Balanced Weighing Matrices

Hadi KharaghaniDepartment of Mathematics and Computer Science, University of Lethbridge,
Lethbridge, Alberta, T1K 3M4, Canada.
Thomas PenderDepartment of Mathematics and Computer Science, University of Lethbridge,
Lethbridge, Alberta, T1K 3M4, Canada.
Sho SudaDepartment of Mathematics,  National Defense Academy of Japan, Yokosuka, Kanagawa 239-8686, Japan.

A unified approach to the construction of weighing matrices and certain symmetric designs is presented.
Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a
for each positive integer $m$. The case of $n=p+1$ reduces to the balanced weighing matrices with classical parameters
The equivalence with certain classes of association schemes is discussed in detail.

§ INTRODUCTION

A weighing matrix of order $n$ and weight $p$, denoted $W(n,p)$, is a $(0,\pm 1)$-matrix $W$ of order $n$ such that $WW^t = pI_n$. The special cases in which $n=p+1$ is called a conference matrix and $n=p$ is a Hadamard matrix. A weighing matrix $W(n,p)$ is said to be balanced if, upon setting each non-zero entry to unity, it provides the incidence matrix of a symmetric $(n,p,\lambda)$ balanced incomplete block design.

Balanced  weighing matrices with classical parameters include weighing matrices
$W\left(\frac{p^{m+2}-1}{p-1},p^{m+1}\right)$,  for each positive integer $m$. Upon squaring all the entries it reduces to the symmetric

Most of the existing constructive methods for weighing matrices relate to orders which are multiples of four. Indeed,  it is conjectured that there is a $W(4n,k)$  for positive integers $n$ and $1\le k \le 4n$, see <cit.>. In contrast to that case, the progress for the remaining cases of $n\equiv 1,2,3\pmod 4$ have been quite limited.

In this paper,  a new method is presented that is suitable for the remaining three cases. The unifying method of construction establishes a link between the construction and applications of weighing matrices,  and the construction of certain symmetric designs. The outcome is the existence of new weighing matrices,  and a new method of construction for balanced weighing matrices and balanced generalized weighing matrices.

The basic assumption on which most of the results found herein depend is that the weight $p$ in the $W(n,p)$ is a prime power. With the restricted value of the weight $p$,  many powerful tools such as balanced generalized weighing  matrices and  orthogonal arrays  will be available and instrumental in the development of  some new techniques presented in this paper.

The main references for the paper are <cit.>.

§ PRELIMINARIES

§.§ BGWs over $\mathbb{Z}_n$

Let $G$ be a multiplicatively written group, and let $W=[w_{ij}]$ be a ($0,{G}$)-matrix of order $v$. The matrix $W$ is a balanced generalized weighing matrix over $G$ with parameters $(v,k,\lambda)$, denoted $BGW(v,k,\lambda;G)$, if each row of $W$ contains exactly $k$ nonzero entries, and if the multiset $\{w_{ik}w_{jk}^{-1}\mid 1\leq k\leq v, w_{ik} \neq 0 \neq w_{j,k}\}$ contains exactly $\lambda/|G|$ copies of every element of $G$, for any distinct $i,j\in\{1,\ldots,v\}$.

A  balanced weighing matrix of order $v$ and weight $k$ is a

Every conference matrix of order $v$ is a $BGW(v,v-1,v-2;\{1,-1\})$.

For the prime power $q$, the $BGW(\frac{q^{m+1}-1}{q-1},q^m,q^{m-1}(q-1))$ over the cyclic group $G$ whose order divides $q-1$, for each positive integer $m$, is the largest class of known balanced generalized weighing matrices with the classical parameters, see Theorem 10.2.5 of  <cit.>. It follows that for any odd $q$ there is a balanced weighing matrix with classical parameters.

A weighing matrix $W(n,p)$ is said to be in normal form  if  it has the block configuration

\begin{bmatrix} \mathbf{0} & R \\ \mathbf{1} & D  \end{bmatrix},

for some matrices $(0,\pm 1)$-matrices $R$ and $D$, where $\mathbf{0}$ and $\mathbf{1}$  are the $(n-p)\times 1$ column vector with all entries $0$ and the $p\times 1$ column vector of all entries $1$, respectively.  We call $R$ and $D$, respectively, the residual and the derived parts of the weighing matrix. Note that $RR^t = pI_{n-p}$, and $DD^t = pI_{p} - J_p$, and $RD^t = DR^t = 0$.  By permuting and negating some rows, if necessary, every weighing matrix can be assumed to be in normal form.

Two well-known results will be used throughout the paper. The first is a result related to the existence of balanced generalized weighing matrices <cit.>.

For every prime power $p$ there is a $BGW(p+1,p,p-1)$ over a cyclic group $\mathbb{Z}_k$ for any $k$ dividing $p-1$.

We introduce the following terminology.

A weighing matrix $W(p+1,p)$ is called a seed weighing matrix, and a $BGW(p+1,p,p-1)$ over a cyclic group whose order divides $p-1$ is called a seed balanced generalized weighing matrix.

The use  of a particular family of orthogonal arrays whose existence is predicated on the existence of prime powers is the key to the method presented in this paper. For details and definitions concerning orthogonal arrays see <cit.>.

For the prime power $p$ and the positive integer $m$,  there is an array ${\bf O}$ in $p$ symbols of  dimensions
$p^{m+1}\times (\frac{p^{m+1}-1}{p-1})$
for which any two distinct rows agree in $\frac{p^m-1}{p-1}$ columns.

§ WEIGHING MATRICES

The first result relates to weighing matrices $W(n,p)$ for $p$ a prime power.

If there is a $W(n,p)$, where $p$ is a prime power, then there is a
for each positive integer $m$.

$$W=\begin{bmatrix} \mathbf{0} & R \\ \mathbf{1} & D  \end{bmatrix}$$
be the assumed $W(n,p)$.
Let $\mathcal{W}$ be a $W\left(\frac{p^{m+1}-1}{p-1},p^{m}\right)$,  and consider the matrix

$$\mathcal{R}=\mathcal{W}\otimes {R}.$$

It is easy to see that the rows of the $(0,\pm 1)$-matrix $\mathcal{R}$
are mutually orthogonal. Let $\bf {O}$ be the array in Proposition <ref> and let $\mathcal{D}$ be the matrix obtained from $\bf {O}$ by replacing the $p$ symbols by the $p$ rows of the matrix $\mathbf{D}$ in any order. It follows that

The matrix
$$\begin{bmatrix} \mathbf{0} & \mathcal{R} \\ \mathbf{1} & \mathcal {D}  \end{bmatrix}$$

is a
for each positive integer $m$.

The existence of $W(43,25)$ is not known. In order to illustrate the construction above a $W(43,25)$ is constructed next.

We Start with a $W(8,5)$

\[
\left[\begin{array}{c|ccccccc}
1&1&1&1 & 1 & 0&0 &0\\
1&1&-&-  &  0 &1&0&0\\
1&-&1&-  & 0 &0 & 1&0\\
1&-&-&1  &0 &0 &0 &1\\
1 & 0&0 &0 &-&-&-&-\\\hline
0 &1&0&0&-&-&1&1 \\
0 &0 & 1&0&-&1&-&1\\
0 &0 &0 &1&-&1&1&-
\end{array}\right].
\]

\begin{bmatrix}
1&-&-&0&1&0&0 \\
\end{bmatrix},

for which $DD^t = 5I - J$, and

R =
\begin{bmatrix}
\end{bmatrix},

for which $RR^t = 5I$.

Replacing the five symbols in the $OA(5,2)$ shown in the Appendix with  the five rows of $R$ (in any order) the matrix  $\mathcal D$ is obtained. It is seen that $\D\D^t = 25{I}-J$.

If $W$ is any normal $W(6,5)$, then

\[
\begin{array}{cc}
\R = W \otimes R=
\begin{bmatrix}
\end{bmatrix}\otimes R=
\end{bmatrix}\end{array}\]

where $\bar{R}=-R$,. Then $\R\R^t = 25I$, and
moreover,  $\D\R^t = 0$.
Consequently, the matrix

\left[\begin{array}{c|c}
\mathbf{1} & \D \\\hline \mathbf{0} & \R
\end{array}\right]

forms a $W(43,25)$, which is shown in the Appendix.

The existence of weighing matrices of small orders has been verified through a number of papers. Most of the weighing matrices constructed by Theorem <ref> are new. Table 1 below shows the orders of the constructed weighing matrices for small order.

We will use the superscripts $(v,k)^\dag$ and $(v,k)^\ddag$ to denote a new order/weight pair and balancedness over $\mathbb{Z}_2$, respectively.
[c]| c | c |
Consequential order/weight pairs for $v \leq 1000$.

Seed $(v,k)$     Succident $(v',k')$

2cContinuation of Table <ref>

Seed $(v,k)$     Succident $(v',k')$

$(7,4)$     $(31,16)^\ddag$, $(127, 64)$, $(511, 256)$
$(6,5)^\ddag$     $(31, 25)^\ddag$, $(156, 125)^\ddag$, $(781, 625)^\ddag$
$(8,5)$     $(43, 25)^\dag$, $(218, 125)$
$(10,5)$     $(55, 25)$, $(280, 125)$
$(12,5)$     $(67, 25)$, $(342, 125)$
$(8,7)^\ddag$     $(57, 49)^\ddag$, $(400, 343)^\ddag$
$(12,7)$     $(89, 49)^\dag$, $(628, 343)$
$(16,7)$     $(121, 49)$, $(856, 343)$
$(20,7)$     $(153, 49)$
$(10,8)$     $(82, 64)$, $(658, 512)$
$(12,8)$     $(100, 64)$, $(804, 512)$
$(14,8)$     $(118, 64)$, $(950, 512)$
$(16,3)$     $(69,9)$, $(196,27)$, $(601, 81)$
$(16,4)$     $(76, 16)$, $(316, 64)$
$(16,5)$     $(91,25)$, $(466,125)$
$(16,7)$     $(121, 49)$, $(856, 343)$
$(16,8)$     $(136, 64)$
$(16,11)$     $(181, 121)^\dag$
$(16,13)$     $(211, 169)^\dag$
$(10,9)^\ddag$     $(91, 81)^\ddag$, $(820, 729)^\ddag$
$(12,9)$     $(111, 81)^\dag$
$(13,9)$     $(121,81)^\ddag$
$(14,9)$     $(131, 81)$
$(16,9)$     $(151, 81)$
$(14,13)^\ddag$     $(183, 169)^\ddag$
$(16,13)$     $(211, 169)^\dag$
$(18,13)$     $(239, 169)^\dag$
$(19,9)^\ddag$     $(181, 81)^\ddag$
$(20,13)$     $(267, 169)^\dag$

§ AN APPLICATION OF WEIGHING MATRICES TO SYMMETRIC DESIGNS

The balanced incomplete block design $BIBD(v,b,r,k,\lambda)$ is said to have a twin mate  if the complement design is also a $BIBD(v,b,r,k,\lambda)$.

The $BIBD(6,10,5,3,2)$ with the incidence matrix

\[
\begin{array}{cccccccccc}
{1}&0& {1}& {1}& {1}&0&0&0& {1}&0\\
0& {1}&0& {1}& {1}& {1}&0&0&0& {1}\\
{1}& {1}& {1}&0&0&0& {1}&0&0& {1}\\
{1}&0&0& {1}&0& {1}& {1}& {1}&0&0\\
0& {1}&0&0& {1}&0& {1}& {1}& {1}&0\\
0&0& {1}&0&0& {1}&0& {1}& {1}& {1}
\end{array}
\right]
\]
has a twin mate with the incidence matrix

\[
\begin{array}{cccccccccc}
{0}&  {1}&  {0}&  {0}&  {0}&  {1}&  {1}&  {1}&  {0}&  {1}\\
{1}&  {0}&  {1}&  {0}&  {0}&  {0}&  {1}&  {1}&  {1}&  {0}\\
{0}&  {0}&  {0}&  {1}&  {1}&  {1}&  {0}&  {1}&  {1}&  {0}\\
{0}&  {1}&  {1}&  {0}&  {1}&  {0}&  {0}&  {0}&  {1}&  {1}\\
{1}&  {0}&  {1}&  {1}&  {0}&  {1}&  {0}&  {0}&  {0}&  {1}\\
{1}&  {1}&  {0}&  {1}&  {1}&  {0}&  {1}&  {0}&  {0}&  {0}
\end{array}
\right].\]

The construction method in the following theorem, perhaps surprisingly, is similar to the that of Theorem <ref>.

For the prime power $p$,  the residual design
$$R=BIBD\left(p+1,2p,p,\frac{p+1}{2},\frac{p-1}{2}\right) $$
and the derived design
$$D=BIBD\left(p,2p,p-1,\frac{p-1}{2},\frac{p-3}{2}\right) $$
of a symmetric
are the seed designs  used to construct a symmetric

for any positive integer $m$.

Let $W$ be a balanced $W(\frac{p^{m+1}}{p-1},p^m)$ and consider $\mathcal{R}=W\otimes R$. Let $W=W^+-W^-$, where $W^+$ and $W^-$ are  $(0,1)$-matrices. Let $R'$ be the twin mate to $R$, and consider $\mathcal{R}=W^+\otimes R+W^-\otimes R'$. It is routine, but tedious, to show that $\mathcal{R}$ is a quasi-residual

$$BIBD\left((p+1)(\frac{p^{m+1}-1}{p-1}), 2p(\frac{p^{m+1}-1}{p-1}),p^{m+1},p^m(\frac{p+1}{2}),p^m(\frac{p-1}{2})\right).$$

Let $\bf {O}$ be the array in Proposition <ref>,  and let $\mathcal{D}$ be the matrix obtained from $\bf {O}$ by replacing the $p$ symbols by the $p$ rows of the matrix ${D}$ in any order.

By carefully counting, it follows that

The last step is to add the appropriate column  so that

\begin{align*}&\begin{bmatrix} \mathbf{0} & \mathcal{R} \\ \mathbf{1} & \mathcal {D}  \end{bmatrix}
\end{align*}

is the required matrix.

The construction above makes use of the balanced weighing matrix $W$. Although the existence of such a balanced weighing matrix is known, by a method similar to the above construction and the use of the seed $W(p+1,p)$ weighing matrix, a balanced weighing matrix $W(\frac{p^{m+1}}{p-1},p^m)$ is reconstructed here.  The parameters of the design  are not new. See <cit.> in which a generalized Hadamard matrix is used instead of an orthogonal array.

§ BALANCED GENERALIZED WEIGHING MATRICES

One of the  most frequently used tools in the construction of many new symmetric designs  is the classical balanced generalized weighing matrices. We refer the reader to <cit.> for the most comprehensive coverage of the subject.

In this section, assuming the existence  of seed balanced generalized weighing matrices, the unifying technique used in previous sections is employed to offer a new construction for the classical BGW matrices.

For the prime power $p$, let $W$  be a $BGW(p+1,p,p-1)$ over a cyclic group $G$ of order  $p-1$. Then there is a
$$BGW\left(\frac{p^{m+1}-1}{p-1}, p^m,p^{m-1}(p-1)\right)$$
over $G$ for any positive integer $m$.

Without loss of generality  the matrix $W$ is assumed to be in the normal form

$$W=\begin{bmatrix} \mathbf{0} & \mathbf{1}^t \\ \mathbf{1} & \mathcal {D}  \end{bmatrix}.$$

As in previous two cases, let $\bf {O}$ be the array in Proposition <ref>, and let $\mathcal{D}$ be the matrix obtained from $\bf {O}$ by replacing the $p$ symbols by the $p$ rows of the matrix ${D}$ in any order.
The $(0,G)$-matrix $\mathcal{D}$ has dimensions  $p^{m+1}\times p\left(\frac{p^{m+1}-1}{p-1}\right)$ with $p^{m+1}-1$ ones in each row. Noting that any two distinct rows of $\bf O$ have  exactly $\frac{p^m-1}{p-1}$ in the same column, there are $(\frac{p^m-1}{p-1})(p-1)=p^m-1$ identity elements of $G$ upon taking the conjugate inner product between the rows. The remaining $p^{m+1}$ columns in the two rows contain different rows of $W$ and so there are $p^{m}$ times of each element of the group except for the identity element which has one less.
$\mathcal{R}=W\otimes \mathbf{1}^t$.  Then

$$\begin{bmatrix} \mathbf{0} & \mathcal{R} \\ \mathbf{1} & \mathcal{D}  \end{bmatrix}$$

is a  BGW
$$\left(\frac{p^{m+1}-1}{p-1}, p^m,p^{m-1}(p-1)\right)$$
over $G$.

Let $p=3$ and consider

\[
W=\begin{bmatrix} 0 & 1 & 1 &1\\1& 0 & 1 & -\\1 & - & 0 & 1\\1 & 1 & - & 0\end{bmatrix}.
\]

By applying Theorem <ref>, the balanced weighing matrices
$$BGW({(3^{m+1}-1)/2},3^m, {2\cdot 3^{m-1}})$$
over $\mathbb{Z}_2$
are constructed.

The above construction is  further illuminated in an example for $p=5$ in the Appendix.

§ ASSOCIATION SCHEMES

The main reference for this section is <cit.>.
Let $d$ be a positive integer.
Let $X$ be a finite set and $R_i$ ($i\in\{0,1,\ldots,d\}$) a nonempty subset of $X\times X$.

The adjacency matrix $A_i$ of the graph with vertex set $X$ and edge set $R_i$ is a $(0,1)$-matrix indexed by the elements of $X$ such that $(A_i)_{xy}=1$ if $(x,y)\in R_i$, and $(A_i)_{xy}=0$ otherwise.

A commutative association scheme with $d$ classes is a pair $(X,\{R_i\}_{i=0}^d)$ satisfying the following:

* $A_0=I_{|X|}$.

* $\sum_{i=0}^d A_i = J_{|X|}$.

* $A_i^t\in\{A_1,\ldots,A_d\}$ for any $i\in\{1,\ldots,d\}$.

* For all $i$ and $j$, $A_i A_j$ is a linear combination of $A_0,A_1,\ldots,A_d$.

* $A_i A_j=A_j A_i$ for all $i,j$.

We will also refer to $(0,1)$-matrices $A_0,A_1,\ldots,A_d$ satisfying 1-5 as a commutative association scheme.

If an association scheme satisfies $A_i^t=A_i$,  for any $i\in\{1,\ldots,d\}$, then the association scheme is said to be symmetric, in which case it is necessarily commutative.

The vector space spanned by the collection $\{A_i\}$ forms a commutative algebra, denoted by $\mathcal{A}$, and is called the Bose-Mesner algebra.

There exists a basis of $\mathcal{A}$ consisting of primitive idempotents, say, $E_0=(1/|X|)J_{|X|},E_1,\ldots,E_d$.
Note that the collection $\{E_i\}$ are projections onto maximal common eigenspaces of $A_0,A_1,\ldots,A_d$.
Since  $\{A_0,A_1,\ldots,A_d\}$ and $\{E_0,E_1,\ldots,E_d\}$ are two bases in $\mathcal{A}$, there exist the change-of-bases matrices $P=(p_{ij})_{i,j=0}^d$, $Q=(q_{ij})_{i,j=0}^d$ so that

\begin{align*}
A_j=\sum_{i=0}^d p_{ij}E_i,\quad E_j=\frac{1}{|X|}\sum_{i=0}^d q_{ij}A_i.
\end{align*}

The matrices $P$ and $Q$ are said to be the first and second eigenmatrices, respectively.

Let $\mathbb{Z}_n=\{\bar{0},\bar{1},\ldots,\overline{n-1}\}$ be an additive cyclic group of order $n$, and let
$W=(w_{ij})$ be a balanced generalized weighing matrix over $\mathbb{Z}_n$ with parameters $(v,k,\lambda)$.
Define the $v\times v$ $(0,1)$-matrices $B_{i}$ by

\begin{align*}
W=\sum_{i=0}^{n-1} \bar{i} B_i \in \text{Mat}_v(\mathbb{C}[\mathbb{Z}_n]),
\end{align*}

where $\text{Mat}_v(\mathbb{C}[\mathbb{Z}_n])$ is the matrix algebra of $v \times v$ matrices over the group ring $\mathbb{C}[\mathbb{Z}_n]$.

Let $W^*$ denote the matrix with $(i,j)$-entry given by $-w_{j i}$ if $w_{ij}\in \mathbb{Z}_n$,  and $0$ if $w_{i j}=0$.
If $W$ is a balanced generalized weighing matrix, then so is $W^*$.
Thus, we have

\begin{align*}
\sum_{i,j=0}^{n-1} \overline{i-j}B_{i} B_{j}^t=\sum_{i,j=1}^n \overline{i-j}B_{j}^t B_{i}=k I_{v}+\frac{\lambda}{n}\mathbb{Z}_n(J_v-I_v).
\end{align*}

Comparing entries with $\bar{h}$, for each $\bar{h}\in \mathbb{Z}_n$, yields

\begin{align}
\sum_{i=0}^{n-1} B_i B_i^t&=\sum_{i=0}^{n-1} B_i^t B_i=k I_{v}+\frac{\lambda}{n}(J_v-I_v),\label{eq:1}\\
\sum_{i=0}^{n-1} B_i B_{i-h}^t&=\sum_{i=0}^{n-1} B_{i-h}^t B_i=\frac{\lambda}{n}(J_v-I_v) \text{ for }h\neq 0.\label{eq:2}
\end{align}

Conversely, disjoint $(0,1)$-matrices $B_i$ ($i\in\{0,1,\ldots,n-1\}$) satisfying (<ref>) and (<ref>) give rise to a $BGW(v,k,\lambda)$ $W=\sum_{i=0}^{n-1} \bar{i} B_i$ over the cyclic group $\mathbb{Z}_n$ of order $n$.

Let $P=\text{circ}(010\cdots0)$ be a cyclic matrix of order $n$.
Define the adjacency matrices as follows:

\begin{align*}
P^i\otimes I_v&0\\
0& P^i\otimes I_v
\end{bmatrix},\quad (0\leq i \leq n-1) ,\\
J_n \otimes (J_v-I_v)&0\\
0&J_n \otimes (J_v-I_v)
\end{bmatrix},\\
0&\sum_{j=0}^{n-1}P^j\otimes B_{i+j}\\
\sum_{j=0}^{n-1}P^j \otimes B_{-i-j}^t&0
\end{bmatrix},\quad (0\leq i \leq n-1),\\
J_n\otimes(J_v-\sum_{j=0}^{n-1} B_j^t)&0
\end{bmatrix}.
\end{align*}

Note that when the weight $k$ is equal to the order $d$, namely, $W$ is a generalized Hadamard matrix, the matrix $A_3=0$.

Let $W$ be a $BGW(v,k,\lambda;\mathbb{Z}_n)$.

Then the following hold:

* If $k=v$, then $\{A_{0,i},A_1,A_{2,i} \mid 0\leq i\leq n-1 \}$ is a commutative association scheme.

* If $k<v$, then $\{A_{0,i},A_1,A_{2,i},A_3 \mid 0\leq i\leq n-1\}$ is a commutative association scheme.

* The association schemes in $1,2$ are symmetric if and only if $n=2$.

1, 2:
It is easily checked that $A_{0,0}$ is the identity matrix,  the sum of all adjacency matrices is equal to the all-ones matrix and, $A_{0,i}^t=A_{0,-i}$ for $0\leq i\leq n-1$, $A_1^t=A_1$, $A_{2,i}^t=A_{2,-i}$ and $A_3^t=A_3$, where the second index $-i$ is taken modulo $n$.
Set $\mathcal{A}=\text{span}_{\mathbb{C}}\{A_{0,i},A_1,A_{2,i},A_3 \mid 0\leq i \leq n-1\}$.
It is enough to show that $\mathcal{A}$ is closed under multiplication and is commutative.

For $0\leq i,j\leq n-1$, it is easy to see that

\begin{align*}
A_{0,i} A_{0,j}&=A_{0,j} A_{0,i}=A_{0,i+j}\in\mathcal{A},\\
%A_{0,i} A_{0,j}&=A_{0,j} A_{0,i}=\begin{bmatrix}
%P^{i+j}\otimes I_d&0\\
%0& P^{i+j}\otimes I_d
A_{0,i} A_1&=A_1 A_{0,i},%\begin{bmatrix}
%P^i J_n\otimes (J_d-I_d)&0\\
%0& P^i J_n\otimes (J_d-I_d)
%J_n\otimes (J_d-I_d)&0\\
%0& J_n\otimes (J_d-I_d)
=A_1\in\mathcal{A}, \\
%n J_n\otimes ((d-2)J_d+I_d)&0\\
%0& n J_n\otimes ((d-2)J_d+I_d)
=n(v-1)\sum_{i=0}^{n-1}A_{0,i} +n(v-2)A_1\in\mathcal{A}.
\end{align*}

For $0\leq i,j\leq n-1$,

\begin{align*}
A_{0,i} A_{2,j}&=A_{2,j} A_{0,i}=\begin{bmatrix}
0&\sum_{k=0}^{n-1} P^{i+k}\otimes B_{j+k}\\
\sum_{k=0}^{n-1} P^{i+k}\otimes B_{-j-k}^t &0
\end{bmatrix}\\
0&\sum_{k=0}^{n-1} P^{k}\otimes B_{j-i+k}\\
\sum_{k=0}^{n-1} P^{k}\otimes B_{-j+i-k}^t &0
\end{bmatrix}\\
&=A_{2,-i+j}\in \mathcal{A}.
\end{align*}

For $0\leq i\leq n-1$,

\begin{align*}
A_1 A_{2,i}&=A_{2,i}A_1 =\begin{bmatrix}
0&J_n\otimes(k J_v-\sum_{k=0}^{n-1}B_k)\\
J_n\otimes(k J_v-\sum_{k=0}^{n-1}B_k^t)&0
\end{bmatrix}\\
&=(k-1)\sum_{j=0}^{n-1} A_{2,j}+k A_3\in \mathcal{A}.
\end{align*}

For $0\leq i,j \leq n-1$, we use the following equality:

\begin{align*}
\sum_{h,\ell=0}^{n-1}P^{h+\ell}\otimes B_{i+h}B_{-j-\ell}^t
&=\sum_{\alpha,h=0}^{n-1} P^\alpha \otimes B_{i+h}B_{-j-(\alpha-h)}^t \\
&=\sum_{\alpha,\beta=0}^{n-1} P^\alpha \otimes B_\beta B_{-j-(\alpha-(\beta-i))}^t \\
&=\sum_{\alpha,\beta=0}^{n-1} P^\alpha \otimes B_\beta B_{-j-i-\alpha+\beta}^t \\
&=\sum_{\alpha=0}^{n-1} P^\alpha \otimes (\sum_{\beta=0}^{n-1} B_\beta B_{-j-i-\alpha+\beta}^t) \\
&=P^{i+j} \otimes (\sum_{\beta=0}^{n-1} B_\beta B_\beta^t)+\sum_{\substack{0\leq \alpha\leq n-1\\ \alpha\neq i+j}} P^\alpha \otimes (\sum_{\beta=0}^{n-1} B_\beta B_{-j-i-\alpha+\beta}^t) \\
&=P^{i+j} \otimes (k I_v+\frac{\lambda}{n}(J_v-I_v))+\sum_{\substack{0\leq \alpha\leq n-1\\ \alpha\neq i+j}} P^\alpha \otimes \frac{\lambda}{n}(J_v-I_v) \\
&=k P^{i+j} \otimes I_v+\frac{\lambda}{n} \sum_{\alpha=0}^{n-1} P^\alpha \otimes (J_v-I_v) \\
&=k P^{i+j} \otimes I_v+\frac{\lambda}{n} J_n \otimes (J_v-I_v).
\end{align*}

Similarly, we have

\begin{align*}
\sum_{h,\ell=0}^{n-1}P^{h+\ell}\otimes B_{-i-h}^t B_{j+\ell}
=k P^{i+j} \otimes I_v+\frac{\lambda}{n} J_n \otimes (J_v-I_v).
\end{align*}

\begin{align*}
\sum_{h,\ell=0}^{n-1}P^{h+\ell}\otimes B_{i+h}B_{j-\ell}^t  &0\\
0&\sum_{h,\ell=0}^{n-1}P^{h+\ell}\otimes B_{i-h}^t B_{j+\ell}
\end{bmatrix}\\
P^{i+j} \otimes I_v  &0\\
0 & P^{i+j} \otimes I_v
\end{bmatrix}
J_n\otimes (J_v-I_v)  &0\\
0&J_n\otimes (J_v-I_v)
\end{bmatrix}\\
&=k A_{0,i+j}+\frac{\lambda}{n}A_1 \in \mathcal{A}.
\end{align*}

Finally, we have the following equalities:

\begin{align*}
A&_{0,i}A_3=A_3 A_{0,i}=A_3,\\
A&_1 A_2=A_2A_1%\\
%0&n J_n\otimes((v-k-1)J_v+\sum_{j=0}^{n-1}B_j)\\
%n J_n\otimes((v-k-1)J_v+\sum_{j=0}^{n-1}B_j^t)&0
%(k-\lambda) J_n\otimes(J_v-I_v)&0\\
%0&(k-\lambda) J_n\otimes(J_v-I_v)
n(v-k)\sum_{j=0}^{n-1} A_{0,j}+n(v-2k+\lambda)A_1,
\end{align*}

each of which belongs to $\mathcal{A}$.
Thus, the vector space $\mathcal{A}$ is closed under multiplication and is commutative.

3: The association scheme is symmetric if and only if  $A_{0,i}$ and $A_{2,i}$ are symmetric, for each $i=1,\ldots,n-1$.
The latter condition is equivalent to $x=-x$ for each $g\in \mathbb{Z}_n\setminus \{\bar{0}\}$, that is $n$ is $2$.

\begin{align*}
E_{3,i}&=\frac{1}{2nv}\sum_{j=0}^{n-1}w^{ij}(vA_{0,j}+\frac{v}{\sqrt{k}}A_{2,j}),\quad (i\in\{1,\ldots,n-1\}),\\
E_{4,i}&=\frac{1}{2nv}\sum_{j=0}^{n-1}w^{ij}(vA_{0,j}-\frac{v}{\sqrt{k}}A_{2,j}),\quad (i\in\{1,\ldots,n-1\}),
\end{align*}

where $w=e^{2\pi\sqrt{-1}/n}$.
It readily follows from the intersection numbers of the association schemes that

* $E_0,E_1,E_{2,1},E_{2,2},E_{3,i},E_{4,i}$ $(i\in\{1,\ldots,n-1\})$ are the primitive idempotents if $v>k$, and

* $E_0,E_1,E_2=E_{2,1}+E_{2,2},E_{3,i},E_{4,i}$ $(i\in\{1,\ldots,n-1\})$ are the primitive idempotents if $v=k$.

Therefore, we have the following.

The eigenmatrices $P,Q$ of the association schemes in Theorem <ref> are given as follows.

* If $k=v$, then

\begin{align*}
& A_{0,i} & A_1 & A_{2,i}   \cr
E_0 & 1 & n(v-1)  & k   \cr
E_1 & 1 & n(v-1) &  -k  \cr
E_2 & 1 & -n &  0  \cr
E_{3,j} & w^{-ij} & 0 &  \sqrt{k}w^{-ij}  \cr
E_{4,j} & w^{-ij} & 0 & -\sqrt{k}w^{-ij}
& E_{0} & E_1 & E_2 & E_{3,j} & E_{4,j}  \cr
A_{0,i} & 1 & 1 & 2(v-1)  & v w^{ij} & v w^{ij} \cr
A_1    & 1 & 1 & -2 & 0 & 0 \cr
A_{2,i} & 1 & -1 & 0 & \frac{v}{\sqrt{k}}w^{ij} & -\frac{v}{\sqrt{k}}w^{ij}
\end{align*}

* If $k<v$, then

\begin{align*}
& A_{0,i} & A_1 & A_{2,i}  & A_{3}  \cr
E_0 & 1 & n(v-1)  & k & n(v-k) \cr
E_1 & 1 & n(v-1) &  -k & -n(v-k) \cr
E_{2,1} & 1 & -n & \sqrt{\frac{k(v-k)}{v-1}}  & -n\sqrt{\frac{k(v-k)}{v-1}} \cr
E_{2,2} & 1 & -n & -\sqrt{\frac{k(v-k)}{v-1}}  & n\sqrt{\frac{k(v-k)}{v-1}} \cr
E_{3,j} & w^{-ij} & 0 &  \sqrt{k}w^{-ij} & 0 \cr
E_{4,j} & w^{-ij} & 0 & -\sqrt{k}w^{-ij} & 0
& E_{0} & E_1 & E_{2,1}  & E_{2,2} & E_{3,j} & E_{4,j}  \cr
A_{0,i} & 1 & 1 & v-1 & v-1 & v w^{ij} & v w^{ij} \cr
A_1    & 1 & 1 & -1 & -1 & 0 & 0 \cr
A_{2,i} & 1 & -1 & \sqrt{\frac{(v-1)(v-k)}{k}} & -\sqrt{\frac{(v-1)(v-k)}{k}} & \frac{v}{\sqrt{k}}w^{ij} & -\frac{v}{\sqrt{k}}w^{ij} \cr
A_3    & 1 & -1 & -\sqrt{\frac{(v-1)k}{v-k}} & \sqrt{\frac{(v-1)k}{v-k}} & 0 & 0
\end{align*}

We now show the converse of the theorem, that is, a BGW over a cyclic group can be constructed by the association scheme as follows.

If there exists a commutative association scheme with the same eigenmatrices in Theorem <ref>, then there exists a $BGW(v,k,\lambda)$ over $\mathbb{Z}_n$.

We deal with 1 and 2 simultaneously.

Since $\sum_{i=0}^{n-1}A_{0,i}+A_1$ has the eigenvalues $nv,0$ with multiplicities $2,2nv-2$, respectively.
Therefore, after suitably permuting the vertices, we may assume that

\sum_{i=0}^{n-1}A_{0,i}+A_1=\begin{bmatrix} J_{nv} & 0 \\ 0 & J_{nv} \end{bmatrix}.

Similarly $\sum_{i=0}^{n-1}A_{0,i}$ has the eigenvalues $n,0$ with multiplicities $2v,2nv-2v$, respectively.
Then we may assume that

\sum_{i=0}^{n-1}A_{0,i}=\begin{bmatrix} I_v\otimes J_n & 0 \\ 0 & I_v\otimes J_n \end{bmatrix}.

The set $\{A_{0,0},A_{0,1},\ldots,A_{0,n-1}\}$ forms a cyclic group of order $n$ and the matrix $A_{0,1}$ generates the group.
Then $A_{0,1}$ is of the form

$$\begin{bmatrix}U_1 & 0 & \cdots & 0 \\ 0 & U_2 & \cdots  & 0 \\ \vdots & \vdots & \ddots  & 0 \\0 & 0 & \cdots  & U_{2v} \end{bmatrix}, $$
where each $U_i$ is a permutation matrix of order $n$.
Then, after suitably changing the ordering of the vertices, we may assume that $U_i=P:=\text{circ}(010\cdots0)$ for each $i$.
Therefore $A_{0,1}=\begin{bmatrix} I_v\otimes P&  0 \\ 0 & I_v\otimes P\end{bmatrix}$.

By considering the map $(x,y)\mapsto (y,x)$,  we have $A_{0,1}=\begin{bmatrix} P\otimes I_v & 0 \\ 0 & P\otimes I_v\end{bmatrix}$.
Then it is shown that $A_{0,i}$ ($i\in\{0,1,\ldots,n-1\}$) and $A_1$ have the desired form.

By the intersection numbers, we have $A_{0,1}^{-1}A_{2,0}A_{0,1}=A_{2,0}$, which shows that $A_{2,0}$ is of the desired for some $(0,1)$-matrices $B_0,B_1,\ldots,B_{n-1}$.
Again, by the intersection numbers, we have $A_{0,1}A_{2,j}=A_{2,j-1}$, which implies that $A_{2,j}$ is of the desired form.

Finally, by the intersection numbers, $A_{2,i}A_{2,j}=kA_{0,i+j}+\frac{\lambda}{n}A_1$ where $\lambda=\frac{k(k-1)}{v-1}$.
This shows that (<ref>) and (<ref>) hold for $B_0,B_1,\ldots,B_{n-1}$.
Therefore, $W=\sum_{k=0}^{n-1}\bar{k}B_k$ is a BGW over the cyclic group, where $\bar{k}$ is an element of $\mathbb{Z}_n$.

It is important to remember that the results of this section are predicated upon the group in question being cyclic. This leaves out the case of those generalized Hadamard matrices over the elementary abelian groups whose orders are non-prime. Though we do not explicitly show it here, one may replace the circulant matrices $\{P^i \mid 0 \leq i < n\}$ with the collection $\{U_g \mid g \in G\}$, where $G$ is a finite abelian group, and where $U_g$ is a regular linear representation of the group element $g$. This still omits the sporadic cases of BGW matrices over non-abelian groups, but all of the known infinite families are now encapsulated by the result.

§ ACKNOWLEDGMENTS.

Hadi Kharaghani is supported by the Natural Sciences and
Engineering  Research Council of Canada (NSERC).  Sho Suda is supported by JSPS KAKENHI Grant Number 18K03395.

author = Eiichi Bannai  and Tatsuro Ito,
TITLE = Algebraic combinatorics. I,
NOTE = Association schemes,
PUBLISHER = The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA,
YEAR = 1984,
PAGES = xxiv+425,
ISBN = 0-8053-0490-8,

author=Hadi Kharaghani, ,
title=New class of weighing matrices,
journal=Ars Combin.,

author = Jennifer Seberry  and Mieko Yamada,
TITLE = Hadamard matrices, sequences, and block designs,
BOOKTITLE = Contemporary design theory,
SERIES = Wiley-Intersci. Ser. Discrete Math. Optim.,
PAGES = 431–560,
PUBLISHER = Wiley, New York,
YEAR = 1992,

author=Seberry, Jennifer,
title=Orthogonal designs,
note=Hadamard matrices, quadratic forms and algebras;
Revised and updated edition of the 1979 original [MR0534614],
publisher=Springer, Cham,

author = A. S. Hedayat and N. J. A. Sloane and John Stufken,
title = Orthogonal arrays,
series= Springer Series in Statistics,
note = Theory and applications,
With a foreword by C. R. Rao,
publisher = Springer-Verlag, New York,
date = 1999,
pages = xxiv+416,
isbn= 0-387-98766-5,

author = Yury J. Ionin and Mohan S. Shrikhande,
TITLE = Combinatorics of symmetric designs,
SERIES = New Mathematical Monographs,
VOLUME = 5,
PUBLISHER = Cambridge University Press, Cambridge,
YEAR = 2006,
PAGES = xiv+520,
ISBN = 978-0-521-81833-9; 0-521-81833-8,

§ APPENDIX

We begin with the orthogonal array of dimension $25 \times 6$ over the alphabet $\{1, \dots, 5\}$ shown transposed below.

\[
\left[
\begin{array}{ccccccccccccccccccccccccc}
\end{array}
\right].
\]

Using this array, we construct a $W(43,25)$ as

\[
\arraycolsep=1.0pt\def\arraystretch{1.0}
\left[
\begin{array}{ccccccccccccccccccccccccccccccccccccccccccc}
\end{array}
\right]
\]

By a similar process, a $BGW(30,25,20; \mathbb{Z}_4)$ is constructed as

\[
\arraycolsep=1.0pt\def\arraystretch{1.0}
\left[
\begin{array}{ccccccccccccccccccccccccccccccc}
\end{array}
\right].
\]

\[
\arraycolsep=1.0pt\def\arraystretch{1.0}
\left[
\begin{array}{cccccc}
\end{array}
\right]
\]

§ ACKNOWLEDGMENTS.

Hadi Kharaghani is supported by the Natural Sciences and
Engineering  Research Council of Canada (NSERC).  Sho Suda is supported by JSPS KAKENHI Grant Number 18K03395.

[1]
E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes,
Benjamin/Cummings, Menlo Park, CA, 1984.

[2]
A. E. Brouwer, A. E. Cohen and A. Neumaier,
Distance-regular graphs,
Springer-Verlag, Berlin, 1989. xviii+495.

[3]
W. de Launey, and D. G. Sarvate, Non-existence of certain GBRDs, Ars Combin., 18, (1983), 5–20.

[4] P. B. Gibbons, R. Mathon, Construction methods for Bhaskar Rao and related designs, J. Aust. Math. Soc., Ser. A 42,  (1987), 5–30.


# The local solubility for homogeneous polynomials with random coefficients
over thin sets

Heejong Lee  Department of Mathematics, University of Toronto, 40 St. George
Street, Toronto, ON M5S 2E4, Canada<EMAIL_ADDRESS>,  Seungsu
Lee  Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake
City, UT 84112, USA<EMAIL_ADDRESS>and  Kiseok Yeon  Department of
Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN
47907-2067, USA<EMAIL_ADDRESS>
###### Abstract.

Let $d$ and $n$ be natural numbers greater or equal to $2$. Let
$\langle\boldsymbol{a},\nu_{d,n}(\boldsymbol{x})\rangle\in\mathbb{Z}[\boldsymbol{x}]$
be a homogeneous polynomial in $n$ variables of degree $d$ with integer
coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the
inner product, and $\nu_{d,n}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$ denotes
the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety
$V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by
$\langle\boldsymbol{a},\nu_{d,n}(\boldsymbol{x})\rangle=0.$ In this paper, we
examine a set of these varieties defined by

$\mathbb{V}^{P}_{d,n}(A)=\\{V_{\boldsymbol{a}}\subset\mathbb{P}^{n-1}|\
P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\\},$

where $P\in\mathbb{Z}[{\boldsymbol{x}}]$ is a non-singular form in $N$
variables of degree $k$ with $2\leq k\leq C({n,d})$ for some constant
$C({n,d})$ depending at most on $n$ and $d$. Suppose that
$P(\boldsymbol{a})=0$ has a nontrivial integer solution. We confirm that the
proportion of varieties $V_{\boldsymbol{a}}$ in $\mathbb{V}^{P}_{d,n}(A)$,
which are everywhere locally soluble, converges to a constant $c_{P}$ as
$A\rightarrow\infty.$ In particular, if there exists
$\boldsymbol{b}\in\mathbb{Z}^{N}$ such that $P(\boldsymbol{b})=0$ and the
variety $V_{\boldsymbol{b}}$ in $\mathbb{P}^{n-1}$ admits a smooth
$\mathbb{Q}$-rational point, the constant $c_{P}$ is positive.

###### Key words and phrases:

Projective variety, Local solubility

###### 2020 Mathematics Subject Classification:

11E76,14G25

##  1\. Introduction

In this article, we study the $p$-adic and real solubility for projective
varieties defined by forms with integer coefficients. Denote by
$f({\boldsymbol{x}})\in{\mathbb{Z}}[x_{1},x_{2},\ldots,x_{n}]$ a homogeneous
polynomial of degree $d$ and let us write $V\subset{\mathbb{P}}^{n-1}$ for a
projective variety defined by $f({\boldsymbol{x}})=0$. A conjecture of Artin
[References] asserts that the variety $V$ admits a ${\mathbb{Q}}_{p}$-point
for every $p$ if $n\geq d^{2}+1$. However, this conjecture is known to be
false in general, and the first counter example has been founded by Terjanian
[References]. Alternatively, one may expect that there exists
$n_{0}:=n_{0}(d)\in{\mathbb{N}}$ such that whenever $n>n_{0}(d),$ the variety
$V$ admits a ${\mathbb{Q}}_{p}$-point. The current knowledge is accessible to
this expectation. In particular, it is known by Wooley [References] that it
suffices to take $n_{0}=d^{2^{d}}$ (see also [References], [References],
[References]). As a different approach concerning the $p$-adic solubility for
the variety $V$, the Ax–Kochen theorem [References] shows that a homogeneous
polynomial $f({\boldsymbol{x}})\in{\mathbb{Z}}[x_{1},\ldots,x_{n}]$ of degree
$d$ with $n\geq d^{2}+1$ has a solution in
${\mathbb{Q}}_{p}^{n}\setminus\\{{\bf 0}\\}$ for sufficiently large prime $p$
with respect to $d$ (see also $[\ref{ref801}]$). It is also known that if
$f({\boldsymbol{x}})\in{\mathbb{Z}}[x_{1},x_{2},\ldots,x_{n}]$ is an
absolutely irreducible form of degree $d$ over $\mathbb{F}_{p}$ with a prime
$p$ sufficiently large in terms of degree $d$, it follows by applying the
Lang–Weil estimate (see [References] and $[\ref{ref100.0},\text{Theorem}\ 3]$)
and the Hensel’s lemma that the equation $f({\boldsymbol{x}})=0$ has a
solution in ${\mathbb{Q}}_{p}^{n}\setminus\\{{\bf 0}\\}$. As for the real
solubility for the variety $V$, one sees that if the degree $d$ is odd, the
variety $V$ always admits a real point. When $d$ is even, the real solubility
for the variety $V$ merely depends on the choice of the coefficients of
$f({\boldsymbol{x}}).$

One infers from the previous paragraph that if the number of variables $n$ is
not large enough, the main difficulties for verifying the $p$-adic solubility
for the variety $V$ occur from the case of small prime $p.$ Nevertheless,
thanks to the density lemma introduced in [References, Lemma 20], we are
capable of obtaining some information about the $p$-adic solubility for the
variety $V$ even with small primes $p.$ In order to describe this information,
we temporarily pause and introduce some notation. Let $n$ and $d$ be natural
numbers with $d\geq 2.$ Let $N:=N_{d,n}=\binom{n+d-1}{d}.$ Let
$\nu_{d,n}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{N}$ denote the Veronese
embedding, defined by listing all the monomials of degree $d$ in $n$ variables
with the lexicographical ordering. We denote a homogeneous polynomial in $n$
variables of degree $d$ with integer coefficients, by
$\langle{\boldsymbol{a}},\nu_{d,n}({\boldsymbol{x}})\rangle$ with integer
vectors ${\boldsymbol{a}}\in{\mathbb{Z}}^{N}$ where
$\langle\cdot,\cdot\rangle$ is the inner product. Here and throughout this
paper, we write
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=\langle{\boldsymbol{a}},\nu_{d,n}({\boldsymbol{x}})\rangle$
for simplicity. Consider a variety
$V_{{\boldsymbol{a}}}\subset\mathbb{P}^{n-1}$ defined by
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0$. For $A\in{\mathbb{R}}_{>0}$, we
define a set of varieties $V_{{\boldsymbol{a}}}$ given by

(1.1) $\mathbb{V}_{d,n}(A):=\\{V_{{\boldsymbol{a}}}\subset\mathbb{P}^{n-1}|\
{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\\}.$

For any $V_{{\boldsymbol{a}}}$, write
$V_{{\boldsymbol{a}}}\biggl{(}{\mathbb{R}}^{n}\times\displaystyle\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}^{n}\biggr{)}$ for the set of points in
${\mathbb{R}}^{n}\times\displaystyle\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}^{n}$ that the variety $V_{{\boldsymbol{a}}}$
admits. Define a quantity

(1.2)
$\varrho_{d,n}^{\text{loc}}(A):=\frac{\\#\biggl{\\{}V_{{\boldsymbol{a}}}\in\mathbb{V}_{d,n}(A)\bigg{|}\
V_{{\boldsymbol{a}}}\biggl{(}{\mathbb{R}}^{n}\times\displaystyle\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}^{n}\biggr{)}\neq\emptyset\biggr{\\}}}{\\#\mathbb{V}_{d,n}(A)}.$

We notice here that the quantity $\varrho_{d,n}^{\text{loc}}(A)$ is the
proportion of varieties $V_{{\boldsymbol{a}}}$ in $\mathbb{V}_{d,n}(A)$ that
are locally soluble, i.e. that admit a real point and a $p$-adic point for all
primes $p$. Thus, the behavior of $\varrho_{d,n}^{\text{loc}}(A)$ provides the
information about $p$-adic solubility for the varieties $V_{{\boldsymbol{a}}}$
in $\mathbb{V}_{d,n}(A)$ even with small primes $p$. We also define
$c_{\infty}$ (resp. $c_{p}$) to be the proportion of varieties
$V_{{\boldsymbol{a}}}$ with ${\boldsymbol{a}}$ varying in $[-1,1]^{N}$ (resp.
in ${\mathbb{Z}}_{p}^{N}$) admitting a real point (resp. a $p$-adic point). By
using the density lemmas [References, Lemmas 20 and 21], Poonen and Voloch
[References, Theorem 3.6] proved that whenever $n,d\geq 2$, one has

(1.3) $\lim_{A\rightarrow\infty}\varrho_{d,n}^{\text{loc}}(A)=c,$

where $c$ is the product of $c_{\infty}$ and $c_{p}$ for all primes $p$.

In this paper, we investigate the proportion of locally soluble varieties
$V_{{\boldsymbol{a}}}$ in a thinner set than the set $\mathbb{V}_{d,n}(A).$ To
describe the thin set of our interest, let
$P(\mathbf{t})\in{\mathbb{Z}}[t_{1},\dots,t_{N}]$ be a non-singular form in
$N$ variables of degree $k\geq 2$. On recalling the definition of
$V_{{\boldsymbol{a}}}\subset\mathbb{P}^{n-1}$, we examine a set of varieties
$V_{{\boldsymbol{a}}}$ defined by

$\mathbb{V}^{P}_{d,n}(A):=\\{V_{{\boldsymbol{a}}}\subset\mathbb{P}^{n-1}|\
{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N},\ P({\boldsymbol{a}})=0\\}.$

Analogous to the quantity $\varrho_{d,n}^{\text{loc}}(A)$, we have the
proportion of locally soluble varieties $V_{{\boldsymbol{a}}}$ of our concern
defined as

(1.4)
$\varrho_{d,n}^{P,\text{loc}}(A):=\frac{\\#\biggl{\\{}V_{{\boldsymbol{a}}}\in\mathbb{V}^{P}_{d,n}(A)\bigg{|}\
V_{{\boldsymbol{a}}}\biggl{(}{\mathbb{R}}^{n}\times\displaystyle\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}^{n}\biggr{)}\neq\emptyset\biggr{\\}}}{\\#\mathbb{V}^{P}_{d,n}(A)}.$

In order to describe our main theorems, we define

(1.5)
$T_{\infty}:=\left\\{{\boldsymbol{a}}\in[-1,1]^{N}\cap{\mathbb{R}}^{N}\middle|\
\exists\ {\boldsymbol{x}}\in{\mathbb{R}}^{n}\setminus\\{\boldsymbol{0}\\}\
\text{such that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\}$

and

(1.6) $T_{p}:=\left\\{{\boldsymbol{a}}\in\mathbb{Z}_{p}^{N}\middle|\ \exists\
{\boldsymbol{x}}\in{\mathbb{Z}}_{p}^{n}\setminus\\{\boldsymbol{0}\\}\
\text{such that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\},$

for every prime $p.$ Let $\mu_{p}$ denote the Haar measure on
${\mathbb{Z}}_{p}^{N}$ normalized to have total mass $1$. For given measurable
sets $S_{p}\subseteq{\mathbb{Z}}_{p}^{N}$ and
$S_{\infty}\subseteq{\mathbb{R}}^{N}$ with the Haar measure $\mu_{p}$ and the
Lebesgue measure, we define

$\sigma_{p}(S_{p}):=\lim_{r\rightarrow\infty}p^{-r(N-1)}\\#\left\\{{\boldsymbol{a}}\
(\text{mod}\ p^{r})\middle|\ {\boldsymbol{a}}\in S_{p}\ \text{and}\
P({\boldsymbol{a}})\equiv 0\ (\text{mod}\ p^{r})\right\\}$

and

$\sigma_{\infty}(S_{\infty})=\lim_{\eta\rightarrow
0+}(2\eta)^{-1}V_{\infty}(\eta),$

where $V_{\infty}(\eta)$ is the volume of the subset of ${\boldsymbol{y}}\in
S_{\infty}$ satisfying $|P({\boldsymbol{y}})|<\eta$. Furthermore, for
$d_{1},d_{2}\in{\mathbb{N}}$, we define $C_{n,d}(d_{1},d_{2})$ a rational
number such that

$C_{n,d}(d_{1},d_{2})=\frac{d!(n+d_{1}-1)!}{d_{1}!(n+d-1)!}+\frac{d!(n+d_{2}-1)!}{d_{2}!(n+d-1)!}.$

$C_{n,d}(d_{1},d_{2})$ belongs to $(0,1)$ and maximizes at $(1,d-1)$ and
$(d-1,1)$, see Lemma 2.5. Our first main theorem shows that the proportion of
locally soluble varieties $V_{{\boldsymbol{a}}}$ in the thin set
$\mathbb{V}^{P}_{d,n}(A)$ converges to the product of “local proportions” as
$A\rightarrow\infty$.

######  Theorem 1.1.

Suppose that $P(\mathbf{t})\in{\mathbb{Z}}[t_{1},\dots,t_{N}]$ is a non-
singular form in $N$ variables of degree $k$ with $2\leq
k<\lfloor(1-C_{n,d}(1,d-1))N\rfloor$ and $(k-1)2^{k}<N$ where
$C_{n,d}(d_{1},d_{2})$ is as above. Suppose that $P(\mathbf{t})=0$ has a
nontrivial integer solution. Then, whenever $n\geq 2$ and $d\geq 2$, one has

$\lim_{A\rightarrow\infty}\varrho_{d,n}^{P,\text{loc}}(A)=c_{P},$

where

$c_{P}:=\frac{\sigma_{\infty}(T_{\infty})\cdot\prod_{p}\sigma_{p}(T_{p})}{\sigma_{\infty}([-1,1]^{N})\cdot\prod_{p}\sigma_{p}({\mathbb{Z}}_{p})}.$

######  Remark 1.

Since $P({\bf t})$ is a non-singular form in $N$ variables of degree $k$ with
$(k-1)2^{k}<N$ and $P({\bf t})=0$ has a nontrivial integer solution, the
classical argument (see $[\ref{ref8}]$ and $[\ref{ref26},\text{the proof of
Theorem 1.3}]$) reveals that the quantity
$\sigma_{\infty}([-1,1]^{N})\cdot\prod_{p\
\text{prime}}\sigma_{p}({\mathbb{Z}}_{p})$ is convergent and is bounded above
and below by non-zero constants, respectively, depending on the polynomial
$P({\bf t})$. Then, on observing that

$0\leq\sigma_{\infty}(T_{\infty})\leq\sigma_{\infty}([-1,1]^{N})\ \text{and}\
0\leq\sigma_{p}(T_{p})\leq\sigma_{p}({\mathbb{Z}}_{p}),$

we infer that the infinite product
$\sigma_{\infty}(T_{\infty})\cdot\prod_{p}\sigma_{p}(T_{p})$ in the numerator
of $c_{P}$ converges to a non-negative constant.

Our second main theorem shows that under an additional condition on the
polynomial $P$ defining the thin set, the product of local proportions is
strictly positive.

######  Theorem 1.2.

In addition to the setup of Theorem 1.1, suppose that there exists
${\boldsymbol{b}}\in{\mathbb{Z}}^{N}\setminus\\{{\bf 0}\\}$ with
$P({\boldsymbol{b}})=0$ such that the variety $V_{{\boldsymbol{b}}}$ in
${\mathbb{P}}^{n-1}$ admits a smooth ${\mathbb{Q}}$-point. Then, the constant
$c_{P}$ is positive.

## Structure of the paper and notation

In section 2, we provide auxiliary lemmas and a proposition. Lemma 2.2 is used
to prove Lemma $\ref{lem2.42.4}.$ Lemma 2.4 and Lemma 2.5 are required to
prove Proposition 2.6. In section 3, we also provide an auxiliary lemma and a
proposition, and provide the proofs of Theorem 1.1 and Theorem 1.2 by making
use of lemmas and propositions obtained in section 2 and 3. In particular, by
making use of Lemma 2.1, Lemma 2.3, Lemma 3.1 and Proposition 2.6, we prove
Theorem 1.1 in section $\ref{sec3}.$ By utilizing Theorem 1.1, Lemma 2.3 and
Proposition 3.2, we prove Theorem 1.2 in section $\ref{sec3}.$

For a given vector $\boldsymbol{v}\in{\mathbb{R}}^{N}$, we write the $i$-th
coordinate of $\boldsymbol{v}$ by $(\boldsymbol{v})_{i}$ or $v_{i}$. We use
$\langle\cdot,\cdot\rangle$ for the inner product. We write
$0\leq{\boldsymbol{x}}\leq X$ or ${\boldsymbol{x}}\in[0,X]^{s}$ to abbreviate
the condition $0\leq x_{1},\ldots,x_{s}\leq X$. For a prime $p$ and vectors
$\boldsymbol{v}\in{\mathbb{R}}^{n}$, we use $p^{h}\|\boldsymbol{v}$ when one
has $p^{h}|v_{i}$ for all $1\leq i\leq n$ but $p^{h+1}\nmid v_{i}$ for some
$1\leq i\leq n.$ Throughout this paper, we use $\gg$ and $\ll$ to denote
Vinogradov’s well-known notation, and write $e(z)$ for $e^{2\pi iz}$. We use
$A\asymp B$ when both $A\gg B$ and $A\ll B$ hold. We adopt the convention that
when $\epsilon$ appears in a statement, then the statement holds for each
$\epsilon>0$, with implicit constants depending on $\epsilon.$

## Acknowledgement

The authors acknowledge support from NSF grant DMS-2001549 under the
supervision of Trevor Wooley. The third author would like to thank James
Cumberbatch for helpful discussion and suggestion for the proof of Lemma 3.1.
Especially, the third author would like to thank Trevor Wooley for his
constant encouragement to complete this work.

##  2\. Auxiliary lemmas and propositions

Throughout this section, we fix a non-singular form
$P(\mathbf{t})\in{\mathbb{Z}}[t_{1},\dots,t_{N}]$ in $N$ variables of degree
$k\geq 2$. We also assume that $N>(k-1)2^{k}$. Let $\mathfrak{B}$ be a box in
$[-1,1]^{N}\cap{\mathbb{R}}^{N}$. Denote by ${\mathbb{Z}}^{N}_{\text{prim}}$
the set of primitive integer vectors. For given $A,B>0$ and
${\boldsymbol{r}}\in{\mathbb{Z}}^{N}$ with $0\leq{\boldsymbol{r}}\leq B-1$, we
define

$\mathcal{N}(A,\mathfrak{B},B,{\boldsymbol{r}},P)=\\#\left\\{{\boldsymbol{a}}\in
A\mathfrak{B}\cap{\mathbb{Z}}^{N}_{\text{prim}}\middle|\
P({\boldsymbol{a}})=0,\ {\boldsymbol{a}}\equiv{\boldsymbol{r}}\ (\text{mod}\
B)\right\\}.$

The first lemma in this section provides the asymptotic formula for

$\mathcal{N}(A,\mathfrak{B},B,{\boldsymbol{r}},P)\ \text{as}\
A\rightarrow\infty.$

In advance of the statement of Lemma 2.1, we define $v_{p}(L)$ with
$L\in{\mathbb{Z}}$ and $p$ prime as the integer $s$ such that $p^{s}\|L.$

######  Lemma 2.1.

Let $B=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{m}^{r_{m}}$ with $p_{i}$ distinct
prime numbers and $r_{i}\geq 0$ for $i=1,2,\dots,m$. Put
$M=\\{p_{1},p_{2},\dots,p_{m}\\}.$ Then, for a given
${\boldsymbol{r}}\in{\mathbb{Z}}^{N}$ with $0\leq{\boldsymbol{r}}\leq B-1$ and
for sufficiently large $A>0$, there exists $\delta>0$ such that

$\displaystyle\mathcal{N}(A,\mathfrak{B},B,{\boldsymbol{r}},P)=\frac{1}{\zeta(N-k)}\prod_{p\notin
M}\sigma_{p}\cdot\prod_{p\in
M}\sigma^{B,{\boldsymbol{r}}}_{p}\cdot\sigma_{\infty}\cdot
A^{N-k}+O(A^{N-k-\delta}),$

where

$\displaystyle\sigma_{p}$
$\displaystyle:=\lim_{r\rightarrow\infty}p^{-r(N-1)}\\#\left\\{1\leq{\boldsymbol{a}}\leq
p^{r}\middle|\ P({\boldsymbol{a}})\equiv 0\ (\text{mod}\ p^{r})\right\\},$
$\displaystyle\sigma^{B,{\boldsymbol{r}}}_{p}$
$\displaystyle:=\lim_{r\rightarrow\infty}p^{-r(N-1)}\\#\left\\{1\leq{\boldsymbol{a}}\leq
p^{r}\middle|\ P({\boldsymbol{a}})\equiv 0\ (\text{mod}\ p^{r}),\
{\boldsymbol{a}}\equiv{\boldsymbol{r}}\ (\text{mod}\ p^{v_{p}(B)})\right\\}$

and

$\sigma_{\infty}:=\sigma_{\infty}(\mathfrak{B})=\lim_{\eta\rightarrow
0+}(2\eta)^{-1}V_{\infty}(\eta)$

in which $V_{\infty}(\eta)$ is the volume of the subset of
${\boldsymbol{y}}\subseteq\mathfrak{B}$ satisfying
$|P({\boldsymbol{y}})|<\eta.$ In particular, the implicit constant in
$O(A^{N-k-\delta})$ depends on $B$ and $P(\mathbf{t}).$

We record this lemma without proof because it is readily obtained by the
previous results as follows. By repeating the argument of Birch in
[References] replaced with the variables ${\boldsymbol{x}}$ imposed on the
congruence condition ${\boldsymbol{x}}\equiv{\boldsymbol{r}}\ (\text{mod}\
B)$, we obtain an asymptotic formula for the number of integer solutions
${\boldsymbol{x}}\in[-A,A]^{N}$ of $P({\boldsymbol{x}})=0$ with the congruence
condition ${\boldsymbol{x}}\equiv{\boldsymbol{r}}\ (\text{mod}\ B)$ as
$A\rightarrow\infty$. Then, by the Möbius inversion formula, one has the
asymptotic formula for $\mathcal{N}(A,\mathfrak{B},B,{\boldsymbol{r}},P)$ (see
[References, Lemma 2.2] for an exposition of the application of the Möbius
inversion formula). The main term of this asymptotic formula includes the
product of $p$-adic densities and the singular integral. By applying the
strategy proposed by Schmidt [References, References] (see also a refined
version [References, section 9]), the singular integral can be replaced by the
real density $\sigma_{\infty}$. Furthermore, we readily deduce by the
coprimality between $p$ and $B$ that the product of $p$-adic densities in the
main term becomes $\prod_{p\notin M}\sigma_{p}\cdot\prod_{p\in
M}\sigma^{B,{\boldsymbol{r}}}_{p}$. Thus, this yields the asymptotic formula
for $N(A,\mathfrak{B},B,{\boldsymbol{r}},P)$ as desired in Lemma 2.1.

######  Lemma 2.2.

Suppose that $C$ is a sufficiently large constant and that $A$ and $Q$ are
positive numbers with $CQ\leq A$. Then, for a given
${\boldsymbol{c}}\in{\mathbb{Z}}^{N}$ with $1\leq{\boldsymbol{c}}\leq Q$, we
have

$\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
P({\boldsymbol{a}})=0\ \text{and}\ {\boldsymbol{a}}\equiv{\boldsymbol{c}}\
(\text{mod}\ Q)\right\\}\ll(A/Q)^{N-k},$

where the implicit constant may depend on $P(\mathbf{t}).$

###### Proof.

By orthogonality, we have

(2.1)
$\displaystyle\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
P({\boldsymbol{a}})=0\ \text{and}\ {\boldsymbol{a}}\equiv{\boldsymbol{c}}\
(\text{mod}\ Q)\right\\}$ $\displaystyle=\int_{0}^{1}\sum_{-A\leq
Q{\boldsymbol{y}}+{\boldsymbol{c}}\leq A}e(\alpha
P(Q{\boldsymbol{y}}+{\boldsymbol{c}}))d\alpha.$

By change of variable $\alpha=\beta/Q^{k},$ the last expression is seen to be

(2.2) $\displaystyle Q^{-k}\int_{0}^{Q^{k}}\sum_{-A\leq
Q{\boldsymbol{y}}+{\boldsymbol{c}}\leq A}e(\beta
P({\boldsymbol{y}}+{\boldsymbol{c}}/Q))d\beta$
$\displaystyle\ll\sup_{\begin{subarray}{c}1\leq l\leq Q^{k}\\\
l\in{\mathbb{N}}\end{subarray}}\int_{l-1}^{l}\sum_{-A\leq
Q{\boldsymbol{y}}+{\boldsymbol{c}}\leq A}e(\beta
P({\boldsymbol{y}}+{\boldsymbol{c}}/Q))d\beta$
$\displaystyle=\sup_{\begin{subarray}{c}1\leq l\leq Q^{k}\\\
l\in{\mathbb{N}}\end{subarray}}\int_{l-1}^{l}\sum_{-A\leq
Q{\boldsymbol{y}}+{\boldsymbol{c}}\leq
A}e(\beta(P({\boldsymbol{y}})+g({\boldsymbol{y}})))d\beta,$

where $g\in{\mathbb{Q}}[{\boldsymbol{y}}]$ is a polynomial of degree at most
$k-1.$

For a given $l\in{\mathbb{Z}},$ we define the major arcs
$\mathfrak{M}^{l}_{\delta}$ by

$\mathfrak{M}^{l}_{\delta}=\bigcup_{\begin{subarray}{c}0\leq a\leq
q\leq(A/Q)^{\delta}\\\ (q,a)=1\end{subarray}}\mathfrak{M}^{l}(q,a),$

where

$\mathfrak{M}^{l}(q,a)=\left\\{\beta\in[l-1,l)|\
\left|\beta-(l-1)-\frac{a}{q}\right|\leq\frac{(A/Q)^{\delta}}{q(A/Q)^{k}}\right\\}.$

Furthermore, we define the minor arcs
$\mathfrak{m}^{l}_{\delta}:=[l-1,l)\setminus\mathfrak{M}^{l}_{\delta}.$ Then,
on writing that

(2.3) $S(\beta):=\displaystyle\sum_{-A\leq
Q{\boldsymbol{y}}+{\boldsymbol{c}}\leq
A}e(\beta(P({\boldsymbol{y}})+g({\boldsymbol{y}}))),$

we see that

$\displaystyle\int_{l-1}^{l}S(\beta)d\beta=\displaystyle\int_{\mathfrak{M}^{l}_{\delta}}S(\beta)d\beta+\displaystyle\int_{\mathfrak{m}^{l}_{\delta}}S(\beta)d\beta.$

Note by [References, Lemma 3.6] and the fact that $P(\mathbf{t})$ is a non-
singular form that

(2.4)
$\sup_{\beta\in\mathfrak{m}^{l}_{\delta}}S(\beta)\ll(A/Q)^{N-N\delta/(2^{k-1}(k-1))+\epsilon}.$

One infers by repeating the argument in [References] with the bound
$(\ref{ineq})$ that whenever $N>(k-1)2^{k}$ and $A/Q>0$ is sufficiently large,
we have

(2.5) $\displaystyle\int_{l-1}^{l}S(\beta)d\beta\ll(A/Q)^{N-k}.$

Recall that

$\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
P({\boldsymbol{a}})=0\ \text{and}\ {\boldsymbol{a}}\equiv{\boldsymbol{c}}\
(\text{mod}\ Q)\right\\}$

is bounded above by the last expression in $(\ref{exp2.17})$ and recall the
definition $(\ref{S2.20})$ of $S(\beta)$. Therefore, by substituting the bound
$(\ref{bound2.20})$ into the last expression of $(\ref{exp2.17})$, we conclude
that

$\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
P({\boldsymbol{a}})=0\ \text{and}\ {\boldsymbol{a}}\equiv{\boldsymbol{c}}\
(\text{mod}\ Q)\right\\}\ll(A/Q)^{N-k}.$

∎

The following lemma provides an upper asymptotic estimate for the number of
integer points in the variety cut by the polynomial
$P(\mathbf{t})\in{\mathbb{Z}}[\mathbf{t}]$, that reduce modulo $p$, for some
sufficiently large $p>M,$ to an $\mathbb{F}_{p}$-point of an another given
variety $Y\subset\mathbb{A}^{n}$ defined over ${\mathbb{Z}}$.

######  Lemma 2.3.

Let $\mathfrak{B}$ be a compact region in ${\mathbb{R}}^{N}$ having a finite
measure, and let $Y$ be any closed subscheme of
$\mathbb{A}^{N}_{{\mathbb{Z}}}$ of codimension $r\geq 1.$ Let $A$ and $M$ be
positive real numbers. Suppose that $r-1>k$ and that $P(\mathbf{t})=0$ has a
non-trivial integer solution. Then, there exists
$A_{0}:=A_{0}(P(\mathbf{t}))\in{\mathbb{R}}_{>0}$ such that whenever
$A>A_{0}$, we have

(2.6) $\displaystyle\\#\left\\{{\boldsymbol{a}}\in
A\mathfrak{B}\cap{\mathbb{Z}}^{N}\middle|\ \begin{aligned} &(i)\
{\boldsymbol{a}}\ (\text{mod}\ p)\in Y(\mathbb{F}_{p})\ \text{for some prime}\
p>M\\\ &(ii)\ P({\boldsymbol{a}})=0\end{aligned}\right\\}$
$\displaystyle\ll\frac{A^{N-k}}{M^{r-k-1}\log M}+A^{N-r+1},$

where the implicit constant may depend on $\mathfrak{B}$ and $Y$.

In [References, Theorem 3.3], Bhargava provided an upper asymptotic estimate
that

(2.7) $\displaystyle\\#\left\\{{\boldsymbol{a}}\in
A\mathfrak{B}\cap{\mathbb{Z}}^{N}\middle|\ {\boldsymbol{a}}\ (\text{mod}\
p)\in Y(\mathbb{F}_{p})\ \text{for some prime}\ p>M\right\\}$
$\displaystyle\ll\frac{A^{N}}{M^{r-1}\log M}+A^{N-r+1}.$

Furthermore, as alluded in [References, Remark 3.4], the bound in
$(\ref{ineq2.16})$ can be achieved for suitable choices of $Y$. Thus, this
bound is essentially optimal.

In the proof of Lemma 2.3, we mainly adopt the argument in [References,
Theorem 3.3] though, we freely admit that the bound in $(\ref{ineq2.162.16})$
seems not optimal order of magnitude of the bound. Especially, one finds that
the second term is trivially obtained by that in $(\ref{ineq2.16})$. We are
independently interested in a sharper bound in (2.6) and expect that one might
be able to improve this bound. Nevertheless, since the strength of the upper
asymptotic estimate in Lemma 2.3 is enough for our purpose, we do not put our
effort into optimizing this upper bound in this paper.

###### Proof of Lemma 2.3.

We can and do assume that $Y$ is irreducible. Otherwise, we can take its
irreducible components and add up the equation (2.6) to deduce general cases.
Since $Y$ has codimension $r$, there exists
$f_{1},\dots,f_{r}\in{\mathbb{Z}}[t_{1},\dots,t_{N}]$ such that the vanishing
locus $V(f_{1},\dots,f_{r})$ contains an irreducible component of codimension
$r$ containing $Y$. Indeed, we can assume that $Y$ equals to the irreducible
component, as they have the same underlying reduced subscheme, and we only
consider ${\mathbb{Z}}$ or $\mathbb{F}_{p}$-points of them. By [References,
Lemma 3.1], the number of ${\boldsymbol{a}}\in A\mathfrak{B}\cap
Y({\mathbb{Z}})$ is $\ll A^{N-r}$. (Note that $A\mathfrak{B}\cap
Y({\mathbb{Z}})$ equals to $A\mathfrak{B}\cap{\mathbb{Z}}^{N}\cap
Y({\mathbb{R}})$) Thus, it suffices to find an upper bound of the size of the
set by

$\displaystyle\\#\left\\{{\boldsymbol{a}}\in
A\mathfrak{B}\cap{\mathbb{Z}}^{N}\middle|\ \begin{aligned} &(i)\
P({\boldsymbol{a}})=0\\\ &(ii)\ {\boldsymbol{a}}\ (\text{mod}\ p)\in
Y(\mathbb{F}_{p})\ \text{for some prime}\ p>M\\\ &(iii)\
{\boldsymbol{a}}\notin Y({\mathbb{Z}})\end{aligned}\right\\}$
$\displaystyle\ll\frac{A^{N-k}}{M^{r-k-1}\log M}+A^{N-r+1}.$

We may assume that $r>k+1$, since $r-1>k$ from the hypothesis in the statement
of Lemma 2.3. We shall find an upper bound of a slightly larger set by

(2.8) $\displaystyle\\#\left\\{({\boldsymbol{a}},p)\middle|\begin{aligned}
&(i)\ {\boldsymbol{a}}\in A\mathfrak{B}\cap{\mathbb{Z}}^{N}\ \text{and}\
P({\boldsymbol{a}})=0\\\ &(ii)\ p>M\ \text{a prime and}\ {\boldsymbol{a}}\
(\text{mod}\ p)\in Y(\mathbb{F}_{p})\\\ &(iii)\ {\boldsymbol{a}}\notin
Y({\mathbb{Z}})\end{aligned}\right\\}$
$\displaystyle\ll\frac{A^{N-k}}{M^{r-k-1}\log M}+A^{N-r+1}.$

First, we count pairs (${\boldsymbol{a}},p$) on the left hand side of
$(\ref{ineq2.18})$ for each prime $p$ satisfying $p\leq A$; such primes arise
only when $A>M.$ Meanwhile, by Lemma 2.2 we find that for a given
${\boldsymbol{c}}\in[1,p]^{N}$, the number of integer solutions
${\boldsymbol{a}}\in[-A,A]^{N}$ of $P({\boldsymbol{a}})=0$ with the congruence
condition ${\boldsymbol{a}}\equiv{\boldsymbol{c}}\ (\text{mod}\ p)$ is
$O((A/p)^{N-k}).$ Then, since $\\#Y(\mathbb{F}_{p})=O(p^{N-r}),$ we see that
the number of ${\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}$ such that
${\boldsymbol{a}}\ (\text{mod}\ p)$ is in $Y(\mathbb{F}_{p})$ is
$O(p^{N-r})\cdot O((A/p)^{N-k})=O(A^{N-k}/p^{r-k}).$ Thus the total number of
desired pairs $({\boldsymbol{a}},p)$ with $p\leq A$ is at most

(2.9) $\displaystyle\\#$
$\displaystyle\left\\{({\boldsymbol{a}},p)\middle|\begin{aligned} &(i)\
{\boldsymbol{a}}\in A\mathfrak{B}\cap{\mathbb{Z}}^{N}\ \text{and}\
P({\boldsymbol{a}})=0\\\ &(ii)\ A\geq p>M\ \text{a prime and}\
{\boldsymbol{a}}\ (\text{mod}\ p)\in Y(\mathbb{F}_{p})\\\ &(iii)\
{\boldsymbol{a}}\notin Y({\mathbb{Z}})\end{aligned}\right\\}$
$\displaystyle\ll\sum_{M<p\leq
A}O\left(\frac{A^{N-k}}{p^{r-k}}\right)=O\left(\frac{A^{N-k}}{M^{r-k-1}\log
M}\right).$

Next, we count pairs $({\boldsymbol{a}},p)$ with $p>A.$ It follows from (see
the equation $(17)$ in the proof of $[\ref{ref13},\text{Theorem}\ 3.3]$) that

(2.10) $\displaystyle\\#\left\\{({\boldsymbol{a}},p)\middle|\begin{aligned}
&(i)\ {\boldsymbol{a}}\in A\mathfrak{B}\cap{\mathbb{Z}}^{N}\ \text{and}\
P({\boldsymbol{a}})=0\\\ &(ii)\ p>A\ \text{a prime and}\ {\boldsymbol{a}}\
(\text{mod}\ p)\in Y(\mathbb{F}_{p})\\\ &(iii)\ {\boldsymbol{a}}\notin
Y({\mathbb{Z}})\end{aligned}\right\\}$ $\displaystyle\ll$ $\displaystyle\
\\#\left\\{({\boldsymbol{a}},p)\middle|\begin{aligned} &(i)\
{\boldsymbol{a}}\in A\mathfrak{B}\cap{\mathbb{Z}}^{N}\\\ &(ii)\ p>A\ \text{a
prime and}\ {\boldsymbol{a}}\ (\text{mod}\ p)\in Y(\mathbb{F}_{p})\\\ &(iii)\
{\boldsymbol{a}}\notin Y({\mathbb{Z}})\end{aligned}\right\\}\ll A^{N-r+1}.$

Therefore, we find by $(\ref{ineq2.19})$ and $(\ref{ineq2.20})$ that the
inequality (2.8) holds. Hence, we complete the proof of Lemma
$\ref{lem2.42.4}.$∎

Next, we will prove that most of $f_{{\boldsymbol{a}}}({\boldsymbol{x}})$ are
irreducible. Let $Y$ be a subset of ${\mathbb{A}}_{\mathbb{Z}}^{N}$ defined to
be

(2.11) $Y:=\left\\{{\boldsymbol{a}}\in{\mathbb{A}}_{\mathbb{Z}}^{N}\middle|\
f_{{\boldsymbol{a}}}({\boldsymbol{x}})\text{ is reducible over}\
{\mathbb{C}}\right\\}.$

Our goal here is to show that $Y$ is, in fact, an algebraic variety and that
the codimension of $Y$ is strictly greater than a constant depending on $n$
and $d$. Here, we fix
$r:=\operatorname{codim}_{\mathbb{A}_{\mathbb{Z}}^{N}}Y$. To show the claim,
we record the following two useful lemmas:

######  Lemma 2.4.

For any integers $n\geq 3$ and $d\geq 3$,

$\frac{(n+d)(n+d-1)}{n-1}<{n+d\choose d}$

###### Proof.

We do this by induction on $n$. When $n=3$, we have

$\begin{split}3&<3+d-2\\\
\frac{3(3+d)(3+d-1)}{2}&<\frac{(3+d)(3+d-1)(3+d-2)}{2}\\\
\frac{(3+d)(3+d-1)}{2}&<\frac{(3+d)(3+d-1)(3+d-2)}{3\cdot 2}={n+d\choose
d}\end{split}$

Suppose that the lemma is true for $n$. Then,

$\begin{split}\frac{(n+d+1)(n+d)}{n}&=\frac{(n+d+1)(n+d)(n+d-1)(n-1)}{n(n-1)(n+d-1)}\\\
&<\frac{(n+d)!}{n!d!}\cdot\frac{(n+d+1)(n-1)}{n(n+d-1)}\\\
&=\frac{(n+d+1)!}{(n+1)!d!}\cdot\frac{(n+1)(n-1)}{n(n+d-1)}\\\
&<\frac{(n+d+1)!}{(n+1)!d!}\end{split}$

The last inequality follows from $\frac{(n-1)(n+1)}{n(n+d-1)}<1$. ∎

######  Lemma 2.5.

Let $n\geq 3$ and $d\geq 3$ be integers. Suppose that $d_{1}$ and $d_{2}$ are
natural numbers with $d=d_{1}+d_{2}$. Let $C_{n,d}(d_{1},d_{2})$ be a rational
number defined as

$C_{n,d}(d_{1},d_{2})=\frac{d!(n+d_{1}-1)!}{d_{1}!(n+d-1)!}+\frac{d!(n+d_{2}-1)!}{d_{2}!(n+d-1)!}$

Then, for given $n$ and $d,$ the quantity $C_{n,d}$ attains the maximum value
when $(d_{1},d_{2})=(1,d-1)$ or $(d_{1},d_{2})=(d-1,1)$. Furthermore, its
maximum is strictly less than 1.

###### Proof.

Without loss of generality, we assume that $d_{1}\leq d_{2}.$ We shall first
show that one has

(2.12) $C_{n,d}(d_{1},d_{2})\leq C_{n,d}(d_{1}-1,d_{2}+1).$

In order to verify the inequality $(\ref{1.8}),$ we observe that whenever
$n\geq 3$ one has

$(n-1)\cdot\frac{(n+d_{1}-2)!}{d_{1}!}\leq(n-1)\cdot\frac{(n+d_{2}-1)!}{(d_{2}+1)!}.$

Equivalently, this is seen to be

$\frac{(n+d_{1}-1)!}{d_{1}!}-\frac{(n+d_{1}-2)!}{(d_{1}-1)!}\leq\frac{(n+d_{2})!}{(d_{2}+1)!}-\frac{(n+d_{2}-1)!}{d_{2}!},$

and thus

(2.13)
$\frac{(n+d_{1}-1)!}{d_{1}!}+\frac{(n+d_{2}-1)!}{d_{2}!}\leq\frac{(n+d_{2})!}{(d_{2}+1)!}+\frac{(n+d_{1}-2)!}{(d_{1}-1)!}.$

Therefore, we find from $(\ref{1.9})$ that

$\frac{(n+d-1)!}{d!}C_{n,d}(d_{1},d_{2})\leq\frac{(n+d-1)!}{d!}C_{n,d}(d_{1}-1,d_{2}+1).$

This confirms the inequality $(\ref{1.8}).$ Then, by applying $(\ref{1.8})$
iteratively, we conclude that the quantity $C_{n,d}$ attains the maximum value
when $(d_{1},d_{2})=(1,d-1)$ or $(d_{1},d_{2})=(d-1,1)$.

Next, we deduce by applying Lemma 2.4 that

(2.14)
$\begin{split}C_{n,d}(1,d-1)&=\frac{d!n!}{(n+d-1)!}+\frac{d!(n+d-2)!}{(d-1)!(n+d-1)!}\\\
&=\frac{d!n!}{(n+d-1)!}+\frac{d}{n+d-1}\\\ &=\frac{n+d}{{n+d\choose
d}}+\frac{d}{n+d-1}\\\ &<\frac{(n-1)(n+d)}{(n+d)(n+d-1)}+\frac{d}{n+d-1}\\\
&=\frac{n-1}{n+d-1}+\frac{d}{n+d-1}=1.\end{split}$

Therefore, this completes the proof of Lemma 2.5. ∎

######  Proposition 2.6.

Recall the definition of $Y\subseteq\mathbb{A}^{N}_{\mathbb{Z}}$ in (2.11).
Then, $Y$ is an affine variety. Further, let $r$ be the codimension of $Y$.
Then, one has $r>\lfloor(1-C_{n,d}(1,d-1))N\rfloor$.

###### Proof.

Let $\langle{\boldsymbol{a}},v_{d,n}({\boldsymbol{x}})\rangle$ be a reducible
polynomial. Then, we write
$\langle{\boldsymbol{a}},v_{d,n}({\boldsymbol{x}})\rangle=f^{(1)}({\boldsymbol{x}})f^{(2)}({\boldsymbol{x}})$.
Here, $f^{(1)}({\boldsymbol{x}})$ and $f^{(2)}({\boldsymbol{x}})$ are
homogeneous whose degrees are strictly less than $d$. Let
$d_{i}=\deg(f^{(i)}({\boldsymbol{x}}))$ and $t={n+d_{1}-1\choose n-1}$. Then,

(2.15)
$\langle{\boldsymbol{a}},v_{d,n}({\boldsymbol{x}})\rangle=\underbrace{(u_{1}x_{1}^{d_{1}}+\cdots
u_{t}x_{n}^{d_{1}})}_{=f^{(1)}({\boldsymbol{x}})}\underbrace{(u_{t+1}x_{1}^{d_{2}}+\cdots+u_{M(d_{1},d_{2})}x_{n}^{d_{2}})}_{=f^{(2)}({\boldsymbol{x}})}$

Let $Y_{(d_{1},d_{2})}\subset Y$ be a subset where
$\langle{\boldsymbol{a}},v_{d,n}({\boldsymbol{x}})\rangle$ seperates as
(2.15). Comparing the coefficients in (2.15) for both sides, we attain
$a_{i}$’s as a polynomial of $u_{1},\dots,u_{M}$. Now, let us write
$a_{i}=g_{i}(u_{1},\dots,u_{M(d_{1},d_{2})})$. Consider a map
$\varphi_{(d_{1},d_{2})}:{\mathbb{Z}}[t_{1},\dots,t_{N}]\to{\mathbb{Z}}[u_{1},\dots,u_{M(d_{1},d_{2})}]$
sending $t_{i}\mapsto g_{i}(u_{1},\dots,u_{M(d_{1},d_{2})})$. Then, by the
construction, $Y_{(d_{1},d_{2})}=V(\ker\varphi_{(d_{1},d_{2})})$, and so
$Y=\bigcup Y_{(d_{1},d_{2})}$ is an affine variety.

Now, we prove $r>k+1$. We will instead find the upper bound of the dimension
of $Y$. Let $M=\max M(d_{1},d_{2})$. Since
${\mathbb{Z}}[t_{1},\dots,t_{N}]/\ker\varphi_{(d_{1},d_{2})}$ injects into
${\mathbb{Z}}[u_{1},\dots,u_{M(d_{1},d_{2})}]$, $\dim Y$ is strictly less than
$M$. Hence, it suffices to show $M\leq N-\lfloor(1-C_{n,d}(1,d-1))N\rfloor$.
Note that we have $M(d_{1},d_{2})=C_{d,n}(d_{1},d_{2})N\leq
N-\lfloor(1-C_{n,d}(1,d-1))N\rfloor$. Indeed, we have

$\begin{split}M(d_{1},d_{2})&={n+d_{1}-1\choose n-1}+{n+d_{2}-1\choose n-1}\\\
&=\frac{(n+d_{1}-1)!}{(n-1)!d_{1}!}+\frac{(n+d_{2}-1)!}{(n-1)!d_{2}!}\\\
&=\frac{(n+d-1)!}{d!(n-1)!}\underbrace{\left(\frac{d!(n+d_{1}-1)!}{d_{1}!(n+d-1)!}+\frac{d!(n+d_{2}-1)!}{d_{2}!(n+d-1)!}\right)}_{C_{n,d}(d_{1},d_{2})}\\\
&=C_{n,d}(d_{1},d_{2})N\leq N-\lfloor(1-C_{n,d}(1,d-1))N\rfloor\end{split}$

The latter inequality is by Lemma 2.5. Now, since $\dim Y<M\leq
N-\lfloor(1-C_{n,d}(1,d-1))N\rfloor$, we have
$\lfloor(1-C_{n,d}(1,d-1))N\rfloor<N-\dim Y=r$, which is desired. ∎

##  3\. Proof of Theorem 1.1 and Theorem 1.3

In this section, we provide the proofs of Theorem 1.1 and 1.2. Recall the
definition (1.5) and (1.6) of sets $T_{\infty}$ and $T_{p}$ for every prime
$p$. We begin this section with a lemma which says that the sets $T_{\infty}$
and $T_{p}$ are measurable in ${\mathbb{R}}^{N}$ and ${\mathbb{Z}}_{p}^{N}$
with the Lebesgue measure and the Haar measure $\mu_{p}$, respectively.

######  Lemma 3.1.

The sets $T_{\infty}\subseteq{\mathbb{R}}^{N}$ and
$T_{p}\subseteq{\mathbb{Z}}_{p}^{N}$ are measurable with the Lebesgue measure
and the Haar measure $\mu_{p},$ respectively.

###### Proof.

We shall first prove that $T_{\infty}$ is Lebesgue measurable. On observing
that
$f_{{\boldsymbol{a}}}(c{\boldsymbol{x}})=c^{d}f_{{\boldsymbol{a}}}({\boldsymbol{x}})$
for any $c\in{\mathbb{R}}$, we note that

$T_{\infty}=\left\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}\cap[-1,1]^{N}\middle|\
\exists\ {\boldsymbol{x}}\in[-1,1]^{n}\cap{\mathbb{R}}^{n}\ \text{such that}\
f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\}.$

We claim that

(3.1)
$T_{\infty}=\bigcap_{k\in{\mathbb{N}}}\bigcup_{{\boldsymbol{x}}\in[-1,1]^{n}\cap{\mathbb{R}}^{n}}\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|<1/k\\},$

where $|\cdot|$ is the euclidean norm in ${\mathbb{R}}$. Obviously, the set
$T_{\infty}$ is included in the set on the right-hand side of
$(\ref{Tinfinityrep}).$ Conversely, for any element
${\boldsymbol{a}}\in{\mathbb{R}}^{N}$ contained in the set on the right hand
side of $(\ref{Tinfinityrep})$, there exists a sequence
${\boldsymbol{x}}_{m}\in[-1,1]^{n}\cap{\mathbb{R}}^{n}$ such that
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})$ converges to $0$ as
$m\rightarrow\infty$. Since $[-1,1]^{n}\cap{\mathbb{R}}^{n}$ is a compact set,
we see that there exists a limit point
${\boldsymbol{x}}\in[-1,1]^{n}\cap{\mathbb{R}}^{n}$ such that
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$ This means that
${\boldsymbol{a}}\in T_{\infty}.$ Hence, we confirm that
$(\ref{Tinfinityrep})$ holds. Meanwhile, since the set
$[-1,1]^{n}\cap{\mathbb{Q}}^{n}$ is a dense in
$[-1,1]^{n}\cap{\mathbb{R}}^{n}$, we see that

(3.2)
$T_{\infty}=\bigcap_{k\in{\mathbb{N}}}\bigcup_{{\boldsymbol{x}}\in[-1,1]^{n}\cap{\mathbb{Q}}^{n}}\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|<1/k\\}.$

For fixed $k\in{\mathbb{N}}$ and
${\boldsymbol{x}}\in[-1,1]^{n}\cap{\mathbb{Q}}^{n}$, the set
$\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|<1/k\\}$ is an open set. Therefore, on
noting from $(\ref{Tinfinityrep2})$ that $T_{\infty}$ is obtained by taking
countable unions and countable intersections of the set
$\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|<1/k\\}$, one finds that the set
$T_{\infty}$ is Lebesgue measurable.

Next, we shall prove that $T_{p}$ is measurable with $\mu_{p},$ by following
the same method used in the previous paragraph. We claim that

(3.3)
$T_{p}=\bigcap_{k\in{\mathbb{N}}}\bigcup_{\begin{subarray}{c}{\boldsymbol{x}}\in{\mathbb{Z}}_{p}^{n}\end{subarray}}\\{{\boldsymbol{a}}\in{\mathbb{Z}}_{p}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|_{p}<p^{-k}\\}.$

Clearly, the set $T_{p}$ is included in the set on the right-hand side of
$(\ref{Tprep})$. Conversely, for any element
${\boldsymbol{a}}\in{\mathbb{Z}}_{p}^{N}$ in the set on the right hand side of
$(\ref{Tprep}),$ there exists a sequence
${\boldsymbol{x}}_{m}\in{\mathbb{Z}}_{p}^{n}$ such that
$|f_{{\boldsymbol{a}}}({\boldsymbol{x}}_{m})|_{p}$ converges to $0$ as
$m\rightarrow\infty.$ Hence, there exists a point
${\boldsymbol{x}}\in{\mathbb{Z}}_{p}^{N}$ such that
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$ This means that
${\boldsymbol{a}}\in T_{p}.$ Thus, we confirm that $(\ref{Tprep})$ holds.
Meanwhile, since the set ${\mathbb{Z}}^{n}$ is a dense in
${\mathbb{Z}}_{p}^{n}$, we find that

(3.4)
$T_{p}=\bigcap_{k\in{\mathbb{N}}}\bigcup_{\begin{subarray}{c}{\boldsymbol{x}}\in{\mathbb{Z}}^{n}\end{subarray}}\\{{\boldsymbol{a}}\in{\mathbb{Z}}_{p}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|_{p}<p^{-k}\\}.$

For fixed $k\in{\mathbb{N}}$ and ${\boldsymbol{x}}\in{\mathbb{Z}}^{n}$, the
set $\\{{\boldsymbol{a}}\in{\mathbb{Z}}_{p}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|_{p}<p^{-k}\\}$ is open and thus is
measurable with $\mu_{p}$. Therefore, on noting from (3.4) that $T_{p}$ is
obtained by taking countable unions and countable intersections of the
$\\{{\boldsymbol{a}}\in{\mathbb{Z}}_{p}^{N}|\
|f_{{\boldsymbol{a}}}({\boldsymbol{x}})|_{p}<p^{-k}\\}$, we discern that the
set $T_{p}$ is measurable with $\mu_{p}$.

∎

In advance of the proofs of Theorem 1.1 and Theorem 1.2, we provide some
definitions and observations. We consider the natural map

$\displaystyle\Phi^{A}:\ [-A,A]^{N}\cap{\mathbb{Z}}^{N}_{\text{prim}}$
$\displaystyle\rightarrow[-1,1]^{N}\times\displaystyle\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}^{N}$ $\displaystyle{\boldsymbol{a}}$
$\displaystyle\mapsto\left(\frac{{\boldsymbol{a}}}{A},{\boldsymbol{a}},\ldots,{\boldsymbol{a}},\ldots\right).$

We sometimes use a different order of primes in the product $\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}$, for notational convenience.

Furthermore, for a given subset $\mathcal{U}$ of
$[-1,1]^{N}\times\displaystyle\prod_{p\ \text{prime}}{\mathbb{Z}}_{p}^{N}$ and
a given polynomial $P(\mathbf{t})\in{\mathbb{Z}}[t_{1},\dots,t_{N}]$, we
define a quantity ${\boldsymbol{d}}(\mathcal{U},A;P)$ by

(3.5)
${\boldsymbol{d}}(\mathcal{U},A;P):=\frac{\\#\left\\{{\boldsymbol{a}}\in(\Phi^{A})^{-1}(\mathcal{U})\middle|\
P({\boldsymbol{a}})=0\right\\}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}_{\text{prim}}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

We say that a subset of ${\mathbb{Z}}_{p}$ is an open interval if it has the
form $\\{x\in{\mathbb{Z}}_{p}|\ |x-a|_{p}\leq b\\}$ for some
$a\in{\mathbb{Z}}_{p}$ and $b\in{\mathbb{R}}.$ Furthermore, by an open box
$I_{p}$ in ${\mathbb{Z}}_{p}^{N}$ with a given prime $p,$ we mean a Cartesian
product of open intervals. Suppose that $\mathfrak{B}$ be an open box in
$[-1,1]^{N}\cap{\mathbb{R}}^{N}.$ For a given set $\mathfrak{p}$ of prime
numbers, it follows that

$\displaystyle{\boldsymbol{d}}\biggl{(}\mathfrak{B}\times\prod_{p\in\mathfrak{p}}I_{p}\times\prod_{p\notin\mathfrak{p}}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}$
$\displaystyle=\frac{\\#\biggl{\\{}{\boldsymbol{a}}\in(\Phi^{A})^{-1}\biggl{(}\mathfrak{B}\times\displaystyle\prod_{p\in\mathfrak{p}}I_{p}\times\displaystyle\prod_{p\notin\mathfrak{p}}{\mathbb{Z}}_{p}^{N}\biggr{)}\biggl{|}\
P({\boldsymbol{a}})=0\biggr{\\}}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}_{\text{prim}}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

We observe that the set
$(\Phi^{A})^{-1}\biggl{(}\mathfrak{B}\times\displaystyle\prod_{p\in\mathfrak{p}}I_{p}\times\displaystyle\prod_{p\notin\mathfrak{p}}{\mathbb{Z}}_{p}^{N}\biggr{)}$
can be viewed by a set of integers in
$A\mathfrak{B}\cap{\mathbb{Z}}^{N}_{\text{prim}}$ satisfying certain
congruence conditions associated with the radius and the center of the open
intervals defining $I_{p}$. Thus, we infer from the Chinese remainder theorem
that there exists $B\in{\mathbb{Z}}$ whose prime divisors are in
$\mathfrak{p}$, and ${\boldsymbol{r}}\in{\mathbb{Z}}^{N}$ such that

${\boldsymbol{d}}\biggl{(}\mathfrak{B}\times\displaystyle\prod_{p\in\mathfrak{p}}I_{p}\times\displaystyle\prod_{p\notin\mathfrak{p}}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}=\frac{\\#\biggl{\\{}{\boldsymbol{a}}\in
A\mathfrak{B}\cap{\mathbb{Z}}^{N}_{\text{prim}}\biggl{|}\
P({\boldsymbol{a}})=0,\ {\boldsymbol{a}}\equiv{\boldsymbol{r}}\ \text{mod}\
B\biggr{\\}}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}_{\text{prim}}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

Then, by applying Lemma 2.1, we obtain

(3.6)
$\displaystyle{\boldsymbol{d}}\biggl{(}\mathfrak{B}\times\prod_{p\in\mathfrak{p}}I_{p}\times\prod_{p\notin\mathfrak{p}}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}$
$\displaystyle=\frac{\frac{1}{\zeta(N-k)}\prod_{p\notin\mathfrak{p}}\sigma_{p}\cdot\prod_{p\in\mathfrak{p}}\sigma_{p}^{B,{\boldsymbol{r}}}\cdot\sigma_{\infty}(\mathfrak{B})+O(A^{-\delta})}{\frac{1}{\zeta(N-k)}\prod_{p}\sigma_{p}\cdot\sigma_{\infty}([-1,1]^{N})+O(A^{-\delta})}.$

For a given measurable set $S_{p}\subseteq{\mathbb{Z}}_{p}^{N},$ recall the
definition of $\sigma_{p}(S_{p})$ in the preamble to the statement of Theorem
$\ref{thm1.2}.$ We observe that
$\sigma_{p}^{B,{\boldsymbol{r}}}=\sigma_{p}(I_{p})$ and
$\sigma_{p}=\sigma_{p}({\mathbb{Z}}_{p})$. Therefore, we find from (3.6) that

(3.7)
$\lim_{A\rightarrow\infty}{\boldsymbol{d}}\biggl{(}\mathfrak{B}\times\displaystyle\prod_{p\in\mathfrak{p}}I_{p}\times\displaystyle\prod_{p\notin\mathfrak{p}}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}=\frac{\prod_{p\in\mathfrak{p}}\sigma_{p}(I_{p})\cdot\sigma_{\infty}(\mathfrak{B})}{\prod_{p\in\mathfrak{p}}\sigma_{p}({\mathbb{Z}}_{p})\cdot\sigma_{\infty}([-1,1]^{N})}$

###### Proof of Theorem 1.1.

Recall that

$\displaystyle T_{\infty}$
$\displaystyle=\left\\{{\boldsymbol{a}}\in[-1,1]^{N}\cap{\mathbb{R}}^{N}\middle|\
\exists\ {\boldsymbol{x}}\in{\mathbb{R}}^{n}\setminus\\{\boldsymbol{0}\\}\
\text{such that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\}$
$\displaystyle T_{p}$
$\displaystyle=\left\\{{\boldsymbol{a}}\in\mathbb{Z}_{p}^{N}\middle|\ \exists\
{\boldsymbol{x}}\in{\mathbb{Z}}_{p}^{n}\setminus\\{\boldsymbol{0}\\}\
\text{such that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\}.$

On recalling the definition (3.5) of $\boldsymbol{d}(\cdot,A;P),$ we infer
that

$\varrho_{d,n}^{P,\text{loc}}(A)=\boldsymbol{d}\biggl{(}T_{\infty}\times\displaystyle\prod_{p\
\text{prime}}T_{p},A;P\biggr{)}.$

Thus, it suffices to show that

(3.8)
$\lim_{A\rightarrow\infty}\boldsymbol{d}\biggl{(}T_{\infty}\times\displaystyle\prod_{p\
\text{prime}}T_{p},A;P\biggr{)}=c_{P}.$

In order to verify that the equality $(\ref{goal})$ holds, we introduce

$\boldsymbol{d}(A,M)=\boldsymbol{d}\biggl{(}T_{\infty}\times\displaystyle\prod_{p<M}T_{p}\times\displaystyle\prod_{p\geq
M}{\mathbb{Z}}_{p},A;P\biggr{)}.$

One sees by applying the triangle inequality that

(3.9)
$\displaystyle\lim_{A\rightarrow\infty}\biggl{|}\boldsymbol{d}\biggl{(}T_{\infty}\times\prod_{p\
\text{prime}}T_{p},A;P\biggr{)}-c_{P}\biggr{|}\leq\lim_{A\rightarrow\infty}|d_{1}(A,M)|+\lim_{A\rightarrow\infty}|d_{2}(A,M)|,$

where

$d_{1}(A,M)=\boldsymbol{d}\biggl{(}T_{\infty}\times\displaystyle\prod_{p\
\text{prime}}T_{p},A;P\biggr{)}-\boldsymbol{d}(A,M)$

and

$d_{2}(A,M)=\boldsymbol{d}(A,M)-c_{P}.$

First, we analyze the quantity

$\lim_{A\rightarrow\infty}|d_{1}(A,M)|.$

We readily see from the definition of $\boldsymbol{d}(A,M)$ that

$|d_{1}(A,M)|=\boldsymbol{d}\biggl{(}T_{\infty}\times\prod_{p<M}T_{p}\times\biggl{(}\prod_{p\geq
M}T_{p}\biggr{)}^{c},A;P\biggr{)}.$

Furthermore, we find that

(3.10) $\displaystyle|d_{1}(A,M)|$
$\displaystyle\leq\boldsymbol{d}\biggl{(}[-1,1]^{N}\times\prod_{p<M}{\mathbb{Z}}_{p}\times\biggl{(}\prod_{p\geq
M}T_{p}\biggr{)}^{c},A;P\biggr{)}$
$\displaystyle\leq\frac{\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
\begin{aligned} &(i)\ \exists\ p>M\ \text{s.t.}\
f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\ \text{has no solution in}\
{\mathbb{Z}}_{p}^{n}\\\ &(ii)\
P({\boldsymbol{a}})=0\end{aligned}\right\\}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}_{\text{prim}}^{N}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

Meanwhile, for sufficiently large prime $p$, whenever
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})$ is irreducible over
$\overline{{\mathbb{Z}}/p{\mathbb{Z}}}$, the Lang-Weil estimate [References]
(see also $[\ref{ref100.0},\text{Theorem}\ 3]$) ensures the existence of a
smooth point ${\boldsymbol{x}}\in({\mathbb{Z}}/p{\mathbb{Z}})^{n}$ satisfying
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0$. Then, by Hensel’s lemma, we have a
point ${\boldsymbol{x}}$ in ${\mathbb{Q}}_{p}$ satisfying
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$ Therefore, we conclude from
$(\ref{d1AM})$ that for sufficiently large $M>0$, one has

(3.11) $\displaystyle|d_{1}(A,M)|$
$\displaystyle\leq\frac{\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
\begin{aligned} &(i)\ \exists\ p>M\ \text{s.t.}\
f_{{\boldsymbol{a}}}({\boldsymbol{x}})\ \text{is reducible over}\
\overline{{\mathbb{Z}}/p{\mathbb{Z}}}\\\ &(ii)\
P({\boldsymbol{a}})=0\end{aligned}\right\\}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}_{\text{prim}}^{N}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

We shall apply Lemma 2.3 with $Y$ defined in (2.11). With this $Y$ in mind, we
find from $(\ref{3.113.11})$ that

(3.12) $\displaystyle|d_{1}(A,M)|$
$\displaystyle\leq\frac{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
\begin{aligned} &(i)\ {\boldsymbol{a}}\ (\text{mod}\ p)\in Y(\mathbb{F}_{p})\
\text{for some prime}\ p>M\\\ &(ii)\
P({\boldsymbol{a}})=0\end{aligned}\right\\}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}_{\text{prim}}^{N}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

Note by the classical argument (see $[\ref{ref8}]$ and $[\ref{ref26},\text{the
proof of Theorem 1.3}]$) that the fact that $P({\bf t})=0$ has a nontrivial
integer solution implies that
$\sigma_{\infty}([-1,1]^{N})\cdot\prod_{p}\sigma_{p}({\mathbb{Z}}_{p})\asymp
1$. Then, one infers by Lemma 2.1 that

$\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}_{\text{prim}}^{N}\middle|\
P({\boldsymbol{a}})=0\right\\}\asymp A^{N-k}.$

Proposition 2.6 together with the hypothesis in the statement of Theorem 1.1
that $k<\lfloor(1-C_{n,d}(1,d-1))N\rfloor$ reveals that the codimension $r$ of
$Y$ is strictly greater than $k+1$. Hence, we find by applying Lemma 2.3 that

$|d_{1}(A,M)|\ll\frac{1}{M^{r-k-1}\log M}+A^{k-r+1}.$

Therefore, we obtain

(3.13) $\lim_{A\rightarrow\infty}|d_{1}(A,M)|\ll\frac{1}{M^{r-k-1}\log M}.$

Next, we turn to estimate $\lim_{A\rightarrow\infty}|d_{2}(A,M)|.$ For
simplicity, we temporarily write

(3.14) $\lim_{A\rightarrow\infty}|d_{2}(A,M)|=\varphi(M).$

One infers by $(\ref{key})$ together with Lemma 3.1 that

$d_{2}(A,M)=\frac{\prod_{p\leq
M}\sigma_{p}(T_{p})\cdot\sigma_{\infty}(T_{\infty})}{\prod_{p\in\mathfrak{p}}\sigma_{p}({\mathbb{Z}}_{p})\cdot\sigma_{\infty}([-1,1]^{N})},$

and thus by the definition of $c_{P}$, we discern that

(3.15) $\varphi(M)\rightarrow 0,$

as $M\rightarrow\infty.$

Hence, we conclude from $(\ref{triangle}),$ $(\ref{3.123.12})$ and
$(\ref{3.83.8})$ that

$\lim_{A\rightarrow\infty}\biggl{|}\boldsymbol{d}\biggl{(}T_{\infty}\times\displaystyle\prod_{p\
\text{prime}}T_{p},A;P\biggr{)}-c_{P}\biggr{|}\ll\frac{1}{M^{r-k-1}\log
M}+\varphi(M).$

By letting $M\rightarrow\infty$, we see from $(\ref{3.103.10})$ that

$\lim_{A\rightarrow\infty}\biggl{|}\boldsymbol{d}\biggl{(}T_{\infty}\times\displaystyle\prod_{p\
\text{prime}}T_{p},A;P\biggr{)}-c_{P}\biggr{|}=0,$

which gives $(\ref{goal})$. This completes the proof of Theorem
$\ref{thm1.2}.$ ∎

For the proof of Theorem 1.2, we require a proposition that plays an important
role in guaranteeing the positiveness of $c_{P}$. In order to describe this
proposition, it is convenient to define

$S_{\infty}:=\left\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}\middle|\ \exists\
{\boldsymbol{x}}\in{\mathbb{R}}^{n}\setminus\\{\boldsymbol{0}\\}\ \text{such
that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\}$

and recall the definition $(\ref{Tp})$ of $T_{p}$ for every prime $p.$ Also,
for a given ${\boldsymbol{b}}\in{\mathbb{Z}}^{N},$ we define

$\displaystyle B_{\infty}({\boldsymbol{b}},\eta)$
$\displaystyle=\left\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}\middle|\
|a_{i}-b_{i}|<\eta\ \text{for $1\leq i\leq N$}\right\\}$ $\displaystyle
B_{p}({\boldsymbol{b}},\eta)$
$\displaystyle=\left\\{{\boldsymbol{a}}\in{\mathbb{Z}}_{p}^{N}\middle|\
|a_{i}-b_{i}|_{p}<\eta\ \text{for $1\leq i\leq N$}\right\\},$

for every prime $p.$ Furthermore, we consider two functions
$\Psi_{\infty}:{\mathbb{R}}^{N}\times{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}$
and
$\Psi_{p}:{\mathbb{Z}}_{p}^{N}\times{\mathbb{Z}}_{p}^{n}\rightarrow{\mathbb{Z}}_{p}$
defined by

(3.16)
$\Psi_{\infty}({\boldsymbol{a}},{\boldsymbol{x}}):=f_{{\boldsymbol{a}}}({\boldsymbol{x}})\
\text{and}\
\Psi_{p}({\boldsymbol{a}},{\boldsymbol{x}}):=f_{{\boldsymbol{a}}}({\boldsymbol{x}}),$

for every prime $p.$ For the gradient vectors associated with the polynomial
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})$, we use the notation

(3.17) $\displaystyle\nabla
f_{{\boldsymbol{a}}}({\boldsymbol{x}})=(\partial_{x_{1}}f_{{\boldsymbol{a}}}({\boldsymbol{x}}),\ldots,\partial_{x_{n}}f_{{\boldsymbol{a}}}({\boldsymbol{x}})).$

######  Proposition 3.2.

Supoose that there exists ${\boldsymbol{b}}\in{\mathbb{Z}}^{N}\setminus\\{{\bf
0}\\}$ such that the variety $V_{{\boldsymbol{b}}}$ in ${\mathbb{P}}^{n-1}$
admits a smooth ${\mathbb{Q}}$-rational point. Then, for each prime $p$, there
exists positive numbers $\eta_{p}$ and $\eta_{\infty}$ less than or equal to
$1$ such that

$\displaystyle B_{p}({\boldsymbol{b}},\eta_{p})$ $\displaystyle\subseteq
T_{p}$ $\displaystyle B_{\infty}({\boldsymbol{b}},\eta_{\infty})$
$\displaystyle\subseteq S_{\infty}.$

For the proof of Proposition 3.2, we mainly use a version of Hensel’s lemma in
$p$-adic numbers [References, Theorem 25] and use the implicit function
theorem in real numbers.

###### Proof.

First, we shall show that for each prime $p$, there exists $\eta_{p}>0$ such
that

(3.18) $B_{p}({\boldsymbol{b}},\eta_{p})\subseteq T_{p}.$

Recall the definition $(\ref{funcs1.1})$ of $\Psi_{p}$ and the notation
$(\ref{notat1.2})$ for the gradient vector. Note from the hypothesis that
there exists ${\boldsymbol{b}}\in{\mathbb{Z}}^{N}$ with
$P({\boldsymbol{b}})=0$ such that the variety $V_{{\boldsymbol{b}}}$ in
${\mathbb{P}}^{n-1}$ admits a smooth ${\mathbb{Q}}$-rational point. Since
$V_{{\boldsymbol{b}}}$ is a projective variety, it also admits a smooth
integer point. Then, if we write this integer point by
${\boldsymbol{y}}\in{\mathbb{Z}}^{n},$ we find that

(3.19)
$\Psi_{p}({\boldsymbol{b}},{\boldsymbol{y}})=f_{{\boldsymbol{b}}}({\boldsymbol{y}})=0\
\text{and}\ \nabla f_{{\boldsymbol{b}}}({\boldsymbol{y}})\neq{\bf
0}\in{\mathbb{Z}}_{p}^{n}.$

Then, we infer that there exists $j_{0}$ with $1\leq j_{0}\leq n$ such that

(3.20) $\left|\left(\nabla
f_{{\boldsymbol{b}}}({\boldsymbol{y}})\right)_{j_{0}}\right|_{p}\neq 0.$

Let us write $\left|\left(\nabla
f_{{\boldsymbol{b}}}({\boldsymbol{y}})\right)_{j_{0}}\right|_{p}=p^{-\alpha}$
with $\alpha\in{\mathbb{N}}\cup\\{0\\}.$ We find by applying [References,
Theorem 25] with the function $\Psi_{p}$ that for all $p$-adic numbers
$a_{i}\in{\mathbb{Z}}_{p}\ (1\leq i\leq N)$ with
$|a_{i}-b_{i}|_{p}<p^{-2\alpha}$ and for all $p$-adic numbers
$x_{j}\in{\mathbb{Z}}_{p}\ (1\leq j\leq n,\ j\neq j_{0})$ with
$|x_{j}-y_{j}|_{p}<p^{-2\alpha}$, there is a unique $p$-adic number
$x_{j_{0}}$ with $|x_{j_{0}}-y_{j_{0}}|_{p}<p^{-\alpha}$ such that

(3.21)
$\Psi_{p}({\boldsymbol{a}},{\boldsymbol{x}})=f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$

Therefore, by setting $\eta_{p}=p^{-2\alpha},$ we conclude that for all
${\boldsymbol{a}}\in B_{p}({\boldsymbol{b}},\eta_{p})$, there exists
${\boldsymbol{x}}\in{\mathbb{Z}}_{p}^{n}$ such that
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$ In other words, we have

(3.22) $B_{p}({\boldsymbol{b}},\eta_{p})\subseteq T_{p}.$

Next, we shall show that there exists $\eta_{\infty}>0$ such that

$B_{\infty}({\boldsymbol{b}},\eta_{\infty})\subseteq S_{\infty}.$

In order to do this, we use the same argument leading from $(\ref{claim1})$ to
$(\ref{1.6})$ with the implicit function theorem in real numbers in place of
[References, Theorem 25]. By using the same notation ${\boldsymbol{y}}$ for a
smooth ${\mathbb{Q}}$-rational point that the variety
$V_{{\boldsymbol{b}}}\in{\mathbb{P}}^{n-1}$ admits, we infer by applying the
argument leading from (3.18) to $(\ref{1.5})$ that there exists $j_{0}$ with
$1\leq j_{0}\leq n$ such that

$\left|\left(\nabla
f_{{\boldsymbol{b}}}({\boldsymbol{y}})\right)_{j_{0}}\right|\neq 0.$

Then, it follows by the implicit function theorem that there exists $\gamma>0$
having the property that for all real numbers $a_{i}\in{\mathbb{R}}$ $(1\leq
i\leq N)$ with $|a_{i}-b_{i}|<\gamma$ and for all real numbers
$x_{j}\in{\mathbb{R}}$ ($1\leq j\leq n,\ j\neq j_{0}$) with
$|x_{j}-y_{j}|<\gamma$ there is a unique real number $x_{j_{0}}$ such that

$\Psi_{\infty}({\boldsymbol{a}},{\boldsymbol{x}})=f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$

Therefore, by setting $\eta_{\infty}=\gamma$, we conclude that for all
${\boldsymbol{a}}\in B_{\infty}({\boldsymbol{b}},\eta_{\infty}),$ there exists
${\boldsymbol{x}}\in{\mathbb{R}}^{n}$ such that
$f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0.$ In other words, we have

$B_{\infty}({\boldsymbol{b}},\eta_{\infty})\subseteq S_{\infty}.$

This completes the proof of Proposition $\ref{lem1.3}.$ ∎

###### Proof of Theorem 1.2.

Recall the natural map

$\displaystyle\Phi^{A}:\ [-A,A]^{N}\cap{\mathbb{Z}}^{N}_{\text{prim}}$
$\displaystyle\rightarrow[-1,1]^{N}\times\displaystyle\prod_{p\
\text{prime}}{\mathbb{Z}}_{p}^{N}$ $\displaystyle{\boldsymbol{a}}$
$\displaystyle\mapsto\left(\frac{{\boldsymbol{a}}}{A},{\boldsymbol{a}},\ldots,{\boldsymbol{a}},\ldots\right).$

Furthermore, we recall the definitions

$\displaystyle S_{\infty}$
$\displaystyle=\left\\{{\boldsymbol{a}}\in{\mathbb{R}}^{N}\middle|\ \exists\
{\boldsymbol{x}}\in{\mathbb{R}}^{n}\setminus\\{\boldsymbol{0}\\}\ \text{such
that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\}$ $\displaystyle
T_{p}$ $\displaystyle=\left\\{{\boldsymbol{a}}\in\mathbb{Z}_{p}^{N}\middle|\
\exists\ {\boldsymbol{x}}\in{\mathbb{Z}}_{p}^{n}\setminus\\{\boldsymbol{0}\\}\
\text{such that}\ f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\right\\},$

for every prime $p.$

Recall from the hypothesis in the statement of Theorem 1.1 that there exists
${\boldsymbol{b}}\in{\mathbb{Z}}^{N}$ with $P({\boldsymbol{b}})=0$ such that
the variety $V_{{\boldsymbol{b}}}$ in ${\mathbb{P}}^{n-1}$ admits a smooth
${\mathbb{Q}}$-rational point. One sees by applying Proposition 3.2 that for
each prime $p$ there exist positive numbers $\eta_{p}$ and $\eta_{\infty}$
less than $1$ such that $B_{p}({\boldsymbol{b}},\eta_{p})\subseteq T_{p}$ and
$B_{\infty}({\boldsymbol{b}},\eta_{\infty})\subseteq S_{\infty}.$ We choose a
sufficiently large number $C:=C({\boldsymbol{b}})>0$ such that
$B_{\infty}({\boldsymbol{b}}/C,\eta_{\infty}/C)\subseteq[-1,1]^{N}.$
Furthermore, on observing the relation that
$f_{{\boldsymbol{b}}/C}({\boldsymbol{x}})=(1/C)\cdot
f_{{\boldsymbol{b}}}({\boldsymbol{x}})$, we infer that
$B_{\infty}({\boldsymbol{b}}/C,\eta_{\infty}/C)\subseteq S_{\infty}.$
Therefore, on noting that

$\displaystyle T_{\infty}=S_{\infty}\cap[-1,1]^{N}.$

one deduces that

$B_{\infty}({\boldsymbol{b}}/C,\eta/C)\subseteq T_{\infty}.$

Meanwhile, it follows by Theorem 1.1 that

$\lim_{A\rightarrow\infty}\varrho_{d,n}^{P,\text{loc}}(A)=c_{P},$

and thus, we find that

(3.23)
$\lim_{A\rightarrow\infty}\varrho_{d,n}^{P,\text{loc}}(A)\geq\frac{\prod_{p<M}\sigma_{p}(B_{p}({\boldsymbol{b}},\eta_{p}))\cdot\prod_{p\geq
M}\sigma_{p}(T_{p})\cdot\sigma_{\infty}(B_{\infty}({\boldsymbol{b}}/C,\eta_{\infty}/C))}{\prod_{p}\sigma_{p}({\mathbb{Z}}_{p})\cdot\sigma_{\infty}([-1,1]^{N})},$

for any $M>0.$ Then, it suffices to show that the right-hand side in
$(\ref{lowerbound})$ is greater than $0$. We shall prove this by showing that
there exists $M>0$ such that

(3.24)
$\frac{\prod_{p<M}\sigma_{p}(B_{p}({\boldsymbol{b}},\eta_{p}))\cdot\sigma_{\infty}(B_{\infty}({\boldsymbol{b}}/C,\eta_{\infty}/C))}{\prod_{p<M}\sigma_{p}({\mathbb{Z}}_{p})\cdot\sigma_{\infty}([-1,1]^{N})}>0$

and

(3.25) $\frac{\prod_{p\geq M}\sigma_{p}(T_{p})}{\prod_{p\geq
M}\sigma_{p}({\mathbb{Z}}_{p})}>0.$

First, we shall show that the inequality $(\ref{11})$ holds. For a given
$B\in{\mathbb{N}}$ and ${\boldsymbol{r}}\in{\mathbb{Z}}^{N}$, we recall the
definition of $\sigma_{p}^{B,{\boldsymbol{r}}}$ in the statement of Lemma 2.1.
Note that there exists $B\in{\mathbb{N}}\cup\\{0\\}$ such that

(3.26)
$\displaystyle\prod_{p<M}\sigma_{p}(B_{p}({\boldsymbol{b}},\eta_{p}))=\displaystyle\prod_{p<M}\sigma_{p}^{B,{\boldsymbol{b}}}.$

Furthermore, the $p$-adic densities $\sigma_{p}({\mathbb{Z}}_{p})$,
$\sigma_{p}^{B,{\boldsymbol{b}}}$ and the real densities
$\sigma_{\infty}([-1,1]^{N})$,
$\sigma_{\infty}(B_{\infty}({\boldsymbol{b}}/C,\eta_{\infty}/C))$ are greater
than $0$ by the application of the Hensel’s lemma and the implicit function
theorem (see [References, the proof of Theorem 1.3] or [References, Lemma
5.7]). Therefore, one sees from (3.26) that the inequality $(\ref{11})$ holds
for any $M>0$.

Next, we shall show that $(\ref{22})$ holds. For any $M_{1}$ with $M_{1}>M,$
we find from $(\ref{key})$ with $[M,M_{1})$ and $[-1,1]^{N}$ in place of
$\mathfrak{p}$ and $\mathfrak{B}$ that

(3.27)
$\displaystyle\lim_{A\rightarrow\infty}{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\prod_{p\in[M,M_{1})}T_{p}\times\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}=\frac{\prod_{p\in[M,M_{1})}\sigma_{p}(T_{p})}{\prod_{p\in[M,M_{1})}\sigma_{p}({\mathbb{Z}}_{p})}.$

Meanwhile, we see that

$\displaystyle{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\biggl{(}\prod_{p\in[M,M_{1})}T_{p}\biggr{)}^{c}\times\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}$
$\displaystyle\leq\frac{\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}^{N}\middle|\
\begin{aligned} &(i)\ \exists\ p>M\ \text{s.t.}\
f_{{\boldsymbol{a}}}({\boldsymbol{x}})=0\ \text{has no solution in}\
{\mathbb{Z}}_{p}^{n}\\\ &(ii)\
P({\boldsymbol{a}})=0\end{aligned}\right\\}}{\\#\left\\{{\boldsymbol{a}}\in[-A,A]^{N}\cap{\mathbb{Z}}_{\text{prim}}^{N}\middle|\
P({\boldsymbol{a}})=0\right\\}}.$

Then, it follows by the same argument leading from (3.10) to
$(\ref{3.123.12})$ that

(3.28)
$\displaystyle\lim_{A\rightarrow\infty}{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\biggl{(}\prod_{p\in[M,M_{1})}T_{p}\biggr{)}^{c}\times\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}\ll\frac{1}{M^{r-k-1}\log
M}.$

Thus, on noting that

$\displaystyle{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\prod_{p\in[M,M_{1})}T_{p}\times\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}$
$\displaystyle=1-{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\biggl{(}\prod_{p\in[M,M_{1})}T_{p}\biggr{)}^{c}\times\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)},$

we find from (3.28) that

(3.29)
$\lim_{A\rightarrow\infty}{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\displaystyle\prod_{p\in[M,M_{1})}T_{p}\times\displaystyle\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}>1/2,$

for sufficiently large $M>0.$ Therefore, it follows from $(\ref{final1})$ and
$(\ref{final2})$ that for sufficiently large $M>0,$ one has

$\displaystyle\frac{\prod_{p\geq M}\sigma_{p}(T_{p})}{\prod_{p\geq
M}\sigma_{p}({\mathbb{Z}}_{p})}$
$\displaystyle=\lim_{M_{1}\rightarrow\infty}\frac{\prod_{p\in[M,M_{1})}\sigma_{p}(T_{p})}{\prod_{p\in[M,M_{1})}\sigma_{p}({\mathbb{Z}}_{p})}$
$\displaystyle=\lim_{M_{1}\rightarrow\infty}\lim_{A\rightarrow\infty}{\boldsymbol{d}}\biggl{(}[-1,1]^{N}\times\prod_{p\in[M,M_{1})}T_{p}\times\prod_{p\notin[M,M_{1})}{\mathbb{Z}}_{p}^{N},A;P\biggr{)}>1/2.$

Hence, the inequality $(\ref{22})$ holds. By using the inequalities
$(\ref{11})$ and $(\ref{22})$, one finds that

$\frac{\prod_{p<M}\sigma_{p}(B_{p}({\boldsymbol{b}},\eta_{p}))\cdot\prod_{p\geq
M}\sigma_{p}(T_{p})\cdot\sigma_{\infty}(B_{\infty}({\boldsymbol{b}}/C,\eta_{\infty}/C))}{\prod_{p}\sigma_{p}({\mathbb{Z}}_{p})\cdot\sigma_{\infty}([-1,1]^{N})}>0,$

and thus we conclude from (3.23) that

$\displaystyle\lim_{A\rightarrow\infty}\varrho_{d,n}^{P,\text{loc}}(A)>0.$

∎

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