}\tilde Q_{\sigma_i(j)}(z)^*\cap V=\emptyset$ for generic points z and for all $i=1,\ldots,n_0$. Denote by $\mathcal S$ the set of all points $z\in\Delta(R)$ such that $\bigcap_{j=0}^{\ell_i}\tilde Q_{\sigma_i(j)}(z)^*\cap V\ne \emptyset$ for some i. Then $\mathcal S$ is a discrete subset of $\Delta(R)$.

By Lemma 2.4, there is a function $A\in\mathcal C_{\mathcal Q}$, chosen common for all σ_i, such that

\begin{equation*} \|{\bf{f}} (z)\|^d\le A(z)\max_{0\le j\le \ell_i}\frac{|\tilde Q_{\sigma_i(j)}({\bf{f}})(z)|}{\|\tilde Q_{\sigma_i(j)}(z)\|}\ \forall i=1,\ldots,n_0.\end{equation*}

Denote by S(i) the set of all z not in $\mathcal S$ such that $\tilde Q_j({\bf{f}})(z)\ne 0$ for all $j=1,\ldots,q$ and

\begin{equation*} \frac{|\tilde Q_{\sigma_i(0)}({\bf{f}})(z)|}{\|\tilde Q_{\sigma_i(0)}(z)\|}\le \frac{|\tilde Q_{\sigma_i(1)}({\bf{f}})(z)|}{\|\tilde Q_{\sigma_i(1)}(z)\|}\le\cdots\le \frac{|\tilde Q_{\sigma_i(q-1)}({\bf{f}})(z)|}{\|\tilde Q_{\sigma_i(q-1)}(z)\|}.\end{equation*}

Therefore, for every generic point $z\in S(i)$, we have

\begin{equation*}\prod_{j=1}^q\dfrac{\|{\bf{f}} (z)\|^d\|\tilde Q_{j}(z)\|}{|\tilde Q_j({\bf{f}})(z)|}\le C(z)\prod_{j=0}^{\ell_j}\dfrac{\|{\bf{f}} (z)\|^d\|\tilde Q_{\sigma_i(j)}(z)\|}{|\tilde Q_{\sigma_i(j)}({\bf{f}})(z)|},\end{equation*}

where $C(z)=\sum_{i=1}^{n_0}A(z)^{q-\ell_i-1}\in\mathcal C_{\mathcal Q}$.

For $z\not\in\mathcal S$, consider the mapping $\Phi_z$ from V into ${\mathbb{P}}^{q-1}({\mathbb{C}})$ defined by

\begin{equation*}\Phi_z(\textbf{x})=(\tilde Q_1(z)(x):\cdots : \tilde Q_{q}(z)(x))\end{equation*}

for every $\textbf{x}=(x_0:\cdots:x_N)\in V$, where $x=(x_0,\ldots,x_N)$. We set

\begin{equation*}\tilde\Phi_z(x)=(\tilde Q_1(z)(x),\ldots ,\tilde Q_{q}(z)(x)).\end{equation*}

Let $Y_z=\Phi_z(V)$. Since $V\cap\bigcap_{j=1}^{q}\tilde Q_j(z)^*=\emptyset$, $\Phi_z$ is a finite morphism on V and Y_z is a projective subvariety of ${\mathbb{P}}^{q-1}({\mathbb{C}})$ with $\dim Y_z=n$ and of degree

\begin{equation*}\delta_z:=\deg Y_z\le d^{n}.\deg V=\delta.\end{equation*}

For every $\textbf{a} = (a_1,\ldots,a_q)\in\mathbb Z^q_{\ge 0}$ and $\textbf{y} = (y_1,\ldots,y_q)$ we denote $\textbf{y}^\textbf{a} = y_{1}^{a_{1}}\ldots y_{q}^{a_{q}}$. Let u be a positive integer. We set $\xi_u:=\binom{q+u}{u}$ and define the ${\mathbb{C}}$-vector space

\begin{equation*} Y_{z,u}:={\mathbb{C}}[y_1,\ldots,y_q]_u/(I_{Y_z})_u.\end{equation*}

Denote by $(I_Y)_u$ the subspace of the $\mathcal K_{\mathcal Q}$-vector space $\mathcal K_{\mathcal Q}[y_1,\ldots,y_q]_u$ consisting of all homogeneous polynomials $P\in \mathcal K_{\mathcal Q}[y_1,\ldots,y_q]_u$ (including the zero polynomial) such that

\begin{equation*}P(z)(\Phi_z({\bf{f}}(z)))\equiv 0.\end{equation*}

Let $(\tilde R_1,\ldots,\tilde R_p)$ be an $\mathcal K_{\mathcal Q}$-basis of $(I_Y)_u$. By enlarging $\mathcal S$ if necessary, we may assume that all zeros and poles of all non-zero coefficients of $\tilde R_i\ (1\le i\le p)$ are contained in $\mathcal S$, also all above assertions for generic points z still hold for all $z\not\in\mathcal S$. Choose $\xi_u-p$ non-zero monic monomial $v_1,\ldots,v_{\xi_u-p}$ of degree of u in variables $y_1,\ldots,y_q$ such that $\{\tilde R_1,\ldots,\tilde R_p,v_1,\ldots,v_{\xi_u-p}\}$ is a $\mathcal K_{\mathcal Q}$-basis of $\mathcal K_{\mathcal Q}[y_1,\ldots,y_q]_u$.

Denote by $\mathcal T=\{T_1,\ldots,T_{\xi_u}\}$ the set of all non-zero monic monomials of degree of u in variables $y_1,\ldots,y_q$. Then $\{T_1,\ldots,T_{\xi_u}\}$ is a $\mathcal K_{\mathcal Q}$-basis of $\mathcal K_{\mathcal Q}[y_1,\ldots,y_q]_u$, and also is an ${\mathbb{C}}$-basis of ${\mathbb{C}}[y_1,\ldots,y_q]_u$.

From [Reference Quang6, Claim 4.3], we have the following claim.

Then, we have $\xi_u-p=H_{Y_z}(u)$ for all z outside $\mathcal S\cup\mathcal S'.$ Now, consider the holomorphic map F from $\Delta(R)$ into ${\mathbb{P}}^{\xi_u-p-1}({\mathbb{C}})$ with the representation

\begin{equation*} {\bf{F}}=(v_1(\tilde\Phi\circ {\bf{f}}),\ldots,v_{\xi_u-p}(\tilde\Phi\circ {\bf{f}})). \end{equation*}

Since f is algebraically non-degenerate over $\mathcal K_{\mathcal Q}$, F is linearly non-degenerate over $\mathcal K_{\mathcal Q}$.

Now, for $z\not\in\mathcal S\cup\mathcal S'$, we set $\textbf{c}_z = (c_{1,z},\ldots,c_{q,z})\in\mathbb Z^{q},$ where

\begin{align*} c_{i,z}:=\log\frac{\|{\bf{f}}(z)\|^d\|\tilde Q_i(z)\|}{|\tilde Q_i({\bf{f}})(z)|}\ge 0,\quad \text{for } i=1,\ldots,q. \end{align*}

By the definition of the Hilbert weight, there are $\textbf{a}_{1,z},\ldots,\textbf{a}_{\xi_u-p,z}\in\mathbb N^{q}$ with

\begin{equation*} \textbf{a}_{i,z}=(a_{i,1,z},\ldots,a_{i,q,z}), \end{equation*}

where $a_{i,j,z}\in\{1,\ldots,\xi_u\},$ such that the residue classes modulo $(I_Y)_u$ of $\textbf{y}^{\textbf{a}_{1,z}},\ldots,$$\textbf{y}^{\textbf{a}_{\xi_u-p,z}}$ form a basic of ${\mathbb{C}}[y_1,\ldots,y_q]_u/(I_{Y_z})_u$ and

\begin{align*} S_Y(u,\textbf{c}_z)=\sum_{i=1}^{\xi_u-p}\textbf{a}_{i,z}\cdot\textbf{c}_z. \end{align*}

Note that $\textbf{y}^{\textbf{a}_{i,z}}\in\mathcal T$ and the set $\{\tilde R_1,\ldots,\tilde R_p,\textbf{y}^{\textbf{a}_{1,z}},\ldots,\textbf{y}^{\textbf{a}_{\xi_u-p,z}}\}$ is a basis of $\mathcal K_{\mathcal Q}[y_1,\ldots,y_q]$ (by Claim 3.1(iii)). Therefore, the set of equivalent classes of $\left\{\textbf{y}^{\textbf{a}_{1,z}},\ldots,\right.$$\left.\textbf{y}^{\textbf{a}_{\xi_u-p,z}}\right\}$ is a basis of $\dfrac{\mathcal K_{\mathcal Q}[y_1,\ldots,y_q]_u}{I(Y)_u}$. Then $ \textbf{y}^{\textbf{a}_{i,z}}=L_{i,z}\left(v_1,\ldots,v_{H_Y(u)}\right)\ \text{modulo}$ $I(Y)_u, $ where $L_{i,z}\ (1\le i\le \xi_u-p)$ are $\mathcal K_{\mathcal Q}$-independent linear forms with coefficients in $\mathcal K_{\mathcal Q}$. We have

\begin{align*} \log\prod_{i=1}^{\xi_u-p} |L_{i,z}({\bf{F}}(z))|&=\log\prod_{i=1}^{\xi_u-p}\prod_{j=1}^q|\tilde Q_j({\bf{f}})(z)|^{a_{i,j,z}}\\ &= -S_Y(u,\textbf{c}_z)+du(\xi_u-p)\log \|{\bf{f}}(z)\| +\log C_1(z), \end{align*}

where $C_1\in\mathcal C_{\mathcal Q}$. Note that the number of these linear forms $L_{i,z}$ is finite, at most ξ_u. Denote by $\mathcal L$ the set of all $L_{i,z}$ occurring in the above inequalities. The above inequality follows that

\begin{align*} \log\prod_{i=1}^{\xi_u-p}\dfrac{\|{\bf{F}}(z)\|\cdot \|L_{i,z}\|}{|L_{i,z}({\bf{F}}(z))|}= &S_Y(u,\textbf{c}_z)-du(\xi_u-p)\log \|{\bf{f}}(z)\| \\ &+(\xi_u-p)\log \|{\bf{F}}(z)\|+\log C_2, \end{align*}

where $C_2(z)\in\mathcal C_{\mathcal Q}$. Then, we have

(3.2)\begin{align} \begin{split} S_Y(u,\textbf{c}_z)\le&\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in \mathcal J}\dfrac{\|{\bf{F}}(z)\|\cdot \|L\|}{|L({\bf{f}}(z))|}+du(\xi_u-p)\log \|{\bf{f}}(z)\|\\ & -(\xi_u-p)\log \|{\bf{F}}(z)\|+\log C_2(z), \end{split} \end{align}

where the maximum is taken over all subsets $\mathcal J\subset\mathcal L$ with $\sharp\mathcal J=\xi_u-p$ and $\{L|L\in\mathcal J\}$ is linearly independent over $\mathcal K$. From Theorem 2.1 we have

(3.3)\begin{align} \dfrac{1}{u(\xi_u-p)}S_{Y_z}(u,\textbf{c}_z)\ge\frac{1}{(n+1)\delta_z}e_{Y_z}(\textbf{c}_z)-\frac{(2n+1)\delta_z}{u}\max_{1\le i\le q}c_{i,z}. \end{align}

Combining (3.2) and (3.3), we get

(3.4)\begin{align} \begin{split} &\frac{1}{(n+1)\delta_z}e_{Y_z}(\textbf{c}_z)\\ &\le\dfrac{1}{u(\xi_u-p)}\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in \mathcal J}\dfrac{\|{\bf{F}}(z)\|\cdot \|L\|}{|L({\bf{F}}(z))|}\\ &\ \ \ \ +\frac{(2n+1)\delta_z}{u}\max_{1\le i\le q}c_{i,z}+\log^+C_3(z)\\ &\le\dfrac{1}{u(\xi_u-p)}\max_{\mathcal J\subset\mathcal L}\prod_{L\in\mathcal J}\dfrac{\|{\bf{F}}(z)\|\cdot \|L\|}{|L({\bf{F}}(z))|}\\ &\ \ \ \ +\frac{(2n+1)\delta_z}{u}\sum_{1\le i\le q}\log\frac{\|{\bf{f}}(z)\|^d\|\tilde Q_i(z)\|}{|\tilde Q_i({\bf{f}})(z)|}+\log^+C_3(z), \end{split} \end{align}

where $C_3(z)\in\mathcal C_{\mathcal Q}$, for every $z\in \Delta (R)$ outside a discrete subset.

Fix a point $z\in\Delta(R)\setminus(\mathcal S\cup \mathcal S')$. Choose $i\in\{1,\ldots,n_0\}$ such that

\begin{equation*} e_{\sigma_i(0),z}\le e_{\sigma_i(1),z}\le\cdots\le e_{\sigma_i(q-1),z}.\end{equation*}

Since $\bigcap_{j=0}^{\ell_i-1}\tilde Q_{\sigma_i(j)}(z)^*\cap V\ne\emptyset$, by Lemma 2.2, we have

(3.5)\begin{align} \begin{split} \Delta_{V}e_{Y_z}(\textbf{c}_z)&\ge (c_{\sigma_i(0),z}+\cdots +c_{\sigma_i(\ell_i),z})\cdot\delta_z\\ &=\delta_z\biggl(\sum_{j=0}^{\ell_i}\log\frac{\|{\bf{f}}(z)\|^d\|\tilde Q_{\sigma_i(j)}(z)\|}{|\tilde Q_{\sigma_i(j)}({\bf{f}})(z)|}\biggl ). \end{split} \end{align}

Then, from (3.2), (3.4) and (3.5) we have

(3.6)\begin{align} \begin{split} \frac{1}{\Delta_{V}}\log &\prod_{i=1}^q\dfrac{\|{\bf{f}} (z)\|^d\|\tilde Q_i(z)\|}{|\tilde Q_i({\bf{f}})(z)|}\\ &\le\dfrac{n+1}{u(\xi_u-p)}\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in\mathcal J}\dfrac{\|{\bf{F}}(z)\|\cdot \|L\|}{|L({\bf{F}}(z))|}\\ &+\frac{(2n+1)(n+1)\delta_z}{u}\sum_{1\le i\le q}\log\frac{\|{\bf{f}}(z)\|^d\|\tilde Q_{i}(z)\|}{|\tilde Q_{i}({\bf{f}})(z)|}\\ &+\frac{1}{\Delta_{V}}\log C(z)+(n+1)\log^+C_3(z), \end{split} \end{align}

for every $z\in \Delta (R)$ outside a discrete subset.

Denote by Ψ the set of all the coefficients of all linear forms $L_{i,z}$ and suppose that $\Psi=\{a_1,\ldots,a_{q_0}\}$. Then, we see that $\sharp\mathcal L\le\xi_u,\sharp\Psi=q_0\le \xi_u(\xi_u-p)$. For each positive integer m, denote by $\mathcal L(\Psi(m))$ the $C-$vector space generated by the set $\{a_1^{i_1}\ldots a_{q_0}^{i_{q_0}}|i_j\ge 0\ \text{and }\sum_{j=1}^{q_0}i_j\le m\}$. By Remark 3.4 in [Reference Thai and Quang10], there exists the smallest integer p ^ʹ such that

\begin{equation*} \frac{\dim\mathcal L(\Psi (p'+1))}{\dim\mathcal L(\Psi (p'))}\le 1+\frac{\epsilon}{2\Delta_{V}(n+1)}.\end{equation*}

Put $s=\dim\mathcal L(\Psi (p')), t=\dim\mathcal L(\Psi (p'+1))$. Again, by [Reference Thai and Quang10, Remark 3.4], we have

\begin{align*} t&\le \left [\bigl(1+\frac{\epsilon}{2(n+1)\Delta_{V}}\bigl)^{\bigl[\frac{\sharp\Psi}{\log^2(1+\frac{\epsilon}{2(n+1)\Delta_{V}})}\bigl]+1}\right ]\\ &\le \left[\bigl(1+\frac{\epsilon}{2(n+1)\Delta_{V}}\bigl)^{\bigl[\frac{d^n\deg V(u+1)^{n+q}}{\log^2(1+\frac{\epsilon}{2(n+1)\Delta_{V}})}\bigl]+1}\right]. \end{align*}

Here, the last inequality comes from the fact that $\xi_u\le (u+1)^q$ and

\begin{equation*}\xi_u-p\le\delta\binom{n+u}{n}\le d^n\deg V\binom{n+u}{n}\le d^n\deg V(u+1)^n.\end{equation*}

Choose $\{b_1, \ldots ,b_s\}$ an ${\mathbb{C}}$-basis of $\mathcal L(\Psi(p'))$ and $\{b_1, \ldots ,b_t\}$ an ${\mathbb{C}}$-basis of $\mathcal L(\Psi(p'+1))$. Consider the holomorphic map $\tilde F:\Delta(R)\rightarrow\mathbb{P}^{t(\xi_u-p)-1}(\mathbb{C})$ with a presentation

\begin{equation*}\tilde{\bf{F}}=(b_1v_1(\tilde\Phi\circ {\bf{f}}),\ldots,b_1v_{\xi_u-p}(\tilde\Phi\circ {\bf{f}}),\ldots, b_tv_1(\tilde\Phi\circ {\bf{f}}),\ldots,b_tv_{\xi_u-p}(\tilde\Phi\circ {\bf{f}})).\end{equation*}

Note that $\|\ T_{\tilde F}(r)=duT_f(r)+o(T_f(r))$ and $c_{\tilde F}=\frac{1}{du}c_f$. By Theorem 2.3, we have

(3.7)\begin{align} \begin{split} &\biggl \|\ s\int_0^{2\pi}\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in\mathcal J}\frac{\|{\bf{F}}\|}{|L({\bf{F}})|}\frac{d\theta}{2\pi}-N_{W(\tilde F)}(r)\\ &\le \int_0^{2\pi}\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in\mathcal J}\prod_{i=1}^s\frac{\|\tilde{\bf{F}}\|}{|b_iL({\bf{F}})|}-N_{W(\tilde F)}(r)+o(T_f(r))\\ &\le t(\xi_u-p)udT_f(r)+\frac{(t(\xi_u-p)-1)t(\xi_u-p)}{2}(c_{\tilde F}+\frac{\epsilon'}{2du})T_{\tilde F}(r), \end{split} \end{align}

where $\max_{\mathcal J\subset\mathcal L}$ is taken over all subsets $\mathcal J$ of the system $\mathcal L$ of linear forms such that $\mathcal J$ is linearly independent over $\mathcal {K}_{\mathcal {Q}}$, $\epsilon'$ is an arbitrary positive number. Then, by integrating (3.6) and using the above inequality, we obtain

\begin{align*} \biggl\|\ &\left(\frac{1}{\Delta_{V}}-\frac{(2n+1)(n+1)\delta}{u}\right)\sum_{i=1}^qm_f(r,Q_i)\\ &\le\frac{d(n+1)t}{s}T_f(r)-\frac{(n+1)}{u(\xi_u-p)s}N_{W(\tilde F)}(r)\\ &+\frac{(t(\xi_u-p)-1)t(n+1)}{2su}\left(c_{\tilde F}+\frac{\epsilon'}{2du}\right)T_{\tilde F}(r). \end{align*}

We now estimate the quantity $N_{W(\tilde F)}(r)$. Let $z\in\Delta(R)$ which is neither zero nor pole of any coefficients of $\tilde Q_i\ (1\le i\le q)$ and $b_i\ (1\le i\le t)$. We set

\begin{equation*}c_{i}=\max\{0,\nu^0_{\tilde Q_i({\bf{f}})}(z)-\xi_u+p+1\}\ (1\le i\le q) \ \text{and } \textbf{c}=(c_{1},\ldots,c_{q})\in\mathbb Z^q_{\ge 0}.\end{equation*}

Then there are

\begin{equation*}\textbf{a}_i=(a_{i,1},\ldots,a_{i,q}),a_{i,s}\in\{1,...,u\}\end{equation*}

such that $\textbf{y}^{\textbf{a}_1},...,\textbf{y}^{\textbf{a}_{H_Y(u)}}$ is a basic of ${\mathbb{C}}[y_1,\ldots,y_q]_u/(I_{Y_z})_u$ and

\begin{equation*} S_{Y_z}(u,\textbf{c})=\sum_{i=1}^{H_Y(u)}\textbf{a}_i\cdot\textbf{c}.\end{equation*}

Similarly as above, we write $\textbf{y}^{\textbf{a}_i}=L_i(v_1,...,v_{\xi_u-p})$, where $L_1,...,L_{\xi_u-p}$ are linearly independent linear forms. We see that

\begin{equation*} \nu^0_{W(\tilde F)}(z)\ge t\sum_{i=1}^{H_Y(u)}\max\{0,\nu^0_{L_i({\bf{F}})}(z)-n_u\},\end{equation*}

where $n_u=(\xi_u-p)t-1$. It is easy to see that

\begin{equation*} \nu^0_{L_i({\bf{F}})}(z)=\sum_{j=1}^qa_{i,j}\nu^0_{\tilde Q_j({\bf{f}})}(z),\end{equation*}

and hence

\begin{equation*} \max\{0,\nu^0_{L_i({\bf{F}})}(z)-n_u\}\ge\sum_{j=1}^qa_{i,j}c_{j}={\textbf{a}_i}\cdot\textbf{c}. \end{equation*}

Thus, we have

(3.8)\begin{align} \nu^0_{W(\tilde F)}(z)\ge t\sum_{i=1}^{H_Y(u)}{\textbf{a}_i}\cdot\textbf{c}=tS_Y(u,\textbf{c}). \end{align}

Choose an index $\sigma_{i_0}$ such that $\nu^0_{\tilde Q_{\sigma_{i_0}(0)}({\bf{f}})}(z)\ge \nu^0_{\tilde Q_{\sigma_{i_0}(1)}({\bf{f}})}(z)\ge\cdots\ge \nu^0_{\tilde Q_{\sigma_{i_0}(q-1)}({\bf{f}})}(z)$. By Lemma 2.2 we have

\begin{align*} \Delta_{V}e_{Y_z}(\textbf{c})&\ge (c_{\sigma_{i_0}(0),z}+\cdots +c_{\sigma_{i_0}(\ell_{i_0}),z})\cdot\delta_z\\ &=\delta_z\cdot\sum_{j=1}^{l_{i_0}}\max\{0,\nu^0_{\tilde Q_{\sigma_{i_0}(j)}({\bf{f}})}(z)-n_u\}\\ &=\delta_z\cdot\sum_{j=1}^{q}\max\{0,\nu^0_{\tilde Q_j({\bf{f}})}(z)-n_u\}+O(\nu_{R^{i_0}}(z)), \end{align*}

where R ⁱ is the resultant of the family $\{Q_{\sigma_i(j)}\}_{j=0}^{\ell_i}$ for $i=1,\ldots,q$.

On the other hand, by Theorem 2.1 we have that

\begin{align*} S_{Y_z}(u,\textbf{c}) &\ge\frac{u(\xi_u-p)}{(n+1)\delta_z}e_Y(\textbf{c})-(2n+1)\delta_z(\xi_u-p)\max_{1\le i\le q}c_{i}+O(\nu_{R^{i_0}}(z))\\ &\ge\left(\frac{u(\xi_u-p)}{\Delta_{V}(n+1)}-(2n+1)\delta_z (\xi_u-p)\right)\\ &\ \ \times \sum_{j=1}^{q}\max\{0,\nu^0_{\tilde Q_j({\bf{f}})}(z)-n_u\}+O(\nu_{R^{i_0}}(z)). \end{align*}

Combining this inequality and (3.8), we have

\begin{align*} \dfrac{(n+1)}{u(\xi_u-p)s}\nu^0_{W(\tilde F)}(z)&\ge\dfrac{t}{us}\left(\frac{u}{\Delta_{V}}-(2n+1)(n+1)\delta_z\right )\\ &\times\sum_{j=1}^{q}\max\{0,\nu^0_{\tilde Q_j({\bf{f}})}(z)-n_u\}+O(\sum_{i=1}^{n_0}\nu_{R^{i}}(z)). \end{align*}

Integrating both sides of this inequality, we obtain

\begin{align*} \biggl \|\ \dfrac{(n+1)}{u(\xi_u-p)s}N_{W(\tilde F)}(r)&\ge \dfrac{t}{s}\left(\frac{1}{\Delta_{V}}-\frac{(2n+1)(n+1)\delta}{u}\right )\\ &\ \ \times\sum_{j=1}^{q}\left(N_{Q_j(f)}(r)-N^{[n_u]}_{Q_j(f)}(r)\right)+o(T_f(r)). \end{align*}

Seting $m_0=\frac{1}{\Delta_{V}}\,-\,\frac{(2n+1)(n+1)\delta}{u}$ and combining inequalities (3.7) with the above inequality, we get

\begin{align*} \biggl\|\ \sum_{i=1}^{q}m_f(r,Q_i)\le &\frac{d(n+1)t}{sm_0}T_f(r)-\frac{t}{s}\sum_{j=1}^{q}\left(N_{Q_i(f)}(r)-N^{[n_u]}_{Q_j(f)}(r)\right)\\ &+\frac{(t(\xi_u-p)-1)t(n+1)}{2sm_0u}(c_{\tilde F}+\frac{\epsilon'}{2du})T_{\tilde F}(r)+o(T_f(r)). \end{align*}

This inequality implies that

(3.9)\begin{align} \begin{split} \bigl\|\ &\bigl(q-\frac{(n+1)t}{sm_0}\bigl)T_f(r)\\ &\le\sum_{j=1}^{q}\frac{1}{d}N^{[n_u]}_{Q_j(f)}(r)+\frac{(t(\xi_u-p)-1)t(n+1)}{2sdm_0u}(c_{f}+\epsilon')T_{f}(r). \end{split} \end{align}

1. a) We choose $u=\lceil 2\Delta_{V}(2n+1)(n+1)d^n\deg V(\Delta_{V}(n+1)+\epsilon)\epsilon^{-1}\rceil$.

   Then $u\ge \biggl\lceil \dfrac{\Delta_{V}(2n+1)(n+1)\delta(\Delta_{V}(n+1)+\epsilon)}{\Delta_{V}(n+1)+\epsilon-\frac{\Delta_{V}(n+1)t}{s}}\biggl\rceil,$ and we have:

   • • $q-\dfrac{(n+1)t}{sm_0}\ge q-\dfrac{\Delta_V(n+1)t}{(1-\Delta_V(2n+1)(n+1)\delta/u)s}$

     $\ge q-\Delta_{V}(n+1)-\epsilon;$

   • • $n_u+1=(\xi_u-p)t$

     $ \le d^n\deg V(u+1)^n\left[\bigl(1+\frac{\epsilon}{2(n+1)\Delta_{V}}\bigl)^{\bigl[\frac{d^n\deg V(u+1)^{n+q}}{\log^2(1+\frac{\epsilon}{2(n+1)\Delta_{V}})}\bigl]+1}\right]=L;$

   • • $\dfrac{(t(\xi_u-p)-1)t(n+1)}{2sdm_0u} \lt \dfrac{t(n+1)}{sm_0}\cdot\dfrac{(L-1)}{2du}$

     $ \le \dfrac{(\Delta_{V}(n+1)+\epsilon)(L-1)}{2du}.$

   Then, from (3.9) we have

   \begin{align*} \bigl\|\ &(q-\Delta_{V}(n+1)-\epsilon)T_f(r)\\ &\le\sum_{j=1}^{q}\frac{1}{d}N^{[L-1]}_{Q_j(f)}(r)+\frac{(\Delta_{V}(n+1)+\epsilon)(c_f+\epsilon')(L-1)}{2du}T_f (r). \end{align*}

   The assertion a) is proved.

2. b) If all Q_i are fixed hypersurfaces then $t=s=1$. Choosing $u'=\lceil \Delta_{V}(2n+1)$$(n+1)d^n\deg V(\Delta_{V}(n+1)+\epsilon)\epsilon^{-1}\rceil$ and replacing u in the above by u ^ʹ, we have

   • • $q-\dfrac{(n+1)}{m_0}\ge q-\Delta_{V}(n+1)-\epsilon,$

   • • $n_{u'}+1=\xi_{u'}-p \lt \delta\left(\begin{matrix}{n+u'}\\{n}\end{matrix}\right)\le d^n\deg V\left(\begin{matrix}{n+u'}\\{n}\end{matrix}\right)$

     $\le \left[d^n\deg Ve^n\left(1+\dfrac{u'}{n}\right)^{n}\right]$

     $\le \left[d^{n^2+n}(\deg V)^{n+1}e^n(2n+5)^n(\Delta^2_V(n+1)\epsilon^{-1}+\Delta_V)^n\right]=L'.$

   Similarly as above, from (3.9) we have the desired inequality of the assertion b).

Hence, the proof of the theorem is completed.$\square$

Remark. For the case $\Delta(R)={\mathbb{C}}$, we have $c_f=0$ if f is non-constant. In this case, (3.7) can be re-written as follows:

(3.10)\begin{align} \begin{split} \biggl \|\ s&\int_0^{2\pi}\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in\mathcal J}\frac{\|{\bf{F}}\|}{|L({\bf{F}})|}\frac{d\theta}{2\pi}-N_{W(\tilde F)}(r)\\ &\le \int_0^{2\pi}\max_{\mathcal J\subset\mathcal L}\log\prod_{L\in\mathcal J}\prod_{i=1}^s\frac{\|\tilde{\bf{F}}\|}{|b_iL({\bf{F}})|}-N_{W(\tilde F)}(r)+o(T_f(r))\\ &\le t(\xi_u-p)udT_f(r)+o(T_f(r)). \end{split} \end{align}

Here, the notation “$\|$” means that the inequality hold for all $r\in [1,+\infty)$ outside a set of finite measure.

Using (3.10) instead of (3.7), from the above proof, we obtain the following second main theorem for holomorphic maps from ${\mathbb{C}}$.

Theorem 3.11. Let V be a smooth subvriety of dimension n of $\mathbb{P}^{N}(\mathbb{C})$. Let $f:{\mathbb{C}}\rightarrow V$ be a holomorphic mapping. Let $\mathcal {Q}=\{Q_{1},\ldots,Q_{q}\}$ be a set of slowly (with respect to f) moving hypersurfaces with the distributive constant $\Delta_{V}$ with respect to V. Assume that f is algebraically non-degenerate over $\mathcal{K}_{\mathcal {Q}}$. Let $d=lcm(\deg Q_1,\ldots,\deg Q_q)$. Then for every $(n+1)\Delta_V \gt \epsilon \gt 0,$

(3.12)\begin{align} \bigl\|\ (q-\Delta_{V}(n+1)-\epsilon)T_f(r)\le\sum_{j=1}^{q}\frac{1}{\deg Q_j}N^{[L]}_{f}(r,Q_j), \end{align}

where

\begin{equation*}L=d^n\deg V(u+1)^n\left[\bigl(1+\frac{\epsilon}{2(n+1)\Delta_{V}}\bigl)^{\bigl[\frac{d^n\deg V(u+1)^{n+q}}{\log^2(1+\frac{\epsilon}{2(n+1)\Delta_{V}})}\bigl]+1}\right],\end{equation*}

with $u=\lceil 2\Delta_{V}(2n+1)(n+1)d^n\deg V(\Delta_{V}(n+1)+\epsilon)\epsilon^{-1}\rceil$.

Moreover, if all $Q_i\ (1\le i\le q)$ are assumed to be fixed hypersurfaces, then for every ϵ > 0 we have the inequality (3.12) with $L=\bigl[d^{n^2+n}(\deg V)^{n+1}e^n(2n+5)^n(\Delta^2_V(n+1)\epsilon^{-1}+\Delta_V)^n\bigl]$.

We also note that our proof is valid for the case of holomorphic maps from higher dimension complex spaces ${\mathbb{C}}^m$ into V. This theorem is an improvement of many previous second main theorem for hypersurface targets, such as [Reference Dethloff and Tan1, Reference Giang3, Reference Quang4, Reference Quang6, Reference Ru7, Reference Shi9].