# A generalized second main theorem for closed subschemes

Liang Wang  Department of Mathematics
Nanchang University
Nanchang 330031, P. R. China<EMAIL_ADDRESS>,  Tingbin Cao
Department of Mathematics
Nanchang University
Nanchang 330031, P. R. China<EMAIL_ADDRESS>(the corresponding author) and
Hongzhe Cao  Department of Mathematics
Nanchang University
Nanchang 330031, P. R. China<EMAIL_ADDRESS>
###### Abstract.

Let $Y_{1},\ldots,Y_{q}$ be closed subschemes which are located in
$\ell$-subgeneral position with index $\kappa$ in a complex projective variety
$X$ of dimension $n.$ Let $A$ be an ample Cartier divisor on $X.$ We obtain
that if a holomorphic curve $f:\mathbb{C}\to X$ is Zariski-dense, then for
every $\epsilon>0,$

$\displaystyle\sum^{q}_{j=1}\epsilon_{Y_{j}}(A)m_{f}(r,Y_{j})\leq_{exc}\left(\frac{(\ell-n+\kappa)(n+1)}{\kappa}+\epsilon\right)T_{f,A}(r).$

This generalizes the second main theorems for general position case due to
Heier-Levin [AM J. Math. 143(2021), no. 1, 213-226] and subgeneral position
case due to He-Ru [J. Number Theory 229(2021), 125-141]. In particular,
whenever all the $Y_{j}$ are reduced to Cartier divisors, we also give a
second main theorem with the distributive constant. The corresponding
Schmidt’s subspace theorem for closed subschemes in Diophantine approximation
is also given.

###### Key words and phrases:

Closed subschemes, Second main theorem, Holomorphic curves, Cartier divisors,
Schmidt’s subspace theorem

###### 2020 Mathematics Subject Classification:

Primary 32H30, 30D35; Secondary 11J87, 11J97, 14C20

The corresponding author: Tingbin Cao

##  1\. Introduction and main results

In 1933, H. Cartan [1] established the important second main theorem for
linearly nondegenerate holomorphic mappings from $\mathbb{C}^{m}$ into
$\mathbb{P}^{N}(\mathbb{C})$ intersecting hyperplanes in general position. He
also conjectured that it should be true for hyperplanes in subgeneral
position. In 1983, Nochka [8] confirmed the Cartan’s conjecture by introducing
the Nochka weight and Nochka constant. In 2009, Ru [11] extended the Cartan’s
second main theorem to the case of Cartier divisors in complex projective
variety as follows (For more background of Nevanlinna theory, we refer to
[12]). Here we use notations from Nevanlinna theory which can be found in
Section 2.

######  Theorem 1.1.

[11] Let $X$ be a complex projective variety of dimension $n$ and
$D_{1},\ldots,D_{q}$ be effective Cartier divisors on $X,$ located in general
position. Suppose that there exists an ample Cartier divisor $A$ on $X$ and
positive integers $d_{j}$ such that $D_{j}\sim d_{j}A$ for $j=1,\ldots,q.$ Let
$f:\mathbb{C}\rightarrow X$ be a holomorphic map with Zariski dense image.
Then, for every $\epsilon>0,$

$\displaystyle\sum^{q}_{j=1}\frac{1}{d_{j}}m_{f}(r,D_{j})\leq_{exc}(n+1+\epsilon)T_{f,A}(r),$

where $\leq_{exc}$ means the inequality holds for all $r\in\mathbb{R}^{\geq
0}$ outside a set of finite Lebesgue measure.

In 2017, Ru and Wang [13] considered the case of closed subschemes and gave
the second main theorem as follows.

######  Theorem 1.2.

[13] Let $X$ be a projective variety. Let $Y_{1},\ldots,Y_{q}$ be closed
subschemes of $X$ such that at most $\ell$ of the closed subschemes meet at
any point $x\in X.$ Let $A$ be a big Cartier divisor on $X.$ Let
$f:\mathbb{C}\rightarrow X$ be a holomorphic curve with Zariski-dense image.
Let

$\displaystyle\beta_{A,Y_{j}}=\lim_{N\rightarrow\infty}\frac{\sum_{m=1}^{\infty}h^{0}(\tilde{X}_{j},N\pi^{*}_{j}A-mE_{i})}{Nh^{0}(X,NA)},~{}j=0,\ldots,q,$

where $\pi_{j}:\tilde{X}_{j}\rightarrow X$ is the blowing-up of $X$ along
$Y_{j},$ with associated exceptional divisor $E_{j}.$ Then, for every
$\epsilon>0,$

$\displaystyle\sum^{q}_{j=1}m_{f}(r,Y_{j})\leq_{exc}\ell(\max_{1\leq j\leq
q}\\{\beta^{-1}_{A,Y_{j}}\\}+\epsilon)T_{f,A}(r),$

where $\leq_{exc}$ means the inequality holds for all $r\in\mathbb{R}^{\geq
0}$ outside a set of finite Lebesgue measure.

When the closed subschemes $Y_{j}=y_{j}$ are distinct points in $X$ and
dim$X=n,$ then one may take $\ell=1$ in Theorem 1.2, Mckinnon and Roth [7]
have shown that the Seshadri constants $\epsilon_{y_{j}}(A)$ and
$\beta_{A,y_{j}}$ have the following relation:

$\beta_{A,y_{j}}\geq\frac{n}{n+1}\epsilon_{y_{j}}(A).$

In order to improve the condition $D_{j}\sim d_{j}A$ in Theorem 1.1, Heier and
Levin [4] used the notion of Seshadri constant $\epsilon_{D_{j}}(L)$ (see
Definition 2.1), and obtained the following generalization of the second main
Theorem for closed subschemes in general position.

######  Theorem 1.3.

[4, Theorem 1.8] Let $X$ be a complex projective variety of dimension $n.$ Let
$Y_{0},\ldots,Y_{q}$ be closed subschemes of $X$ in general position,
$f:\mathbb{C}\rightarrow X$ be a holomorphic map with Zariski dense image. Let
$A$ be an ample Cartier divisor on $X.$ Then, for every $\epsilon>0,$

$\displaystyle\sum^{q}_{j=0}\epsilon_{Y_{j}}(A)m_{f}(r,Y_{j})\leq_{exc}(n+1+\epsilon)T_{f,A}(r),$

where $\leq_{exc}$ means the inequality holds for all $r\in\mathbb{R}^{\geq
0}$ outside a set of finite Lebesgue measure.

In 2021, He and Ru [3] considered the $\ell$-subgeneral position case and
generalized Theorem 1.3 by using somewhat different methods. In this paper, we
continue to generalize them to the case of the index $\kappa$ of
$\ell$-subgeneral position. The concept of the index $\kappa$ of
$\ell$-subgeneral position by Ji-Yan-Yu [5] is generalized from Cartier
divisors to closed subschemes.

######  Definition 1.4.

Let $X$ be a projective variety of dimension $n$ which is defined over an
arbitrary field $k$ with characteristic zero and let $Y_{1},\ldots,Y_{q}$ be
$q$ closed subschemes of $X.$ Let $\ell$ and $\kappa$ be two positive integers
such that $\ell\geq\dim X\geq\kappa.$

(a). The closed subschemes $\\{Y_{1},\ldots,Y_{q}\\}$ are said to be in
general position in $X$ if for any subset $I\subset\\{1,\ldots,q\\}$ with
$\sharp I\leq\dim X+1,$

$codim\left(\bigcap_{i\in I}Y_{i}\cap X\right)\geq\sharp I.$

(b). The closed subschemes $\\{Y_{1},\ldots,Y_{q}\\}$ are said to be in
$\ell$-subgeneral position in $X$ if for any subset $I\subset\\{1,\ldots,q\\}$
with $\sharp I\leq\ell+1,$

$\dim\left(\bigcap_{i\in I}Y_{i}\cap X\right)\leq\ell-\sharp I.$

(c). The closed subschemes $Y_{1},\ldots,Y_{q}$ are said to be in
$\ell$-subgeneral position with index $\kappa$ if $Y_{1},\ldots,Y_{q}$ are in
$\ell$-subgeneral position and for any subset $J\subset\\{1,\ldots,q\\}$ with
$\sharp I\leq\kappa,$

$codim\left(\bigcap_{i\in I}Y_{i}\cap X\right)\geq\sharp I.$

######  Theorem 1.5.

Let $f:\mathbb{C}\rightarrow$X be a meromorphic map with Zariski dense image,
where $X$ is a complex projective variety of dimension $n.$ Let
$Y_{1},\ldots,Y_{q}$ be closed subschemes which is located in
$\ell$-subgeneral position with index $\kappa$ in $X,$ and $\ell\geq n.$ Let
$A$ be an ample Cartier divisor on $X.$ Then, for any $\epsilon>0,$

$\displaystyle\sum^{q}_{j=1}\epsilon_{Y_{j}}(A)m_{f}(r,Y_{j})\leq_{exc}\left(\frac{(\ell-n+\kappa)(n+1)}{\kappa}+\epsilon\right)T_{f,A}(r),$

where $\leq_{exc}$ means the inequality holds for all $r\in\mathbb{R}^{\geq
0}$ outside a set of finite Lebesgue measure.

To prepare for the proof of Theorem 1.5, we need firstly to obtain a second
main theorem for Cartier divisors with distributive constant, which is also an
interesting topic independently. The concept of distributive constant was
originally given by Quang[10], and we extend it to the case of Cartier
divisors as follows.

######  Definition 1.6.

Let $X$ be a projective variety of dimension $n$ defined over over arbitrary
field $k$ with characteristic zero and let $D_{1},\ldots,D_{q}$ be $q$ Cartier
divisors of $X.$ The distributive constant $\Delta$ of
$\\{D_{1},\ldots,D_{q}\\}$ in $X$ is defined by

$\Delta:=\max_{\varGamma\subset\\{1,\ldots,q\\}}\frac{\sharp\varGamma}{n-dim((\cap_{j\in\varGamma}SuppD_{j})\cap
X(\overline{k}))}.$

Here, we note that $dim~{}\emptyset=-\infty.$

The second main theorem for holomorphic curves intersecting Cartier divisors
with distributive constant in a complex projective variety is stated as
follows.

######  Theorem 1.7.

Let $X$ be a complex projective variety of dimension $n.$ Let
$D_{1},\ldots,D_{q}$ be Cartier divisors of $X$ with the distributive constant
$\Delta$ in $X.$ Let $A$ be an ample Cartier divisor on $X.$ Let
$f:\mathbb{C}\to X$ be a holomorphic curve with Zariski-dense image. Then, for
every $\epsilon>0,$

$\displaystyle\sum^{q}_{j=1}\epsilon_{D_{j}}(A)m_{f}(r,D_{j})\leq_{exc}(\Delta(n+1)+\epsilon))T_{f,A}(r),$

where $\leq_{exc}$ means the inequality holds for all $r\in\mathbb{R}^{\geq
0}$ outside a set of finite Lebesgue measure.

If $D_{1},\ldots,D_{q}$ are in $\ell$-subgeneral position with index $\kappa$,
then by Remark 3.3 we have $\Delta\leq\frac{\ell-n+\kappa}{\kappa}.$ Hence,
Theorem 1.7 generalizes Theorem 1.1, Ji-Yan-Yu’s work [5, Theorem 1.2] and He-
Ru [3, Theorem 2.6].

Whether can Definition 1.6 be generalized to the closed subschemes in order to
obtain a generalized second main theorem for closed subschemes with
distributive constant? This is an open question.

The rest of this paper is structured as follows. In section 2, we briefly
recall some basic notations and definitions about the closed subscheme and
Nevanlinna theory. Some preliminary lemmas and the proof of Theorem 1.7 for
Cartier divisors with distributive constant are given in section 3. Then
Theorem 1.5 will be proved in section 4. According to the Vojta’s dictionary
([16, 17]) of the analogy between Nevanlinna theory and Diophantine
approximation, the Schmidt’s subspace theorems will be given in section 5.

##  2\. Preliminaries

###  2.1. Seshadri constants

Let $X$ be a projective variety and $Y$ be a closed subscheme of $X,$
corresponding to a coherent sheaf of ideals $\mathcal{I}_{Y}.$ Let
$\mathcal{S}=\oplus_{d\geq 0}\mathcal{I}_{Y}^{d}$ be the sheaf of graded
algebras, where $\mathcal{I}_{Y}^{d}$ is the $d$-th power of
$\mathcal{I}_{Y},$ with the convention that
$\mathcal{I}_{Y}^{0}=\mathcal{O}_{X}.$ Then $\tilde{X}:=Proj\mathcal{S}$ is
called the blowing-up of $X$ with respect to $\mathcal{I}_{Y},$ or, the
blowing-up of $X$ along $Y.$

Let $\pi:\tilde{X}\rightarrow X$ be the blowing-up along $Y.$ From Proposition
II.7.13(a) in [2], the inverse image ideal sheaf
$\tilde{\mathcal{I}}_{Y}=\pi^{-1}\mathcal{I}_{Y}\cdot\mathcal{O}_{\tilde{X}}$
is an invertible sheaf on $\tilde{X}.$ Let $E$ be an effective Cartier divisor
in$\tilde{X}$ whose associated invertible sheaf is the dual of
$\pi^{-1}\mathcal{I}_{Y}\cdot\mathcal{O}_{\tilde{X}}.$

######  Definition 2.1.

Let $Y$ be a closed subscheme of a projective variety $X.$ Let
$\pi:\tilde{X}\rightarrow X$ be the blowing-up of $X$ along $Y.$ Let $A$ be a
nef Cartier divisor on $X.$ We define the Seshadri constant $\epsilon_{Y}(A)$
of $Y$ with respect to $A$ to be the real number

$\epsilon_{Y}(A)=sup\\{\gamma\in\mathbb{Q}_{\geq 0}|\pi^{*}A-\gamma
E~{}is~{}\mathbb{Q}-nef\\},$

where $E$ is an effective Cartier divisor on $\tilde{X}$ whose associated
invertible sheaf is the dual of
$\pi^{-1}\mathcal{I}_{Y}\cdot\mathcal{O}_{\tilde{X}}.$

###  2.2.  The Weil function and its properties

In this section, we briefly recall the definition of the Weil function and its
properties.

######  Definition 2.2.

Let $D$ be a Cartier divisor on a complex variety $X.$ A local Weil function
for $D$ is a function $\lambda_{D}:(X\setminus suppD)\rightarrow\mathbb{R}$
such that for all $x\in X$ there is an open neighborhood $U$ of $x$ in $X,$ a
nonzero rational function $f$ on $X$ with $D\mid_{U}=(f),$ and a continuous
function $\alpha:U\rightarrow\mathbb{R}$ such that

$\lambda_{D}(x)=-\log|f(x)|+\alpha(x)$

for all $x\in(U\setminus SuppD)$.

A continuous (fiber) metric $||\cdot||$ on the line sheaf $\mathcal{O}_{X}(D)$
determines a Weil function for $D$ given by

$\lambda_{D}(x)=-\log||s(x)||$

where $s$ is the rational section of $\mathcal{O}_{X}(D)$ such that $D=(s).$
For example, the Weil function for the hyperplanes
$H=\\{a_{0}x_{0}+\ldots+a_{n}x_{n}=0\\}$ is given by

$\lambda_{H}(x)=\log\frac{\max_{0\leq i\leq n}|x_{i}|\max_{0\leq i\leq
n}|a_{i}|}{|a_{0}x_{0}+\ldots+a_{n}x_{n}|}.$

The Weil functions with respect to divisors satisfy the following properties.

(a) Additivity: If $\lambda_{1}$ and $\lambda_{2}$ are Weil functions for
Cartier divisors $D_{1}$ and $D_{2}$ on $X,$ respectively, then
$\lambda_{1}+\lambda_{2}$ extends uniquely to a Weil function for
$D_{1}+D_{2}.$

(b) Functoriality: If $\lambda$ is a Weil function for a Cartier divisor $D$
on $X,$ and if $\phi:X^{\prime}\rightarrow X$ is a morphism such that
$\phi(X^{\prime})\not\subset suppD,$ then $x\mapsto\lambda(\phi(x))$ is a Weil
function for the Cartier divisor $\phi^{*}D$ on $X^{\prime}.$

(c) Uniqueness: If both $\lambda_{1}$ and $\lambda_{2}$ are Weil functions for
a Cartier divisor on $X,$ then $\lambda_{1}=\lambda_{2}+O(1).$

(d) Boundedness from below: If $D$ is an effective divisor and $\lambda$ is a
Weil function for $D,$ then $\lambda$ is bounded from below.

For the closed subschemes case, let $Y$ be a closed subscheme of a projective
variety $X.$ Then one can associate a Weil function $\lambda_{Y}:X\setminus
SuppY\rightarrow\mathbb{R},$ well-defined up to $O(1)$. The following lemma
indicates that a closed subscheme can be expressed by some Cartier divisors.

######  Lemma 2.3.

[15, Lemma 2.2] Let $Y$ be a closed subscheme of a projective variety $X.$
There exist effective Cartier divisors $D_{1},\ldots,D_{\ell}$ such that

$Y=\cap_{i=1}^{\ell}D_{i}.$

######  Definition 2.4.

Let $Y$ be a closed subscheme of a projective variety $X.$ We define the Weil
function for $Y$ as

$\lambda_{Y}=\min\\{\lambda_{D_{1}},\ldots,\lambda_{D_{\ell}}\\}+O(1),$

where $Y=\cap_{i=1}^{\ell}D_{i}$ (by Lemma 2.3, such $D_{i}$ exist). Then
there is a Weil function for the closed subscheme $\lambda_{Y}:X\setminus
SuppY\rightarrow\mathbb{R},$ which does not depend on the choice of Cartier
divisors.

We briefly recall the natural operations on subschemes which are similar to
the case of divisors, more details can be found in [15, section 2]. Let $Y,Z$
be closed subschemes of $X$,

(i) The sum of $Y$ and $Z$, denoted by $Y+Z,$ is the subscheme of $X$ with
ideal sheaf $\mathcal{I}_{Y}\mathcal{I}_{Z}$.

(ii) The intersection of $Y$ and $Z$, denoted by $Y\cap Z,$ is the subscheme
of $X$ with ideal sheaf $\mathcal{I}_{Y}+\mathcal{I}_{Z}.$

(iii) The union of $Y$ and $Z$, denoted by $Y\cup Z$, is the subscheme of $X$
with ideal sheaf $\mathcal{I}_{Y}\cap\mathcal{I}_{Z}.$

(iv) Let $\phi:X^{\prime}\rightarrow X$ be a morphism of projective varieties,
the inverse image of $Y$ is the subscheme of $X^{\prime}$ with ideal sheaf
$\phi^{-1}\mathcal{I}_{Y}\cdot\mathcal{O}_{X^{\prime}},$ denoted by
$\phi^{*}Y.$

In addition, some properties of the Weil functions for closed subschemes are
written as follows.

If $Y$ and $Z$ are two closed subschemes of $X$, and
$\phi:X^{\prime}\rightarrow X$ is a morphism of projective varieties,

(i) $\lambda_{Y\cap Z}=\min\\{\lambda_{Y},\lambda_{Z}\\}.$

(ii) $\lambda_{Y+Z}=\lambda_{Y}+\lambda_{Z}.$

(iii) If $Y\subset Z,$ $\lambda_{Y}\leq\lambda_{Z}.$

(iv) $\lambda_{Y}(\phi(x))=\lambda_{\phi^{*}Y}(x).$

######  Lemma 2.5.

[15] Let $Y$ be a closed subscheme of $X,$ and let $\tilde{X}$ be the blowing-
up of $V$ along $Y$ with exceptional divisor $E.$ Then for
$P\in\tilde{X}\setminus SuppE,$

$\lambda_{Y}(\pi(P))=\lambda_{E}(P)+O_{v}(1).$

###  2.3.  Characteristic function and proximity function

Let $X$ be a complex projective variety and $f:\mathbb{C}\rightarrow X$ be a
holomorphic map. Let $L\rightarrow X$ be a positive line bundle. Denote by
$||\cdot||$ a Hermitian metric in $L$ and by $\omega$ its Chern form. We
define the characteristic function of $f$ with respect to $L$ by

$T_{f,L}(r)=\int_{0}^{r}\frac{dt}{t}\int_{|z|<t}f^{*}\omega.$

Since any line bundle can be written as $L=L_{1}\otimes L_{2}^{-1}$ with
$L_{1},L_{2}$ are both positive, we define
$T_{f,L}(r)=T_{f,L_{1}}(r)-T_{f,L_{2}}(r).$

The characteristic function satisfies the following properties:

(i) Functoriality: If $\phi:X\rightarrow X^{\prime}$ is a morphism and if $L$
is a line bundle on $X^{\prime},$ then

$T_{f,\phi^{*}L}=T_{\phi\circ f,L}(r)+O(1).$

(ii) Additivity: If $L_{1}$ and $L_{2}$ are line bundle on $X,$ then

$T_{f,L_{1}\otimes L_{2}(r)}=T_{f,L_{1}}(r)+T_{f,L_{2}}(r)+O(1).$

(iii) Positivity: If $L$ is positive and $f:\mathbb{C}\rightarrow X$ is non-
constant, then

$T_{f,L}(r)\rightarrow+\infty~{}~{}as~{}~{}r\rightarrow+\infty.$

For an effective divisor $D$ on $X,$ we define a proximity function of $r,$
for any holomorphic map $f:\mathbb{C}\rightarrow X$ with
$f(\mathbb{C})\not\subset SuppD,$

$m_{f}(r,D):=\int_{0}^{2\pi}\lambda_{D}(f(re^{i\theta}))\frac{d\theta}{2\pi}.$

Analogously, for a closed subscheme $Y$ on $X,$ we define a proximity function
of $r,$ for a holomorphic curve $f:\mathbb{C}\rightarrow X$ with
$f(\mathbb{C})\not\subset SuppY,$

$m_{f}(r,Y):=\int_{0}^{2\pi}\lambda_{Y}(f(re^{i\theta}))\frac{d\theta}{2\pi}.$

##  3\. Some lemmas for divisors and the proof of Theorem 1.7

The following is a reformulation of [10, Lemma 3.1] by taking the logarithm of
both sides of the inequality.

######  Lemma 3.1.

[10, Lemma 3.1] Let $t_{0},t_{1},\ldots,t_{n}$ be $n+1$ integers such that
$1=t_{0}<t_{1}<\ldots<t_{n},$ and let $\Delta=\max_{1\leq s\leq
n}\frac{t_{s}-t_{0}}{s}.$ Then for arbitrary real numbers
$a_{0},a_{1},\ldots,a_{n-1}$ with $a_{0}\geq a_{1}\geq\ldots\geq a_{n-1}\geq
1,$ we have

$\sum^{n-1}_{i=0}(t_{i+1}-t_{i})\log a_{i}\leq\Delta\sum^{n-1}_{i=0}\log
a_{i}.$

Next we need to introduce the definition of
$(t_{1},t_{2},\ldots,t_{n})$-subgeneral position for Cartier divisors, which
was originally given by Quang[10] for hypersurfaces.

######  Definition 3.2.

Let $X$ be a projective variety of dimension $n$ defined over a number field
$k$ and let $D_{1},\ldots,D_{q}$ be $q$ Cartier divisors of $X.$ We say that
$D_{1},\ldots,D_{q}$ are in $(t_{1},t_{2},\ldots,t_{n})$-subgeneral position
with respect to $X$ if for every $1\leq s\leq n$ and $t_{s}+1$ Cartier
divisors $D_{j_{0}},\ldots,D_{j_{t_{s}}},$ we have

$\dim\cap^{t_{s}}_{i=0}SuppD_{j_{i}}\cap X\leq n-s-1,$

where $t_{0},t_{1},\ldots,t_{n}$ are integers with
$0=t_{0}<t_{1}<\ldots<t_{n}.$

######  Remark 3.3.

(i) If ${D_{1},\ldots,D_{q}}$ are in $(t_{1},\ldots,t_{n})$-subgeneral
position with respect to $X$, then their distributive constant in $X$
satisfies

$\Delta=\max_{1\leq k\leq n}\frac{t_{k}}{n-(n-k)}=\max_{1\leq k\leq
n}\frac{t_{k}}{k}.$

(ii) If $D_{1},\ldots,D_{q}$ are in $\ell$-subgeneral position with index
$\kappa$ with respect to $X,$ then one has

$\dim\left(\cap^{k}_{j=1}D_{i_{j}}\right)\leq
n-\kappa-(k-(\ell-n+\kappa-1))=\ell-k-1.$

Hence, ${D_{1},\ldots,D_{q}}$ are in
$(1,2,\ldots,\kappa-1,\ell-n+\kappa,\ell-n+\kappa+1,\ldots,\ell-1,\ell)$-subgeneral
position with respect to $X$ and thus

$\displaystyle\Delta$ $\displaystyle\leq$
$\displaystyle\max\left\\{\frac{1}{n-(n-1)},\frac{2}{n-(n-2)},\ldots,\frac{\kappa-1}{n-(n-\kappa+1)},\ldots,\right.\,\,$
$\displaystyle\left.\frac{(\ell-n)+\kappa}{n-(n-\kappa)},\frac{(\ell-n)+\kappa+1}{n-(n-\kappa+1)},\ldots,\frac{\ell}{n}\right\\}$
$\displaystyle=$ $\displaystyle\max\left\\{1,\frac{\ell-n}{\kappa}+1\right\\}$
$\displaystyle=$ $\displaystyle\frac{\ell-n+\kappa}{\kappa}.$

The following lemma is just a special case of [10, Lemma 3.2].

######  Lemma 3.4.

[10, Lemma 3.2] Let $k$ be a number field. Let $X\subset\mathbb{P}^{M}_{k}$ be
a projective variety of dimension $n.$ Let $H_{0},\ldots,H_{l}$ be hyperplanes
in $X$ which are in $\\{t_{1},t_{2},\ldots,t_{n}\\}$-subgeneral position on
$X$ where $t_{0},t_{1},\ldots,t_{n}$ are integers with
$0=t_{0}<t_{1}<\ldots<t_{n}=l.$ Let $L_{1},\ldots,L_{q}$ be the normalized
linear forms defining $H_{1},\ldots,H_{q},$ respectively. Then there exist
linear forms $L^{\prime}_{1},\ldots,L^{\prime}_{n+1}$ of $M+1$ variables such
that,

(i) $L^{\prime}_{0}=L_{0};$

(ii) For every $s\in\\{1,\ldots,n\\},$ $L^{\prime}_{s}\in
span_{k}(L_{0},\ldots,L_{t_{s}});$

(iii) Let $H^{\prime}_{j},~{}j=0,\ldots,n,$ be the hyperplanes defined by
$L^{\prime}_{j},~{}j=0,\ldots,n.$ Then they are in general position on $X.$

We need also the following lemma [14, Theorem 1.22], which plays an important
role in the arguments regarding dimension.

######  Lemma 3.5.

[14, 3] Let $X$ be a projective variety over an algebraic closed field $k,$
and $A$ be an ample Cartier divisor on $X.$ Let $F\subset X$ be a proper
irreducible subvariety. Then either $F\subset A$ or $\dim(F\cap A)\leq\dim
F-1.$

For the proof of Theorem 1.5, we need to prove the second main theorem for
Cartier divisors with distributive constant. The basic idea is to consider the
distributive constant for Cartier divisors by making use of methods of Heier-
Levin[4] and He-Ru[3].

###### Proof of Theorem 1.7.

Fix a real number $\epsilon>0,$ choose a rational number $\delta>0$ such that

$\delta\Delta+\delta\Delta(n+1+\delta)<\epsilon,$

and for a sufficiently small positive rational number $\delta^{\prime}$
depending on $\delta,$ $\delta A-\delta^{\prime}D_{i}$ is $\mathbb{Q}$-ample
for all $i=1,\ldots,q.$ By the definition of the Seshadri constant, there
exists a rational number $\epsilon_{i}>0$ such that

$\epsilon_{D_{i}}(A)-\delta^{\prime}\leq\epsilon_{i}\leq\epsilon_{D_{i}}(A)$

and that $A-\epsilon_{i}D_{i}$ is $\mathbb{Q}$-nef for all $i=1,\ldots,q.$
Then we have

$(1+\delta)A-(\epsilon_{i}+\delta^{\prime})D_{i}=(A-\epsilon_{i}D_{i})+(\delta
A-\delta^{\prime}D_{i})$

is a $\mathbb{Q}$-ample divisor for all $i.$ Let $N$ be a large enough natural
number such that $N(1+\delta)A$ and
$N[(1+\delta)A-(\epsilon_{i}+\delta^{\prime})D_{i}]$ become very ample
integral divisors for all $i.$

We claim that if $\\{D_{1},\ldots,D_{q}\\}$ is in
$(t_{1},t_{2},\ldots,t_{n})$-subgeneral position with respect to $X,$ then we
can construct divisors $div(s_{i})$ on $X,$ $s_{i}\in
H^{0}(X,N(1+\delta)A-N(\epsilon_{i}+\delta^{\prime})D_{i}),$ $i=1,\ldots,q,$
such that,

(i) $div(s_{i})\sim N(1+\delta)A,i=1,\ldots,q.$

(ii) The divisors $div(s_{1}),\ldots,div(s_{q})$ are in
$\\{t_{1},t_{2},\ldots,t_{n}\\}$-subgeneral position on $X.$

We define $div(s_{1}),\ldots,div(s_{q})$ by induction as follows. Assume that,
for some $j\in\\{1,\ldots,q\\},$ $div(s_{1}),$ $\ldots,$ $div(s_{j-1})$ with
desired property have been defined and
$div(s_{1}),\ldots,div(s_{j-1}),D_{j},\ldots,D_{q}$ are in
$\\{t_{1},t_{2},\ldots,t_{n}\\}$-subgeneral position on $X$ (for $j=1,$ this
reduces to the hypothesis that $D_{1},\ldots,D_{q}$ are in
$\\{t_{1},t_{2},\ldots,t_{n}\\}$-subgeneral position).

By the definition of $\\{t_{1},t_{2},\ldots,t_{n}\\}$-subgeneral position,

(a) For all $1\leq s\leq
n,I^{\prime}\subset\\{1,\ldots,j-1\\},J^{\prime}\subset\\{j+1,\ldots,q\\},$
and $\sharp I^{\prime}+\sharp J^{\prime}=t_{s},$ we have

$\dim(\cap_{i\in I^{\prime}}div(s_{i})\cap(\cap_{k\in J^{\prime}}D_{k})\cap
X)\leq n-s,$

and

$\dim(D_{j}\cap(\cap_{i\in I^{\prime}}div(s_{i}))\cap(\cap_{k\in
J^{\prime}}D_{k})\cap X)\leq n-s-1.$

We can find a non-zero section $s_{j}\in
H^{0}(X,N(1+\delta)A-N(\epsilon_{j}+\delta^{\prime})D_{j})$ such that $s_{j}$
does not vanish entirely on any irreducible components of $\cap_{i\in
I^{\prime}}div(s_{i})\cap(\cap_{k\in J^{\prime}}D_{k}),$ for all $1\leq s\leq
n,I^{\prime}\subset\\{1,\ldots,j-1\\},J\subset\\{j+1,\ldots,q\\},$ and $\sharp
I^{\prime}+\sharp J^{\prime}=t_{s}.$ Then by Lemma 3.5

$\displaystyle\dim\\{div(s_{j})\cap(\cap_{i\in
I^{\prime}}div(s_{i}))\cap(\cap_{k\in J^{\prime}}D_{k})\cap X\\}$
$\displaystyle\leq$ $\displaystyle\max\left(\dim D_{j}\cap(\cap_{i\in
I^{\prime}}div(s_{i}))\cap(\cap_{k\in J^{\prime}}D_{k})\cap X,n-s-1\right)$
$\displaystyle=$ $\displaystyle n-s-1.$

(b) For all $1\leq s\leq
n,I\subset\\{1,\ldots,j-1\\},J\subset\\{j+1,\ldots,q\\},$ and $\sharp I+\sharp
J=t_{s}+1,$

$\dim(\cap_{i\in I}div(s_{i})\cap(\cap_{k\in J}D_{k})\cap X)\leq n-s-1.$

From this, we get that
$div(s_{1}),\ldots,div(s_{j-1}),div(s_{j}),D_{j+1},\ldots,D_{q}$ are in
$\\{t_{1},\ldots,t_{n}\\}$-subgeneral position on $X.$ Thus, the divisors
$div(s_{1}),\ldots,div(s_{q})$ satisfy the above required properties (i) and
(ii). Hence, the claim is proved.

Denote $F_{i}=div(s_{i})$ for $i=1,\ldots,q.$ It’s easy to see that the
distributive constant of $\\{F_{1},\ldots,F_{q}\\}$ is equal to the
distributive constant of $\\{D_{1},\ldots,D_{q}\\}.$ Now we’re going to prove
the theorem.

It is sufficient for us to consider the case where $\Delta<\frac{q}{n+1}.$
Note that $\Delta\geq 1,$ and hence $q>n+1.$ If there exits
$i\in\\{1,\ldots,q\\}$ such that $\cap^{q}_{j=1,j\neq i}F^{*}_{j}\cap
X\neq\emptyset,$ then

$\Delta\geq\frac{q-1}{n}\geq\frac{q}{n+1}.$

This is a contradiction. Therefore, $\cap^{q}_{j=1,j\neq i}F^{*}_{j}\cap
X=\emptyset$ for all $i\in\\{1,\ldots,q\\}.$

We denote by $\mathcal{I}$ the set of all permutations of the set
$\\{1,\ldots,q\\}.$ Denote by $n_{0}$ the cardinality of $\mathcal{I},$
$n_{0}=q!$ and we write $\mathcal{I}=\\{I_{1},\ldots,I_{n_{0}}\\},$ where
$I_{i}=\\{I_{i}(0),\ldots,I_{i}(q-1)\\}\in\mathbb{N}^{q}$ and
$I_{1}<I_{2}<\ldots<I_{n_{0}}$ in the lexicographic order.

For each $I_{i}\in\mathcal{I},$ since $\cap^{q-1}_{j=1}F_{I_{i}(j)}\cap
X=\emptyset,$ there exists $n+1$ integers $t_{i,0},t_{i,1},\ldots,t_{i,n}$
with $0=t_{i,0}<\ldots<t_{i,n}=l_{i},$ where $l_{i}\leq q-2$ such that
$\cap^{l_{i}}_{j=0}F_{I_{i}(j)}\cap X=\emptyset$ and

$\dim(\cap^{s}_{j=0}F_{I_{i}(j)})\cap X=n-u~{}\forall~{}t_{i,u-1}\leq
s<t_{i,u},~{}1\leq u\leq n.$

Then, $\Delta>\frac{t_{i,u}-t_{i,0}}{u}$ for all $1\leq u\leq n.$

It means that $\\{F_{1},\ldots,F_{q}\\}$ is in
$(t_{i,1},t_{i,2},\ldots,t_{i,n})$-subgeneral position with respect to $X.$
Denote by $\phi:X\to\mathbb{P}^{\tilde{N}}(k)$ the canonical embedding
associated to the very ample divisor $N(1+\delta)A$ and let
$H_{0},\ldots,H_{q-1}$ be the hyperplanes in $\mathbb{P}^{\tilde{N}}(k)$ with
$F_{j}=\phi^{*}H_{j-1}$ for $j=1,\ldots,q.$ We denote $L_{0},\ldots,L_{q-1}$
to be the linear forms defining $H_{0},\ldots,H_{q-1}$ respectively. By Lemma
3.4, there exist hyperplanes $\hat{H}_{0},\ldots,\hat{H}_{n}$ with defining
linear forms $\hat{L}_{0},\ldots,\hat{L}_{n},$ such that $\hat{L}_{0}=L_{0},$
and for every $s\in\\{1,\ldots,n\\},$ $\hat{L}_{s}\in
span_{k}(L_{0},\ldots,L_{t_{s}})$ and
$\phi^{*}\hat{H}_{0},\ldots,\phi^{*}\hat{H}_{n}$ are located in general
position on $X.$ Applying Theorem 1.1 to
$\phi^{*}\hat{H}_{0},\ldots,\phi^{*}\hat{H}_{n},$ we conclude that there
exists a Zariski-closed set $Z$ such that for all $x\in X(k)\backslash Z,$

$\displaystyle\int_{0}^{2\pi}\sum^{n}_{i=0}\lambda_{\phi^{*}\hat{H}_{i}}(x)\frac{d\theta}{2\pi}\leq[(n+1)+\delta]T_{f,N(1+\delta)A}(r).$

Consider a point $P=\phi(x)\in\mathbb{P}^{\tilde{N}}(k),$ fix a element
$I_{i}\in\mathcal{I},$ we arrange such that

$||L_{I_{i}(0)}(P)||\leq||L_{I_{i}(1)}(P)||\leq\ldots\leq||L_{I_{i}(q-1)}||,$

which implies

(3.1)
$\displaystyle\sum^{I_{i}(q-1)}_{j=I_{i}(0)}\lambda_{H_{j}(P)}\leq\sum^{t_{i,n}}_{j=t_{i.0}}\lambda_{H_{j}(P)}+O(1).$

(The proof of (3.1) is similar to that of Lemma 20.7 in [17] which is standard
and is omitted here.) Thus using the construction of
$\hat{L}_{0},\ldots,\hat{L}_{n},$ we have

$||\hat{L}_{u}(P)||\leq B\max_{0\leq j\leq
t_{i,u}}||L_{j}(P)||=B||L_{t_{i,u}}(P)||$

for all $u=0,\ldots,n$ and some constant $B>0.$ Thus, by the definition of
Weil function,

$\lambda_{\hat{H}_{u}}(P)\geq\lambda_{H_{t_{i,u}}}(P)+O(1).$

Therefore, by Lemma 3.1 and (3.1) we have

$\displaystyle\sum^{q-1}_{i=0}\lambda_{H_{j}}(P)$ $\displaystyle=$
$\displaystyle\sum^{I_{i}(q-1)}_{j=t_{i,0}}\lambda_{H_{j}}(P)$
$\displaystyle\leq$
$\displaystyle\sum^{t_{i,1}}_{j=t_{i,0}}\lambda_{H_{j}}(P)+\sum^{t_{i,2}}_{j=t_{i,1}+1}\lambda_{H_{j}}(P)+\cdots+\sum^{t_{i,n}}_{j=t_{i,n-1}+1}\lambda_{H_{j}}(P)+O(1)$
$\displaystyle\leq$
$\displaystyle(t_{i,1}-t_{i,0})\lambda_{H_{t_{i,0}}}(P)+(t_{i,2}-t_{i,1})\lambda_{H_{t_{i,1}}}(P)+\cdots$
$\displaystyle+(t_{i,n}-t_{i,n-1})\lambda_{H_{t_{i,n-1}}}(P)+\lambda_{H_{t_{i,n}}}(P)+O(1)$
$\displaystyle\leq$
$\displaystyle\Delta\sum^{n-1}_{j=0}\lambda_{H_{t_{i,j}}}(P)+\lambda_{H_{t_{i,n}}}(P)+O(1)$
$\displaystyle\leq$
$\displaystyle\Delta\sum^{n}_{j=0}\lambda_{H_{t_{i,j}}}(P)+O(1)$
$\displaystyle\leq$
$\displaystyle\Delta\sum^{n}_{u=0}\lambda_{\hat{H}_{u}}(P)+O(1).$

By the functoriality of Weil function, we have
$\lambda_{H_{j}}(P)=\lambda_{H_{j}}(\phi(x))=\lambda_{\phi^{*}H_{j}}(x)=\lambda_{F_{j+1}}(x),$
thus,

$\displaystyle\int_{0}^{2\pi}\sum^{q}_{i=1}\lambda_{F_{i}}(x)\frac{d\theta}{2\pi}$
$\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\sum^{q-1}_{j=0}\lambda_{H_{j}}(P)\frac{d\theta}{2\pi}$
$\displaystyle\leq$
$\displaystyle\Delta\int_{0}^{2\pi}\sum^{n}_{i=0}\lambda_{\hat{H}_{i}}(P)\frac{d\theta}{2\pi}+O(1)$
$\displaystyle\leq$
$\displaystyle\Delta[(n+1)+\delta]T_{f,N(1+\delta)A}(r)+O(1).$

From the construction of divisors $F_{i}=div(s_{i})$ for $i=1,\ldots,q,$ we
know that $F_{i}-N(\epsilon_{i}+\delta^{\prime})D_{i}$ is effective for each
$i=1,\ldots,q.$ By the definition of the Seshadri constant and the Boundedness
from below of a Weil function,

$\lambda_{F_{i}}(x)\geq
N(\epsilon_{i}+\delta^{\prime})\lambda_{D_{i}}+O(1)\geq
N\epsilon_{D_{i}}(A)\lambda_{D_{i}}(x)+O(1).$

Then,

$\displaystyle
N\int_{0}^{2\pi}\sum^{q}_{i=1}\epsilon_{D_{i}}(A)\lambda_{D_{i}}(x)\frac{d\theta}{2\pi}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{2\pi}\sum^{q}_{i=1}\lambda_{F_{i}}(x)\frac{d\theta}{2\pi}$
$\displaystyle\leq$
$\displaystyle\Delta[(n+1)+\delta]T_{f,N(1+\delta)A}(r)+O(1).$

Note that $T_{f,N(1+\delta)A}(r)=N(1+\delta)T_{f,A}(r),$ hence we get the
inequality as follow,

$\int_{0}^{2\pi}\sum^{q}_{i=1}\epsilon_{D_{i}}(A)\lambda_{D_{i}}(x)\frac{d\theta}{2\pi}\leq\Delta[(n+1)+\delta](1+\delta)T_{f,A}(r).$

It implies that

$\sum^{q}_{i=1}\epsilon_{D_{i}}(A)m_{f}(r,D_{i})\leq\Delta[(n+1)+\delta](1+\delta)T_{f,A}(r).$

Recall the choice of $\delta$, we get for every $\epsilon>0,$

$\displaystyle\sum^{q}_{j=1}\epsilon_{D_{j}}(A)m_{f}(r,D_{j})\leq_{exc}(\Delta(n+1)+\epsilon))T_{f,A}(r).$

∎

##  4\. Proof of theorem 1.5

Before the proof of Theorem 1.5, we need a key lemma.

######  Lemma 4.1.

[6, Lemma 5.4.24] Let $X$ be a projective variety, $\mathcal{I}$ be a coherent
ideal sheaf. Let $\pi:\tilde{X}\rightarrow X$ be the blowing-up of
$\mathcal{I}$ with exceptional divisor $E.$ Then there exists an integer
$p_{0}=p_{0}(\mathcal{I})$ with the property that if $p\geq p_{0},$ then
$\pi_{*}\mathcal{O}_{\tilde{X}}(-pE)=\mathcal{I}^{p},$ and moreover, for any
divisor $D$ on $X,$

$H^{i}(X,\mathcal{I}^{p}(D))=H^{i}(\tilde{X},\mathcal{O}_{\tilde{X}}(\pi^{*}D-pE))$

for all $i\geq 0.$

We now prove the second main theorem for closed subschemes in
$\ell$-subgeneral position with index $\kappa.$

###### Proof of Theorem 1.5.

The proof basically follows Heier-Levin[4] and He-Ru[3]. Denote by
$\mathcal{I}_{i}$ the ideal sheaf of $Y_{i},$ $\pi_{i}:\tilde{X}\rightarrow X$
the blowing-up of $X$ along $Y_{i},$ and $E_{i}$ the exceptional divisor on
$\tilde{X}_{i}.$ Then, fix a real number $\epsilon>0,$ choose a rational
number $\delta>0$ such that

$\frac{\delta(\ell-n+\kappa)}{\kappa}+\frac{\delta(\ell-n+\kappa)}{\kappa}(n+1+\delta)<\epsilon,$

and for a sufficiently small positive rational number $\delta^{\prime}$
depending on $\delta,$ $\delta A-\delta^{\prime}E_{i}$ is $\mathbb{Q}$-ample
for all $i=1,\ldots,q.$ By the definition of the Seshadri constant, there
exists a rational number $\epsilon_{i}>0$ such that

$\epsilon_{E_{i}}(A)-\delta^{\prime}\leq\epsilon_{i}\leq\epsilon_{E_{i}}(A),$

and that $A-\epsilon_{i}E_{i}$ is $\mathbb{Q}$-nef for all $i=1,\ldots,q.$
Then we have

$(1+\delta)A-(\epsilon_{i}+\delta^{\prime})E_{i}=(A-\epsilon_{i}E_{i})+(\delta
A-\delta^{\prime}E_{i})$

is a $\mathbb{Q}$-ample divisor for all $i.$ Let $N$ be a large enough natural
number such that $N(1+\delta)A$ and
$N[(1+\delta)A-(\epsilon_{i}+\delta^{\prime})E_{i}]$ become very ample
integral divisors for all $i.$

We claim that if $\\{Y_{1},\ldots,Y_{q}\\}$ is in $\ell$-subgeneral position
with index $\kappa$ with respect to $X,$ then we can construct divisors
$F_{i}$ on $X,$ $i=1,\ldots,q,$ such that,

(i) $F_{i}\sim N(1+\delta)A,i=1,\ldots,q.$

(ii) $\pi^{*}F_{i}\geq N(\epsilon_{i}+\delta^{\prime})E_{i},i=1,\ldots,q.$

(iii) The divisors $F_{1},\ldots,F_{q}$ are in $\ell$-subgeneral position with
index $\kappa$ on $X.$

Like the special divisor case in the proof of preparation theorem, we can
construct by induction. Assume that, for some $j\in\\{1,\ldots,q\\},$
$div(s_{1}),$ $\ldots,$ $div(s_{j-1})$ with desired property have been defined
and $F_{1},$ $\ldots,$ $F_{j-1},$ $Y_{j},$ $\ldots,$ $Y_{q}$
$(F_{i}=div(s_{i}))$ are in $\ell$-subgeneral position with index $\kappa$ on
$X$ (for $j=1,$ this reduces to the hypothesis that $Y_{1},\ldots,Y_{q}$ are
in $\ell$-subgeneral position with index $\kappa$). To find $F_{j}$, we let
$\tilde{F}^{(j)}_{i}=\pi^{*}_{j}F_{i},i=1,\ldots,j-1$ and
$\tilde{Y}^{j}_{i}=\pi^{*}_{j}Y_{i}$ for $i=j+1,\ldots,q.$ Since in
particular, $F_{1},\ldots,F_{j-1},Y_{j+1},\ldots,Y_{q}$ are in
$\ell$-subgeneral position with index $\kappa$ on $X$, and by noticing that
$\pi^{-1}_{j}$ is an isomorphism outside of $Y_{j}$, we know that
$\tilde{F}_{1},\ldots,\tilde{F}_{j-1},\tilde{Y}_{j+1},\ldots,\tilde{Y}_{q}$
are in $\ell$-subgeneral position with index $\kappa$ on $\tilde{X}_{j}$
outside of $E_{j}.$ Thus it reduces to the construction in the divisor case,
and by the argument in the divisor case, there are sections

$\tilde{s}_{j}\in
H^{0}(\tilde{X}_{j},\mathcal{O}_{\tilde{X}_{j}}(N(1+\delta)\pi^{*}_{j}A-N(\epsilon_{i}+\delta^{\prime})E_{j})),$

such that
$\tilde{F}_{1},\ldots,\tilde{F}_{j-1},div(\tilde{s}_{j}),\tilde{Y}_{j+1},\ldots,\tilde{Y}_{q}$
are in $\ell$-subgeneral position with index $\kappa$ on $\tilde{X}_{j}$
outside of $E_{j}$, where we regard
$H^{0}(\tilde{X}_{j},\mathcal{O}_{\tilde{X}_{j}}(N((1+\delta)\pi^{*}_{j}A-(\epsilon_{j}+\delta^{\prime})E_{j})))$
as a subspace of
$H^{0}(\tilde{X}_{j},\mathcal{O}_{\tilde{X}_{j}}(N(1+\delta)\pi^{*}_{j}A)).$

To guarantee that $F_{j}$ has the required properties, by Lemma 4.1, we have,
for a big enough $N$,

$H^{0}(X,\mathcal{O}_{X}(N(1+\delta)A)\otimes\mathcal{I}^{N(\epsilon_{i}+\delta^{\prime})}_{j})=H^{0}(\tilde{X}_{j},\mathcal{O}_{\tilde{X}_{j}}(N((1+\delta)\pi^{*}_{j}A-(\epsilon_{j}+\delta^{\prime})E_{j}))).$

Therefore there is an effective divisor $F_{j}\sim N(1+\delta)A$ on $X$ such
that $div(\tilde{s}_{j})=\pi^{*}_{j}F_{j}.$ Since $\tilde{s}_{j}\in
H^{0}(\tilde{X}_{j},\mathcal{O}_{\tilde{X}_{j}}(N(1+\delta)\pi^{*}_{j}A-N(\epsilon_{i}+\delta^{\prime})E_{j})),$
we have $\pi^{*}F_{j}\geq N(\epsilon_{i}+\delta^{\prime})E_{j}$ on
$\tilde{X}_{j}.$
$\tilde{F}_{1},\ldots,\tilde{F}_{j-1},div(\tilde{s}_{j}),\tilde{Y}_{j+1},\ldots,\tilde{Y}_{q}$
are in $\ell$-subgeneral position with index $\kappa$ on $\tilde{X}_{j}$
outside of $E_{j},$ and $\pi_{i}$ is an isomorphism above the complement of
$Y_{j},$ thus $F_{1},\ldots,F_{j-1},F_{j},Y_{j+1},\ldots,Y_{q}$ are in
$\ell$-subgeneral position with index $\kappa$ on $X_{j}$ outside of $Y_{j}.$
Since $Y_{j}$ is in $\ell$-subgeneral position with index $\kappa$ with
$F_{1},\ldots,F_{j-1},Y_{j+1},\ldots,Y_{q},$ it implies that
$F_{1},\ldots,F_{j-1},F_{j},Y_{j+1},\ldots,Y_{q}$ are in $\ell$-subgeneral
position with index $\kappa$ on $X.$ Thus, we obtain divisors
$F_{1},\ldots,F_{q}$ with the required properties. Hence, the claim is proved.

Since $\Delta\leq\frac{\ell-n+\kappa}{\kappa},$ as a special case of the
result in the proof of Theorem 1.7 in the above section, we can also get

(4.1) $\displaystyle\hskip
35.0pt\int_{0}^{2\pi}\sum^{q}_{i=1}\lambda_{F_{i}}(x)\frac{d\theta}{2\pi}\leq\left[\frac{\ell-n+\kappa}{\kappa}(n+1+\delta)\right]T_{f,N(1+\delta)A}(r)+O(1)$

on $X(k)\setminus Z$ where $Z$ is a proper Zariski-closed subset of $X.$ By
functoriality, additivity, and the fact that Weil functions of divisors are
bounded from below, for all $P\in\tilde{X}_{i}\setminus SuppE_{i},$

$\displaystyle\lambda_{F_{i}}(\pi_{i}(P))$ $\displaystyle=$
$\displaystyle\lambda_{\pi^{*}_{i}F_{i}}(P)+O(1)$ $\displaystyle\geq$
$\displaystyle
N(\epsilon_{i}+\delta^{\prime})\lambda_{E_{i}}(\pi_{i}(P))+O(1)$
$\displaystyle=$ $\displaystyle
N(\epsilon_{i}+\delta^{\prime})\lambda_{Y_{i}}(\pi_{i}(P))+O(1)$
$\displaystyle\geq$ $\displaystyle
N(\epsilon_{Y_{i}})\lambda_{Y_{i}}(\pi_{i}(P))+O(1).$

Together with (4.1),

$\int_{0}^{2\pi}\sum^{q}_{i=1}N\epsilon_{Y_{i}}\lambda_{Y_{i}}(x)\frac{d\theta}{2\pi}\leq\left[\frac{\ell-n+\kappa}{\kappa}(n+1+\delta)\right]T_{f,N(1+\delta)A}(r)+O(1).$

Then, by the choice of $\epsilon$, we have

$\sum^{q}_{j=1}\epsilon_{Y_{j}}(A)m_{f}(r,Y_{j})\leq_{exc}\left(\frac{(\ell-n+\kappa)(n+1)}{\kappa}+\epsilon\right)T_{f,A}(r).$

∎

##  5\. Schmidt’s subspace theorem

In this section, we give the counterpart in Diophantine approximation of our
main results. The standard notations in Schmidt’s subspace Theorem can be seen
in [4],[3],[16],[17]).

Let $k$ be a number field. Denote by $M_{k}$ the set of places of $k$ and by
$k_{v}$ the completion of $k$ for each $v\in M_{k}.$ Norms $||\cdot||_{v}$ on
$k$ are normalized so that

$||x||_{v}=|\sigma(x)|^{[k_{v}:\mathbb{R}]}\quad
or\quad||p||_{v}=p^{-[k_{v}:\mathbb{Q}_{p}]}$

if $v\in M_{k}$ is an Archimedean place corresponding to an embedding
$\sigma:k\to\mathbb{C}$ or a non-Archimedean place lying above the rational
prime $p$, respectively.

An $M_{k}$-constant is a collection $(c_{v})_{v\in M_{k}}$ of real constants
such that $c_{v}=0$ for all but finitely many $v$. Hights are logarithmic and
relative to the number field used as a base field which is always denoted by
$k.$ For $\mathbf{x}=(x_{0},\ldots,x_{n})\in k^{n+1},$ define

$||\mathbf{x}||_{v}:=\max\\{||x_{0}||_{v},\ldots,||x_{n}||_{v}\\},\quad v\in
M_{k}.$

The absolute logarithmic height of a point
$\mathbf{x}=[x_{0}:\cdots:x_{n}]\in\mathbb{P}^{n}(k)$ is defined by

$h(\mathbf{x}):=\sum_{v\in M_{k}}\log||\mathbf{x}||_{v}.$

For each $v\in M_{k},$ we can associate the local Weil functions
$\lambda_{Y,v}$ which have similar properties as the Weil function introduced
in Section 2.

Similar discussions as in Nevanlinna theory, one can easily obtain the
counterparts of Theorem 1.7, Theorem 1.5 for Schmidt’s subspace theorems in
Diophantine approximation. Thus we omit the details.

######  Theorem 5.1.

Let $X$ be a projective variety of dimension $n$ defined over a number field
$k.$ Let $S$ be a finite set of places of $k.$ For each $v\in S,$ let
$D_{1,v},\ldots,D_{q,v}$ be Cartier divisors of $X$, defined over $k$, and
with the distributive constant $\Delta$. Let $A$ be an ample Cartier divisor
on $X$. Then, for $\epsilon>0,$ there exists a Zariski-closed set $Z\subset X$
such that for all points $x\in X(k)\setminus Z$,

$\displaystyle\sum_{v\in
S}\sum^{q}_{j=1}\epsilon_{D_{j,v}}(A)\lambda_{D_{j,v},v}<(\Delta(n+1)+\epsilon))h_{A}(x).$

Theorem 5.1 generalizes Theorem 2.6 in [3].

######  Theorem 5.2.

Let $X$ be a projective variety of dimension $n$ defined over a number field
$k.$ Let $S$ be a finite set of places of $k.$ For each $v\in S,$ let
$Y_{1,v},\ldots,Y_{q,v}$ be closed subschemes of $X$, defined over $k$, and in
$\ell$-subgeneral position with index $\kappa$ in $X$, and $\ell\geq n.$ Let
$A$ be an ample Cartier divisor on $X$. Then, for $\epsilon>0,$ there exists a
Zariski-closed set $Z\subset X$ such that for all points $x\in X(k)\setminus
Z$,

$\displaystyle\sum_{v\in
S}\sum^{q}_{j=1}\epsilon_{Y_{j,v}}(A)\lambda_{Y_{j,v},v}<\left(\frac{(\ell-n+\kappa)(n+1)}{\kappa}+\epsilon\right)h_{A}(x).$

Theorem 5.2 gives a generalization of Theorem 1.3 in [4] and the Main Theorem
(Arithmetic Part) in [3].

### Acknowledgements

This work was supported by National Nature Science Funds of China (No.
11871260 and No. 12061041). The authors thank to the anonymous referee’s for
his or her suggestions to improve this paper.

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# Non-Uniform Interpolation in Integrated Gradients for Low-Latency
Explainable-AI ††thanks:  Accepted to appear in 56th IEEE International
Conference on Circuits and Systems (ISCAS 2023)

Ashwin Bhat, Arijit Raychowdhury<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Electrical and Computer
Engineering, Georgia Institute of Technology, Atlanta, GA, USA

###### Abstract

There has been a surge in Explainable-AI (XAI) methods that provide insights
into the workings of Deep Neural Network (DNN) models. Integrated Gradients
(IG) is a popular XAI algorithm that attributes relevance scores to input
features commensurate with their contribution to the model’s output. However,
it requires multiple forward & backward passes through the model. Thus,
compared to a single forward-pass inference, there is a significant
computational overhead to generate the explanation which hinders real-time
XAI. This work addresses the aforementioned issue by accelerating IG with a
hardware-aware algorithm optimization. We propose a novel non-uniform
interpolation scheme to compute the IG attribution scores which replaces the
baseline uniform interpolation. Our algorithm significantly reduces the total
interpolation steps required without adversely impacting convergence.
Experiments on the ImageNet dataset using a pre-trained InceptionV3 model
demonstrate 2.6-3.6$\times$ performance speedup on GPU systems for iso-
convergence. This includes the minimal 0.2-3.2% latency overhead introduced by
the pre-processing stage of computing the non-uniform interpolation step-
sizes.

###### Index Terms:

Explainable AI (XAI), Deep Neural Networks (DNN), Hardware-Aware Algorithm
Design, GPU systems

##  I Introduction

There has been a massive growth in the field of Machine Learning (ML) and
Artificial Intelligence (AI). However, the black-box nature of DNN models has
hindered its ubiquitous utilization [1]. Explainable-AI (XAI) provides
insights into the workings of these models to enable adoption in safety-
critical tasks [2] which require transparency and interpretability [3]. Within
the field of XAI, feature attribution methods generate an explanation by
scoring input features proportional to their contribution to the network’s
output [4]. For image based applications, these relevance scores are
visualized as a heatmap [5]. These post-hoc techniques can be applied to
existing pre-trained models [6]. Integrated Gradients (IG), a feature
attribution algorithm, has become popular thanks to its ease of
implementation, axiomatic theoretical underpinnings, and applicability to any
differentiable model [7].

IG accumulates gradients along a straight-line interpolation path between a
baseline and the input. A baseline is indicative of missingness or lack of
input [8]. For example, a black image is a commonly used baseline for vision
tasks. IG requires multiple forward (inference) and backward (gradient
backpropagation) passes through the model for each input. Thus, there is a
large computational overhead in generating the explanation (50-1000$\times$
slower) compared to just evaluating the model’s output which required a single
forward pass. As noted in this XAI deployment study [9], it is necessary to
reduce this overhead to enable real-time low-latency XAI. It is vital to
overcome the technical limitations of computing explanations quickly in
domains like smart healthcare [10], medicine [11], finance [12], and hardware
security [13] where IG is being utilized.

Several optimizations over baseline IG have been proposed to improve the
quality of the generated heatmaps. [8] proposes averaging the attributions
obtained by using several different baselines. Google’s XRAI [14] segments the
input into several regions and applies IG on each segment before stitching the
results together. Captum, a PyTorch based XAI library developed by Meta [15],
uses Noise Tunnel which averages the IG attributions over several noisy copies
of the original input [16]. Despite using baseline IG multiple times in their
pipeline, none of these algorithms attempt to reduce its computational
overhead. Thus, they stand to gain significant performance benefits from an IG
implementation optimized for low-latency on the underlying hardware platform.

Figure 1: Overview of Feature Attribution using the Integrated Gradients (IG)
algorithm (a) Inputs to the algorithm: Image ($x$) and Baseline ($x^{\prime}$)
(b) Interpolated images along straight line path between Baseline ($k=0$) and
Image ($k=m$) (c) Visualization of accumulated gradients to highlight input
features relevant for classifying image as American Chameleon (target class)

All the currently known implementations of IG use uniform interpolation
between baseline and input to approximate the continuous IG integral [Eq. 1]
as Riemann-sum [Eq. 2]. In this work, we propose a non-uniform interpolation
scheme to reduce the discretization steps required while maintaining similar
convergence accuracy. We first identify regions in the IG path with higher
information content. By having smaller step-size in these regions and larger
step-size outside of them, the overall compute overhead of IG is reduced.

In summary, this work makes the following contributions:

  * •

To the best of our knowledge, our work is the first of its kind to design and
employ a non-uniform interpolation scheme along the IG path to compute the
discrete Riemann-sum approximation of the IG integral.

  * •

Based on the experimental observations, we propose and justify the use of
change in classification probability along the IG path as an information
content metric to determine the non-uniform discretization step-size.

  * •

We demonstrate the performance improvement achieved on GPU systems and
quantify the latency overhead introduced by our algorithm compared to the
baseline IG which uses a uniform interpolation scheme.

##  II Integrated Gradients (IG)

The Integrated Gradients (IG) algorithm is a feature attribution method [Fig.
1]. Formally, for a given input ($x$) and a model function ($f$), a feature
attribution method assigns a relevance score ($\phi_{i}(x,f)$) to the ith
input feature. The score is a measure of that feature’s contribution to the
model’s output.

The simplest way to assign a relevance score is to evaluate the gradient of
the model’s output with respect to the input feature. A large gradient value
implies that small changes in the feature value produce a large change in the
model’s output, thereby, indicating higher relevance. However, gradients are a
local explanation method that can suffer from saturation effects. Path
attribution methods (PAM) overcome this issue by accumulating gradients along
a path between a baseline and the actual input [17]. These are mathematically
motivated and satisfy desirable properties such as completeness and
sensitivity. IG is a subset of PAM which considers a straight-line path
between the baseline ($x^{\prime}$) and the input ($x$) as shown in Eq. 1. The
commonly used baselines for computer vision applications include black, white,
or random noise images [8]. They represent the notion of missingness or lack
of any input.

$\phi_{i}\left(f,x,x^{\prime}\right)=(x-x^{\prime})\times\int_{0}^{1}\frac{\partial
f(x^{\prime}+\alpha(x-x^{\prime}))}{\partial x_{i}}\,d\alpha$ (1)

In Eq. 1, $f$ is the model function and $\alpha$ is the interpolation constant
along the straight-line path. In practice, to evaluate IG attributions, the
continuous integral is approximated as a Riemann sum with $m+1$ uniform steps
as shown in Eq.2.

$\phi_{i}\left(f,x,x^{\prime}\right)=(x-x^{\prime})\times\frac{1}{m}\sum_{k=0}^{m}\frac{\partial
f(x^{\prime}+\frac{k}{m}(x-x^{\prime}))}{\partial x_{i}}$ (2)

The number of steps ($m$) typically ranges from 200 to 1000 [18]. Its value is
chosen based on the convergence metric ($\delta$) which is defined (Eq. 3)
using the completeness property [17] satisfied by the continuous integral
formulation of IG.

$\delta=\left|\sum_{i}\phi_{i}(f,x,x^{\prime})-[f(x)-f(x^{\prime})]\right|$
(3)

From Eq. 2, we observe that for every input, the IG algorithm creates several
interpolated versions [Fig. 1(b)]. It then computes the gradient of the
model’s output with respect to input features for each one. This step requires
a forward and a backward pass through the model. These gradients are then
aggregated to assign the overall attribution score [Fig. 1(c))].

##  III Methodology

Background. The run-time latency of IG depends on the number of interpolation
steps [Fig. 2(a)]. More steps yield better convergence $\delta$ [Fig. 2(b)] at
the cost of higher latency.

Figure 2: (a) Latency increases with increase in number of interpolation steps
[values normalized relative to the latency for m=1] (b) Decreasing convergence
$\delta$ with increasing steps implies better convergence at reduced
interpolation step-size. Thus, there is a latency penalty for better
convergence.

Objective. The goal of this work is to reduce the compute overhead of IG for
generating explanations. The typically used range of values for the number of
steps lies beyond the knee-point of the latency v/s step-count graph [Fig.
2(a)]. Thus, for lower latency, the number of interpolation steps must be
decreased without compromising convergence [Fig. 2(b)].

Figure 3: Variation along the IG path of (a) Distinguishability of the image
from its interpolated copies (b) Classification probability of the model (c)
Contribution to the convergence term based on the relative gradient magnitude.
Thus, a small region along the IG path contains most of the information.

Observation. Along the interpolation path, close to the baseline (small values
of $\alpha$), the actual input image is unrecognizable from its interpolated
version. However, after a certain $\alpha$, the input can be identified.
Beyond this threshold, as we move towards the input along the IG path, the
brightness of the interpolated image increases. Intuitively, for a human
observer, the change in classification confidence is not uniform along the IG
path as seen in Fig. 3(a).

We test this intuition on the model. The classification probability of the
model changes sharply as we increase the interpolation constant $\alpha$ [Fig.
3(b)] along the IG path. At $\alpha=0.25$, the classification probability
(0.83) is $>$90% of its final value (0.89) for the input image ($\alpha=1$).
Thus, the model’s confidence about the prediction is built over a small
interval along the IG path with minimal change outside of it.

The IG algorithm accumulates the gradient of the classification probability
with respect to input features for each interpolated image. Thus, the change
in classification probability can be used as a metric for information content
along the IG path. In regions of large change, the gradient values are also
larger [Fig. 3(c)] and contribute more to the overall IG attribution.

Proposed Method. We propose a non-uniform interpolation scheme to replace the
baseline uniform interpolation [Fig. 4(a)] along the IG path. Specifically,
the IG path is divided into multiple intervals and uniform interpolation is
performed within each interval using a different step-size. Based on the
earlier observation [Fig. 3], we can use a smaller step size in the regions
where there is a large change in the classification probability. Outside of
such regions, a larger discretization step size can be used. Overall, the
total steps are non-uniformly distributed along the IG path with a bias
towards regions with higher information content.

Figure 4: (a) Uniform interpolation with constant step size for the entire IG
path (b) Non-Uniform Interpolation with small uniform step size in the region
of large change and large uniform step size in the region of small change.

Algorithm. The proposed algorithm comprises two stages. The first stage takes
the number of intervals ($n_{int}$) as a parametric input to divide the IG
path into $n_{int}$ equal pieces. The classification probability of the
interpolated images at the interval boundaries is evaluated to calculate its
normalized change in each interval [$\Delta f(x_{int})$]. The total number of
steps ($m$) is then distributed across each interval proportional to the
square root of the change [$\sqrt{|\Delta f(x_{int})|}$]. We observed that
linear dependence ($m_{int}\propto\Delta$) allotted negligible discretization
steps to regions with small change. Hence, we use
$m_{int}\propto\sqrt{\Delta}$ to attenuate the bias towards intervals of large
change. In the second stage, we perform uniform IG within each interval with
the respective step count. The IG attributions of different intervals are then
summed up to determine the overall IG attribution. The proposed algorithm is,
therefore, uniform-in-intervals but non-uniform overall along IG path. Fig.
4(b) illustrates this for $n_{int}=4$.

##  IV Results

Experimental Setup. The proposed method can directly replace the baseline IG
algorithm and be applied to any differentiable model. To demonstrate its
efficacy, we test it on the ImageNet dataset [19] using a pre-trained
InceptionV3 model [20]. We consider our baseline to be the existing IG
implementation that employs uniform interpolation. We compare it against our
proposed non-uniform interpolation algorithm and vary the number of intervals
($n_{int}$) parameter.

Figure 5: For different IG interpolation schemes, the variation of (a)
Convergence delta ($\delta$) as we increase the total interpolation steps
($m$) (b) Total interpolation steps required for reaching the convergence
threshold ($\delta_{th}$).

Convergence. In Fig. 5(a), we demonstrate the effect of the total number of
steps on the convergence $\delta$ for different interpolation schemes. For any
given number of steps ($m$), our proposed algorithm achieves better
convergence $\delta$ compared to baseline. Thus, for iso-convergence, the
proposed algorithm is able to reduce the total steps needed.

In practice, the total interpolation steps for IG are determined by fixing a
threshold tolerance for the convergence. For example, in Fig. 5(a), this
threshold ($\delta_{th}$) is set to 0.015. The total number of steps is then
chosen such that the convergence $\delta$ lies below the threshold [Fig.
5(a)]. We vary $\delta_{th}$ and determine the number of steps required for
different interpolation schemes [Fig. 5(b)] to meet the convergence criterion.
For all $\delta_{th}$ values, our proposed algorithm requires fewer steps and
outperforms the baseline. Increasing the number of intervals further reduces
the steps required for convergence. The benefits are more pronounced at
smaller $\delta_{th}$ values. For $\delta_{th}=0.02$, we observe a 2.7$\times$
reduction while for $\delta_{th}=0.005$, we observe a 3.6$\times$ reduction in
the total steps required for convergence.

We further observe that increasing the number of intervals ($n_{int}$) up to a
certain point reduces $\delta$. Further increasing $n_{int}$ causes $\delta$
to increase since certain intervals are allotted negligible discretization
steps which negatively impact convergence. Consequently, it increases the
number of steps required to meet $\delta_{th}$. We observe that $n_{int}>8$
manifests this issue.

Figure 6: For each IG interpolation scheme, the variation of (a) normalized
latency to meet the convergence threshold ($\delta_{th}$) (b) latency overhead
(% of total latency) of the first stage of the non-uniform interpolation
algorithm.

Latency. Since our method is applicable to any differentiable model and the
performance benefit is not specific to a hardware architecture, we report
normalized latency values. The latency is normalized relative to the algorithm
configuration which yields the smallest run-time latency. In our experiments,
we measure the run-time latency for running IG on InceptionV3 model with
batch-size of 16 on a NVIDIA TITAN Xp GPU using the PyTorch benchmark
profiler. The profiler supports CUDA (excludes the overhead of thread
synchronization), performs an initial warm-up, and averages over multiple runs
to determine accurate execution latency.

The normalized latency [Fig. 6(a)] depends on the number of interpolation
steps ($m$) which in turn depends on the convergence threshold
($\delta_{th}$). We make two observations. First, the latency increases as we
decrease $\delta_{th}$ because of an increase in $m$. However, the relative
increase in latency as we reduce $\delta_{th}$ from 0.02 to 0.005 is higher
for baseline (3.3$\times$) when compared to non-uniform interpolation
(2.3-2.5$\times$). Second, in terms of performance, our proposed non-uniform
interpolation scheme outperforms the baseline across all $\delta_{th}$ values.
The latency reduces as we increase $n_{int}$ yielding higher performance
benefits. For $\delta_{th}=0.02$, we achieve a 2.6$\times$ and for
$\delta_{th}=0.005$, we achieve a 3.6$\times$ latency reduction when compared
to the baseline.

Overhead. Determining the number of steps for each interval is the overhead of
our proposed scheme. The memory overhead is minimal since we only store the
classification probability of $n_{int}+1$ interpolated images to determine the
step-sizes. The latency overhead is measured as a fraction of the total
latency of running the IG algorithm [Fig. 6(b)]. The overhead varies between
0.2-3.2% of the total latency.

We observe that the absolute value of the latency overhead depends only on the
parameter $n_{int}$. This is because we run the inference pass through the
network $n_{int}+1$ times to determine the classification probability change
in each interval and distribute the total steps commensurately. Thus, the
absolute overhead increases as we increase $n_{int}$. However, the relative
value of the overhead depends on both $n_{int}$ and $\delta_{th}$ [Fig. 6(b)]
because these parameters affect the total steps required and hence the overall
latency of non-uniform IG.

##  V Discussion & Related Work

With XAI research being in its infancy, there is a paucity of work that
focuses on improving performance (low-latency, high throughput, real-time)
either through (a) specialized hardware architectures or (b) algorithmic
optimizations with a hardware-aware design approach (this work).

Pan et al.[21] accelerates model-distillation based explanation on TPU or
systolic-array hardware substrate. This method is unsuitable for low-latency
explanation generation since it requires training a new model which locally
mimics the input-output behavior of the black-box model for each input. In
this work, we focus on a post-hoc explanation method that is directly
applicable to any off-the-shelf differentiable model. Bhat et al.[22]
accelerates gradient based heatmap visualization on FPGA platform. However,
the implemented feature attribution algorithms suffer from local saturation
effects. In this work, we focus on IG which overcomes this issue via path-
attribution. Although we demonstrate our results on GPU systems, our algorithm
is agnostic to the underlying hardware.

Sotoudeh et al.[18] proposes exact computation of the IG integral but achieves
smaller performance gains (upto 1.7$\times$) compared to this work (upto
3.6$\times$). Kapishnikov et al.[23] avoids high-loss regions by updating a
subset of features with low gradient magnitude at each point along the path.
However, the next step is dynamically determined which limits the performance
on GPUs as batch-size is restricted to 1. In our work, we design a static
processing stage to pre-determine the discretization step size and leverage
batching on GPU systems. Rahman et al.[24] modifies the path by performing a
local gradient ascent around each uniform interpolation point in the IG path.
This magnifies the compute overhead of generating the explanation. Although
both methods modify the baseline IG path to improve the quality of attribution
heatmaps, they require more steps thereby incurring a performance overhead.
Our work modifies the IG path to improve performance while maintaining iso-
convergence with baseline IG.

##  VI Conclusion

In this paper, a novel Non-Uniform Interpolation scheme for computing the
Integrated Gradients attribution is presented. Our methodology is motivated by
the observations we made on the baseline uniform interpolation. Regions of
high-information content along the straight-line IG path are identified using
the change in classification probability. Utilizing classification probability
as an information metric is justified based on its correlation with gradient
magnitude and its contribution to the overall IG attribution. The proposed
algorithm distributes the steps among intervals with a bias towards ones with
higher information content. Compared to the baseline, our algorithm meets iso-
convergence thresholds with fewer total steps. We quantify the performance
benefit on GPU systems using pre-trained models on the ImageNet dataset. Our
experiments show that we can achieve a speed-up of 2.6-3.6$\mathbf{\times}$ at
a very low latency overhead of 0.2-3.2%. In summary, our hardware-aware
algorithm design enables low-latency real-time explainable AI.

##  VII Acknowledgement

This work was supported by Semiconductor Research Corporation (SRC) AI
Hardware (AIHW) Task 2969.001 titled EXPERT: EXPlainable-AI through Efficient
hardware design in EmeRging Technologies.

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