or there is a $k$-dimensional subspace $V\subset\mathbb{R}^{n}$ spanned by
standard basis vectors such that

$\left\lVert
M\bm{v}\right\rVert_{\infty}\gg_{m,n}C^{-1}\left\lVert\bm{v}\right\rVert_{\infty}\quad\text{for
all $\bm{v}\in V$}.$

Next, we are interested in counting the number of integer tuples contained in
the sets $K_{k}(E_{1},\ldots,E_{k+1})$. The next Lemma is taken from [17].

######  Lemma 7.9 (Lemma 4.1 in [17]).

Let $B,C\geq 1$, $\sigma\in\\{0,\ldots,n-1\\}$ and
$k\in\\{0,\ldots,n-\sigma-1\\}$. Assume further $CB\geq E_{1}\geq\ldots\geq
E_{k+1}\geq 1$. Then one of the following alternatives must hold.

  1. (I)$k$

We have the estimate

$\\#(\mathbb{Z}^{n}\cap
K_{k}(E_{1},\ldots,E_{k+1}))\ll_{C,n}B^{\sigma}(E_{1}\cdots
E_{k+1})E_{k+1}^{n-\sigma-k-1}.$

  2. (II)$k$

For some integer $b\in\\{1,\ldots,k\\}$ there exists a
$(\sigma+b+1)$-dimensional subspace $X\subset\mathbb{R}^{n}$ and there exists
$\bm{x}^{(0)}\in K_{b}(E_{1},\ldots,E_{b+1})$ such that $E_{b+1}<C^{-1}E_{b}$
and

$\left\lVert
J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x}^{(0)})\bm{X}\right\rVert_{\infty}\leq
C^{-1}\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}\left\lVert\bm{X}\right\rVert_{\infty}\quad\text{for
all $\bm{X}\in X$.}$

  3. (III)

There exists a $(\sigma+1)$-dimensional subspace $X\subset\mathbb{R}^{n}$ such
that

(7.12) $\left\lVert\widetilde{H}_{\bm{\beta}}(\bm{X})\right\rVert_{\infty}\leq
C^{-1}\left\lVert\bm{X}\right\rVert_{\infty}\quad\text{for all $\bm{X}\in
X$.}$

######  Remark 7.10.

In [17], Lemma 7.9 was stated for $\widetilde{H}_{\bm{\beta}}(\bm{x})$ being a
symmetric matrix, and $\lambda_{\bm{\beta},i}(\bm{x})$ were taken to be the
eigenvalues of $\widetilde{H}_{\bm{\beta}}(\bm{x})$ whose absolute values
coincide with its singular values. However, an inspection of the proof shows
that only the estimates in Lemma 7.8 as well as (7.3) were used, which are
valid for singular values as well as the (absolute values) of the eigenvalues.
Therefore the proof remains valid in our setting.

The next Lemma is similar to Lemma 5.1 in [17], however we need to account for
the fact that $\widetilde{H}_{\bm{\beta}}(\bm{x})$ is not necessarily a
symmetric matrix.

######  Lemma 7.11.

Let $b\in\\{1,\ldots,n-1\\}$ and $\bm{x}^{(0)}\in\mathbb{R}^{n}$ be such that
$\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\neq 0$. Then there exist subspaces
$Y_{1},Y_{2}\subseteq\mathbb{R}^{n}$ with $\dim Y_{1}=\dim Y_{2}=n-b$ such
that for all $\bm{Y}_{1}\in Y_{1}$, $\bm{Y}_{2}\in Y_{2}$ and
$\bm{t}\in\mathbb{R}^{n}$ we have

(7.13)
$\bm{Y}_{1}^{T}\widetilde{H}_{\bm{\beta}}(\bm{t})\bm{Y}_{2}\ll_{n}\left(\frac{\left\lVert
J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x}^{(0)})\bm{t}\right\rVert_{\infty}}{\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}}+\frac{\lambda_{\bm{\beta},b+1}(\bm{x}^{(0)})\cdot\left\lVert\bm{t}\right\rVert_{\infty}}{\lambda_{\bm{\beta},b}(\bm{x}^{(0)})}\right)\left\lVert\bm{Y}_{1}\right\rVert_{\infty}\left\lVert\bm{Y}_{2}\right\rVert_{\infty}$

where the implied constant only depends on $n$ but is otherwise independent
from $\widetilde{H}_{\bm{\beta}}(\bm{t})$

###### Proof.

Given $\bm{x}\in\mathbb{R}^{n}$ define
$\bm{y}^{(1)}_{1}(\bm{x}),\ldots,\bm{y}^{(n-b)}_{1}(\bm{x})$ in the following
way. The $j$-th entries are given by

(7.14)
$(y_{1}^{(i)}(\bm{x}))_{j}=\begin{cases}(-1)^{n-b}\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x})_{k\ell})_{\begin{subarray}{c}k=n-b+1,\ldots,n\\\
\ell=n-b+1,\ldots,n\end{subarray}}\right)\quad&\text{if $j=i$,}\\\
(-1)^{j}\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x})_{k\ell})_{\begin{subarray}{c}k=i,n-b+1,\ldots,n;\;k\neq
j\\\ \ell=n-b+1,\ldots,n\end{subarray}}\right)&\text{if $j>n-b$,}\\\
0&\text{otherwise},\end{cases}$

where $k=i,n-b+1,\ldots,n;\;k\neq j$ denotes that we let the index $k$ run
over the values $i,n-b+1,\ldots,n$ with $k=j$ omitted. Similarly we define
$\bm{y}^{(1)}_{2}(\bm{x}),\ldots,\bm{y}^{(n-b)}_{2}(\bm{x})$ by

$(y_{2}^{(i)}(\bm{x}))_{j}=\begin{cases}(-1)^{n-b}\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x})_{k\ell})_{\begin{subarray}{c}k=n-b+1,\ldots,n\\\
\ell=n-b+1,\ldots,n\end{subarray}}\right)\quad&\text{if $j=i$,}\\\
(-1)^{j}\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x})_{k\ell})_{\begin{subarray}{c}k=n-b+1,\ldots,n\\\
\ell=i,n-b+1,\ldots,n;\;\ell\neq j\end{subarray}}\right)&\text{if $j>n-b$,}\\\
0&\text{otherwise}.\end{cases}$

Using the Laplace expansion of a determinant along columns and rows we thus
obtain

(7.15)
$(\bm{y}_{1}^{(i)}(\bm{x})^{T}\widetilde{H}_{\bm{\beta}}(\bm{x}))_{j}=\begin{cases}(-1)^{n-b}\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x})_{k\ell})_{\begin{subarray}{c}k=i,n-b+1,\ldots,n\\\
\ell=j,n-b+1,\ldots,n\end{subarray}}\right)\quad&\text{if $j\leq n-b$,}\\\
0&\text{otherwise,}\end{cases}$

and

(7.16)
$(\widetilde{H}_{\bm{\beta}}(\bm{x})\bm{y}_{2}^{(i)}(\bm{x}))_{j}=\begin{cases}(-1)^{n-b}\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x})_{k\ell})_{\begin{subarray}{c}k=j,n-b+1,\ldots,n\\\
\ell=i,n-b+1,\ldots,n\end{subarray}}\right)\quad&\text{if $j\leq n-b$,}\\\
0&\text{otherwise,}\end{cases}$

respectively. It follows from (7.14) — (7.16) that there exist matrices
$L_{1}^{(i)}$, $L_{2}^{(i)}$, $M_{1}^{(i)}$ and $M_{2}^{(i)}$ for
$i=1,\ldots,n-b$ with entries only in $\\{0,\pm 1\\}$ such that we obtain

(7.17) $\displaystyle\bm{y}_{1}^{(i)}(\bm{x})$
$\displaystyle=L_{1}^{(i)}\bm{D}^{(\bm{\beta},b)}(\bm{x}),$ (7.18)
$\displaystyle\bm{y}_{2}^{(i)}(\bm{x})$
$\displaystyle=L_{2}^{(i)}\bm{D}^{(\bm{\beta},b)}(\bm{x}),$ (7.19)
$\displaystyle(\bm{y}_{1}^{(i)}(\bm{x}))^{T}\widetilde{H}_{\bm{\beta}}(\bm{x})$
$\displaystyle=[M_{1}^{(i)}\bm{D}^{(\bm{\beta},b+1)}(\bm{x})]^{T},\quad\text{and}$
(7.20)
$\displaystyle\widetilde{H}_{\bm{\beta}}(\bm{x})\bm{y}_{2}^{(i)}(\bm{x})$
$\displaystyle=M_{2}^{(i)}\bm{D}^{(\bm{\beta},b+1)}(\bm{x}).$

Given $\bm{t}\in\mathbb{R}^{n}$ we write $\partial_{\bm{t}}$ for the
directional derivative given by $\sum t_{i}\frac{\partial}{\partial x_{i}}$.
Applying $\partial_{\bm{t}}$ to both sides of (7.20) we obtain

(7.21)
$[\partial_{\bm{t}}\widetilde{H}_{\bm{\beta}}(\bm{x})]\bm{y}_{2}^{(i)}(\bm{x})+\widetilde{H}_{\bm{\beta}}(\bm{x})[\partial_{\bm{t}}\bm{y}_{2}^{(i)}(\bm{x})]=M_{2}^{(i)}[\partial_{\bm{t}}\bm{D}^{(\bm{\beta},b+1)}(\bm{x})].$

Now note

(7.22)
$\partial_{\bm{t}}\bm{D}^{(\bm{\beta},b+1)}(\bm{x})=J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x})\bm{t},\quad\text{and}\quad\partial_{\bm{t}}\widetilde{H}_{\bm{\beta}}(\bm{x})=\widetilde{H}_{\bm{\beta}}(\bm{t}).$

Substituting (7.22) and (7.18) into (7.21) yields

$\widetilde{H}_{\bm{\beta}}(\bm{t})\bm{y}_{2}^{(i)}(\bm{x})=M_{2}^{(i)}J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x})\bm{t}-\widetilde{H}_{\bm{\beta}}(\bm{x})L_{2}^{(i)}\partial_{\bm{t}}\bm{D}^{(\bm{\beta},b)}(\bm{x}).$

If we premultiply this by $\bm{y}_{1}^{(j)}(\bm{x})^{T}$ and use (7.19) then
we obtain

(7.23)
$\bm{y}_{1}^{(j)}(\bm{x})^{T}\widetilde{H}_{\bm{\beta}}(\bm{t})\bm{y}_{2}^{(i)}(\bm{x})=\bm{y}_{1}^{(j)}(\bm{x})^{T}M_{2}^{(i)}J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x})\bm{t}\\\
-[M_{1}^{(j)}\bm{D}^{(\bm{\beta},b+1)}(\bm{x})]^{T}[L_{2}^{(i)}\partial_{\bm{t}}\bm{D}^{(\bm{\beta},b)}(\bm{x})].$

Lemma 7.8 (i) yields the bounds

(7.24)
$\frac{\left\lVert\bm{D}^{(\bm{\beta},b+1)}(\bm{x})\right\rVert_{\infty}}{\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x})\right\rVert_{\infty}}\ll_{n}\lambda_{\bm{\beta},b+1}(\bm{x}),$

and

(7.25)
$\frac{\left\lVert\partial_{\bm{t}}\bm{D}^{(\bm{\beta},b)}(\bm{x})\right\rVert_{\infty}}{\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x})\right\rVert_{\infty}}\ll_{n}\frac{\left\lVert\bm{t}\right\rVert_{\infty}}{\lambda_{\bm{\beta},b}(\bm{x})}.$

Now we specify $\bm{x}=\bm{x}^{(0)}$ so by assumption we have
$\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}>0$.
Thus define

(7.26)
$\bm{Y}_{k}^{(i)}=\frac{\bm{y}_{k}^{(i)}(\bm{x}^{(0)})}{\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}},\quad\text{for
$i=1,\ldots,n-b$ and $k=1,2$.}$

Dividing (7.23) by
$1/\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}^{2}$
and using (7.26) as well as the bounds (7.24) and (7.25) gives

$\left\lvert\bm{Y}_{1}^{(j)}\widetilde{H}_{\bm{\beta}}(\bm{t})\bm{Y}_{2}^{(i)}\right\rvert\ll_{n}\frac{\left\lVert
J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x}^{(0)})\bm{t}\right\rVert_{\infty}}{\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}}+\frac{\lambda_{\bm{\beta},b+1}(\bm{x}^{(0)})\left\lVert\bm{t}\right\rVert_{\infty}}{\lambda_{\bm{\beta},b}(\bm{x}^{(0)})}.$

We claim now that we can take the subspaces $Y_{k}\subseteq\mathbb{R}^{n}$ to
be defined as the span of $\bm{Y}_{k}^{(1)},\ldots,\bm{Y}_{k}^{(n-b)}$ for
$k=1,2$ respectively, so that the Lemma holds. For this we need to show that
(7.13) holds, and also that $\dim Y_{1}=\dim Y_{2}=n-b$. Therefore it suffices
to show the following claim: Given $\bm{\gamma}\in\mathbb{R}^{n-b}$ if we take
$\bm{Y}_{k}=\sum\gamma_{i}Y_{k}^{(i)}$ then
$\left\lVert\bm{\gamma}\right\rVert_{\infty}\ll_{n}\left\lVert\bm{Y}_{k}\right\rVert_{\infty}$,
for $k=1,2$ respectively.

Assume that the $b\times b$ minor of
$\widetilde{H}_{\bm{\beta}}(\bm{x}^{(0)})$ of largest absolute value lies in
the bottom right corner of $\widetilde{H}_{\bm{\beta}}(\bm{x}^{(0)})$. In
other words, we assume

(7.27)
$\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}=\left\lvert\det\left((\widetilde{H}_{\bm{\beta}}(\bm{x}^{(0)})_{k\ell})_{\begin{subarray}{c}k=n-b+1,\ldots,n\\\
\ell=n-b+1,\ldots,n\end{subarray}}\right)\right\rvert.$

After permuting the rows and columns of
$\widetilde{H}_{\bm{\beta}}(\bm{x}^{(0)})$ the identity (7.27) will always be
true. The vectors $\bm{Y}_{k}^{(i)}$ depend on minors of
$\widetilde{H}_{\bm{\beta}}(\bm{x}^{(0)})$. Thus we can apply the same
permutations to $\widetilde{H}_{\bm{\beta}}(\bm{x}^{(0)})$ that ensure that
(7.27) holds to the definition of these vectors. From this we see that we can
always reduce the general case to the case where (7.27) holds.

Now for $k=1,2$ we define matrices

$Q_{k}=\left(\bm{Y}_{k}^{(1)}\Big{|}\cdots\Big{|}\bm{Y}_{k}^{(n-b)}\Big{|}\bm{e}_{n-b+1}\Big{|}\cdots\Big{|}\bm{e}_{n}\right).$

By the definition of $\bm{Y}_{k}^{(i)}$ we see that $Q_{k}$ must be of the
following form

$Q_{k}=\begin{pmatrix}I_{n-b}&0\\\ \widetilde{Q}_{k}&I_{b}\end{pmatrix},$

for some matrix $\widetilde{Q}_{k}$. In particular we find $\det Q_{k}=1$ and
so $\left\lVert Q_{k}^{-1}\right\rVert_{\infty}\ll_{n}1$. Given
$\bm{Y}_{k}=\sum\gamma_{i}Y_{k}^{(i)}$ we thus find

$\left\lVert\bm{\gamma}\right\rVert_{\infty}=\left\lVert
Q_{k}^{-1}\bm{Y}_{k}\right\rVert_{\infty}\ll_{n}\left\lVert\bm{Y}_{k}\right\rVert_{\infty},$

and so the Lemma follows. ∎

The next Corollary is the main technical result from this section, which will
allow us to deduce that either $N_{2}^{\mathrm{aux}}(\bm{\beta},B)$ is small
or a suitable singular locus is large.

######  Corollary 7.12.

Let $B,C\geq 1$ and let $\sigma\in\\{0,\ldots,n-1\\}$. Then one of the
following alternatives is true. Either we have the bound

(7.28) $N_{2}^{\mathrm{aux}}(\bm{\beta},B)\ll_{C,n}B^{n+\sigma}(\log B)^{n},$

or there exist subspaces $X,Y_{1},Y_{2}\subseteq\mathbb{R}^{n}$ with $\dim
X+\dim Y_{1}=\dim X+\dim Y_{2}=n+\sigma+1$, such that

(7.29)
$\left\lvert\bm{Y}_{1}^{T}\widetilde{H}_{\bm{\beta}}(\bm{X})\bm{Y}_{2}\right\rvert\ll_{n}C^{-1}\left\lVert\bm{Y}_{1}\right\rVert_{\infty}\left\lVert\bm{X}\right\rVert_{\infty}\left\lVert\bm{Y}_{2}\right\rVert_{\infty}$

holds for all $\bm{X}\in X,\bm{Y}_{1}\in Y_{1},\bm{Y}_{2}\in Y_{2}$.

###### Proof.

Let $k\in\\{0,\ldots,n-\sigma-1\\}$ and $E_{1},\ldots,E_{k+1}\in\mathbb{R}$ be
such that

$CB\geq E_{1}\geq\ldots\geq E_{k+1}\geq 1.$

We know that one of the alternatives (I)$k$, (II)$k$ or (III) in Lemma 7.9
holds. Assume first that (I)$k$ always holds so that the estimate

(7.30) $\\#(\mathbb{Z}^{n}\cap
K_{k}(E_{1},\ldots,E_{k+1}))\ll_{C,n}B^{\sigma}(E_{1}\cdots
E_{k+1})E_{k+1}^{n-\sigma-k-1}.$

holds for every $k\in\\{0,\ldots,n-\sigma-1\\}$ and
$E_{1},\ldots,E_{k+1}\in\mathbb{R}$ such that $CB\geq E_{1}\geq\ldots\geq
E_{k+1}\geq 1$. From Lemma 7.7 we find that either we have

(7.31) $\frac{N_{2}^{\mathrm{aux}}(\bm{\beta},B)}{B^{n}(\log
B)^{n}}\ll_{n}\\#(\mathbb{Z}^{n}\cap K_{0}(1)),$

or there exist nonnegative integers $e_{1},\ldots,e_{k}$ for some
$k\in\\{1,\ldots,n-1\\}$ such that $\log B\gg_{n}e_{1}\geq\ldots\geq e_{k}$
and

(7.32)
$\frac{2^{e_{1}+\cdots+e_{k}}N_{2}^{\mathrm{aux}}(\bm{\beta},B)}{B^{n}(\log
B)^{n}}\ll_{n}\\#(\mathbb{Z}^{n}\cap K_{k}(2^{e_{1}},\ldots,2^{e_{k}},1)),$

or there exist nonnegative integers $e_{1},\ldots,e_{n}$ such that $\log
B\gg_{n}e_{1}\geq\ldots\geq e_{n}$ and

(7.33)
$\frac{2^{e_{1}+\cdots+e_{n}}N_{2}^{\mathrm{aux}}(\bm{\beta},B)}{B^{n}(\log
B)^{n}}\ll_{n}\\#(\mathbb{Z}^{n}\cap K_{n-1}(2^{e_{1}},\ldots,2^{e_{n}})).$

We may take $C$ to be large enough depending on $n$ such that $CB\geq
2^{e_{1}}$ is satisfied. Then upon sbustituting the bound (7.30) into either
of (7.31), (7.32) or (7.33) gives (7.28).

If (III) holds in Lemma 7.9 we can take $Y_{1}=Y_{2}=\mathbb{R}^{n}$ so that
(7.29) follows from (7.12).

Finally, assume there exist $k\in\\{0,\ldots,n-\sigma-1\\}$ and
$E_{1},\ldots,E_{k+1}\in\mathbb{R}$ with $CB\geq E_{1}\geq\ldots\geq
E_{k+1}\geq 1$ such that (II)$k$ in Lemma 7.9 holds. Recall this means there
exists some integer $b\in\\{1,\ldots,k\\}$, a $(\sigma+b+1)$-dimensional
subspace $X\subset\mathbb{R}^{n}$ and $\bm{x}^{(0)}\in
K_{b}(E_{1},\ldots,E_{b+1})$ such that $E_{b+1}<C^{-1}E_{b}$ and

(7.34) $\left\lVert
J_{\bm{D}^{(\bm{\beta},b+1)}}(\bm{x}^{(0)})\bm{X}\right\rVert_{\infty}\leq
C^{-1}\left\lVert\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\right\rVert_{\infty}\left\lVert\bm{X}\right\rVert_{\infty}\quad\text{for
all $\bm{X}\in X$.}$

As $\bm{x}^{(0)}\in K_{b}(E_{1},\ldots,E_{b+1})$ we have
$E_{i}/2<\lambda_{\bm{\beta},i}(\bm{x}^{(0)})\leq E_{i}$ for $i=1,\ldots,k$
and $\lambda_{\bm{\beta},b+1}(\bm{x}^{(0)})\leq E_{b+1}$. This, together with
the fact that $E_{b+1}<C^{-1}E_{b}$ implies

(7.35)
$\lambda_{\bm{\beta},b+1}(\bm{x}^{(0)})<2C^{-1}\lambda_{\bm{\beta},b}(\bm{x}^{(0)}).$

Also we find $\lambda_{\bm{\beta},b}(\bm{x}^{(0)})\neq 0$, from which it
follows from Lemma 7.8 (i) that $\bm{D}^{(\bm{\beta},b)}(\bm{x}^{(0)})\neq 0$.
Thus we may apply Lemma 7.11 to obtain spaces
$Y_{1},Y_{2}\subseteq\mathbb{R}^{n}$ with $\dim Y_{1}=\dim Y_{2}=n-b$ such
that the estimate (7.13) holds. Now taking $\bm{t}=\bm{X}$ in (7.13) and using
(7.34) and (7.35) then (7.29) follows. Since $\dim X=\sigma+b+1$ we also have
$\dim X+\dim Y_{1}=\dim X+\dim Y_{2}=n+\sigma+1$ as desired. ∎

Recall the definition of the quantity

$s_{\mathbb{R}}^{(2)}\coloneqq\left\lfloor\frac{\max_{\bm{\beta}\in\mathbb{R}^{R}\setminus\\{0\\}}\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})}{2}\right\rfloor+1,$

where $\lfloor x\rfloor$ denotes the largest integer $m$ such that $m\leq x$.
Although we have been assuming $n_{1}=n_{2}$ throughout the definition of this
quantity remains valid if $n_{1}\neq n_{2}$. Note that we have
$\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\subsetneq\mathbb{P}^{n_{1}-1}_{\mathbb{C}}\times\mathbb{P}^{n_{2}-1}_{\mathbb{C}}$
for all $\bm{\beta}\in\mathbb{R}^{R}\setminus\\{0\\}$. For if not, then the
matrix $H_{\bm{\beta}}(\bm{y})$ is identically zero for some
$\bm{\beta}\in\mathbb{R}^{R}\setminus\\{0\\}$ contradicting the fact that
$\mathbb{V}(\bm{F})$ is a complete intersection. In particular this yields
$s_{\mathbb{R}}^{(2)}\leq\frac{n_{1}+n_{2}}{2}-1$.

Before we prove the main result of this section we require another small
Lemma.

######  Lemma 7.13.

Let $\bm{\beta}\in\mathbb{R}\setminus\\{0\\}$. The system of equations

$\bm{y}^{T}\widetilde{H}_{\bm{\beta}}(\bm{e}_{\ell})\bm{x}=0,\;\text{for
$\ell=1,\ldots,n$}\quad\text{and}\quad H_{\bm{\beta}}(\bm{y})\bm{x}=\bm{0}$

define the same variety in
$\mathbb{P}_{\mathbb{C}}^{n-1}\times\mathbb{P}_{\mathbb{C}}^{n-1}$.

###### Proof.

Recall that by definition we have

$\widetilde{H}_{\bm{\beta}}(\bm{z})=\begin{pmatrix}\bm{z}^{T}H_{\bm{\beta}}(\bm{e}_{1})\\\
\vdots\\\ \bm{z}^{T}H_{\bm{\beta}}(\bm{e}_{n}).\end{pmatrix}$

For $\ell\in\\{1,\ldots,n\\}$ we get

$\bm{y}^{T}\widetilde{H}_{\bm{\beta}}(\bm{e}_{\ell})\bm{x}=\bm{y}^{T}\begin{pmatrix}\bm{e}_{\ell}^{T}H_{\bm{\beta}}(\bm{e}_{1})\bm{x}\\\
\vdots\\\
\bm{e}_{\ell}^{T}H_{\bm{\beta}}(\bm{e}_{n})\bm{x}\end{pmatrix}=\sum_{i=1}^{n}y_{i}\bm{e}_{\ell}^{T}H_{\bm{\beta}}(\bm{e}_{i})\bm{x}=\bm{e}_{\ell}^{T}H_{\bm{\beta}}(\bm{y})\bm{x},$

where the last line follows since the entries of $H_{\bm{\beta}}(\bm{y})$ are
linear homogeneous in $\bm{y}$. The result is now immediate. ∎

######  Proposition 7.14.

Let $s_{\mathbb{R}}^{(2)}$ be defined as above and let $B\geq 1$. Then for all
$\bm{\beta}\in\mathbb{R}^{R}\setminus\\{0\\}$ the following holds

$N_{2}^{\mathrm{aux}}(\bm{\beta},B)\ll_{n}B^{n+s_{\mathbb{R}}^{(2)}}(\log
B)^{n}.$

###### Proof.

Suppose for a contradiction the result were false. Then for each positive
integer $N$ there exists some $\bm{\beta}_{N}$ such that

$N_{2}^{\mathrm{aux}}(\bm{\beta}_{N},B)\geq NB^{n+s_{\mathbb{R}}^{(2)}}(\log
B)^{n}.$

From Corollary 7.12 it follows that there are linear subspaces
$X^{(N)},Y_{1}^{(N)},Y_{2}^{(N)}\subset\mathbb{R}^{n}$ with

$\dim X^{(N)}+\dim Y_{i}^{(N)}=n+s_{\mathbb{R}}^{(2)}+1,\quad i=1,2,$

such that for all $\bm{X}\in X^{(N)}$, $\bm{Y}_{i}\in Y_{i}^{(N)}$ we get

$\left\lvert\bm{Y}_{1}^{T}\widetilde{H}_{\bm{\beta}_{N}}(\bm{X})\bm{Y}_{2}\right\rvert\leq
N^{-1}\left\lVert\bm{Y}_{1}\right\rVert_{\infty}\left\lVert\bm{X}\right\rVert_{\infty}\left\lVert\bm{Y}_{2}\right\rVert_{\infty}.$

Note that $\widetilde{H}_{\bm{\beta}_{N}}(\bm{\beta})$ is unchanged when
$\bm{\beta}_{N}$ is multiplied by a constant. Thus we may assume
$\left\lVert\bm{\beta}_{N}\right\rVert_{\infty}=1$ and consider a converging
subsequence of $\bm{\beta}_{N_{r}}$ converging to $\bm{\beta}$, say, as
$N\rightarrow\infty$. This delivers subspaces
$X,Y_{1},Y_{2}\subset\mathbb{R}^{n}$ with $\dim X+\dim
Y_{i}=n+s_{\mathbb{R}}^{(2)}+1$ for $i=1,2$ such that

$\bm{Y}_{1}^{T}\widetilde{H}_{\bm{\beta}}(\bm{X})\bm{Y}_{2}=0\quad\text{for
all $\bm{X}\in X,\bm{Y}_{1}\in Y_{1},\bm{Y}_{2}\in Y_{2}$.}$

There exists some $b\in\\{0,\ldots,n-s_{\mathbb{R}}^{(2)}-1\\}$ such that
$\dim X=n-b$ and $\dim Y_{i}=s_{\mathbb{R}}^{(2)}+b+1$. Now let
$\bm{x}^{(1)},\ldots,\bm{x}^{(n)}$ be a basis for $\mathbb{R}^{n}$ such that
$\bm{x}^{(b+1)},\ldots,\bm{x}^{(n)}$ is a basis for $X$. Write
$[Y_{i}]\subset\mathbb{P}_{\mathbb{C}}^{n-1}$ for the linear subspace of
$\mathbb{P}_{\mathbb{C}}^{n-1}$ associated to $Y_{i}$ for $i=1,2$.

Define the biprojective variety $W\subset[Y_{1}]\times[Y_{2}]$ in the
variables $(\bm{y}_{1},\bm{y}_{2})$ by

$W=\mathbb{V}(\bm{y}_{1}\widetilde{H}_{\bm{\beta}}(\bm{x}^{(i)})\bm{y}_{2})_{i=1,\ldots,b}.$

Since the non-trivial equations defining $W$ have bidegree $(1,1)$ we can
apply Corollary 3.2 to find

(7.36) $\dim W\geq\dim[Y_{1}]\times[Y_{2}]-b=2s_{\mathbb{R}}^{(2)}+b.$

Given $(\bm{y}_{1},\bm{y}_{2})\in W$ we have in particular
$(\bm{y}_{1},\bm{y}_{2})\in[Y_{1}]\times[Y_{2}]$ and so

$\bm{y}_{1}\widetilde{H}_{\bm{\beta}}(\bm{x}^{(i)})\bm{y}_{2}=0,\quad\text{for
$i=b+1,\ldots,n$,}$

and hence $\bm{y}_{1}\widetilde{H}_{\bm{\beta}}(\bm{z})\bm{y}_{2}=0$ for all
$\bm{z}\in\mathbb{R}^{n}$. From Lemma 7.13 we thus see
$H_{\bm{\beta}}(\bm{y}_{1})\bm{y}_{2}=0$ for all $(\bm{y}_{1},\bm{y}_{2})\in
W$. Hence in particular

$\dim W\leq\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\leq
2s_{\mathbb{R}}^{(2)}-1,$

where we regard $\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})$ as a variety in
$\mathbb{P}_{\mathbb{C}}^{n-1}\times\mathbb{P}_{\mathbb{C}}^{n-1}$ in the
variables $(\bm{x},\bm{y})$. This together with (7.36) implies $b\leq-1$,
which is clearly a contradiction. ∎

In the next Lemma we show that $s_{\mathbb{R}}^{(2)}$ is small if
$\mathbb{V}(\bm{F})$ defines a smooth complete intersection. For this we no
longer assume $n_{1}=n_{2}$.

######  Lemma 7.15.

Let $s_{\mathbb{R}}^{(2)}$ be defined as above. If $\mathbb{V}(\bm{F})$ is a
smooth complete intersection in
$\mathbb{P}_{\mathbb{C}}^{n_{1}-1}\times\mathbb{P}_{\mathbb{C}}^{n_{2}-1}$
then we have the bound

(7.37) $\frac{n_{2}-1}{2}\leq s_{\mathbb{R}}^{(2)}\leq\frac{n_{2}+R}{2}.$

###### Proof.

Let $\bm{\beta}\in\mathbb{R}^{R}\setminus\\{\bm{0}\\}$ be such that

$s_{\mathbb{R}}^{(2)}=\left\lfloor\frac{\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})}{2}\right\rfloor+1.$

Note that then

(7.38)
$2s_{\mathbb{R}}^{(2)}-2\leq\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\leq
2s_{\mathbb{R}}^{(2)}-1.$

The variety
$\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\subset\mathbb{P}_{\mathbb{C}}^{n_{1}-1}\times\mathbb{P}_{\mathbb{C}}^{n_{2}-1}$
is defined by $n_{1}$ bilinear polynomials. Using Corollary 3.2 we thus find

$\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\geq n_{2}-2$

so the lower bound in (7.37) follows. We proceed by considering two cases.

Case 1:
$\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}=\emptyset$.
Note that this can only happen if $n_{2}\geq n_{1}$. We can therefore apply
Lemma 3.4 with
$V_{1}=\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}$,
$V_{2}=\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})$ and
$A_{i}=H_{\bm{\beta}}(\bm{e}_{i})$ to find

$\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\leq
n_{2}-1+\dim\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}=n_{2}-2.$

From this and (7.38) the upper bound in (7.37) follows for this case.

Case 2:
$\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}\neq\emptyset$:
By assumption there exists $\bm{x}\in\mathbb{C}^{n_{1}}\setminus\\{\bm{0}\\}$
such that

$\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x}=0,\quad\text{for all
$\ell=1,\ldots,n_{2}$}.$

We claim that there exists $\bm{y}\in\mathbb{C}^{n_{2}}\setminus\\{\bm{0}\\}$
such that $H_{\bm{\beta}}(\bm{y})\bm{x}=\bm{0}$. For this define the vectors

$\bm{u}_{\ell}=H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x},\quad\ell=1,\ldots,n_{2}.$

Note that $\bm{x}\in\langle\bm{u}_{1},\ldots,\bm{u}_{n_{2}}\rangle^{\perp}$ so
these vectors must be linearly dependent. Thus there exist
$y_{1},\ldots,y_{n_{2}}\in\mathbb{C}$ not all zero, such that

$H_{\bm{\beta}}(\bm{y})\bm{x}=\sum_{\ell=1}^{n_{2}}y_{\ell}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x}=\bm{0},$

where the first equality followed since the entries of
$H_{\bm{\beta}}(\bm{y})$ are linear homogeneous in $\bm{y}$. The claim
follows. In particular it follows from this that

$\left(\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}\times\mathbb{P}^{n_{2}-1}\right)\cap\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\neq\emptyset.$

Using Lemma 3.1 and (7.38) we therefore find

(7.39)
$\dim\left[\left(\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}\times\mathbb{P}^{n_{2}-1}\right)\cap\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})\right]\geq\\\
\dim\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x})-n_{2}\geq
2s_{\mathbb{R}}^{(2)}-n_{2}-2.$

Recall $\bm{\beta}\cdot\bm{F}=\bm{x}^{T}H_{\bm{\beta}}(\bm{y})\bm{x}$ so that

$\mathrm{Sing}\mathbb{V}(\bm{\beta}\cdot\bm{F})=\left(\mathbb{V}(\bm{x}^{T}H_{\bm{\beta}}(\bm{e}_{\ell})\bm{x})_{\ell=1,\ldots,n_{2}}\times\mathbb{P}^{n_{2}-1}\right)\cap\mathbb{V}(H_{\bm{\beta}}(\bm{y})\bm{x}).$

Under our assumptions we can apply Lemma 3.3 to find
$\dim\mathrm{Sing}\mathbb{V}(\bm{\beta}\cdot\bm{F})\leq R-2$. The result
follows from this and (7.39). ∎

###### Proof of Theorem 7.1.

Applying Theorem 2.1 it suffices to show

(7.40) $N_{i}^{\mathrm{aux}}(\bm{\beta};B)\leq C_{0}B^{2n-4\mathscr{C}},$

holds for all $\bm{\beta}\in\mathbb{R}^{R}\setminus\\{0\\}$ and $i=1,2$, where
$\mathscr{C}>(2b+u)R$. Let

$s=\max\\{s_{\mathbb{R}}^{(1)},s_{\mathbb{R}}^{(2)}\\},$

where $s_{\mathbb{R}}^{(1)}$ and $s_{\mathbb{R}}^{(2)}$ are as defined in
(1.7) and (1.8), respectively. From Proposition 7.3 and Proposition 7.14 for
any $\varepsilon>0$ we get

$N_{i}^{\mathrm{aux}}(\bm{\beta};B)\ll_{\varepsilon}B^{n+s+\varepsilon},$

with the implied constant not depending on $\bm{\beta}$. Choose
$\varepsilon=\frac{n-s-(8b+4u)R}{2}$, which is a positive real number by our
assumption (7.1). Taking

$\mathscr{C}=\frac{n-s-\varepsilon}{4},$

we see that from the assumption $n-s_{\mathbb{R}}^{(i)}>(8b+4u)R$ for $i=1,2$
we must have $\mathscr{C}>(2b+u)R$ for this choice. Therefore (7.40) holds and
the first part of the theorem follows upon applying Theorem 2.1.

For the second part recall we assume $n>(16b+8u+1)R$ and that the forms
$F_{i}(\bm{x},\bm{y})$ define a smooth complete intersection in
$\mathbb{P}_{\mathbb{C}}^{n-1}\times\mathbb{P}_{\mathbb{C}}^{n-1}$. By Lemma
7.4 in this case we obtain

$s_{\mathbb{R}}^{(1)}\leq R,$

and from Lemma 7.15 we find

$s_{\mathbb{R}}^{(2)}\leq\frac{n+R}{2}.$

Therefore it is easily seen that assuming $n>(16b+8u+1)R$ implies that

$n-s_{\mathbb{R}}^{(i)}>(8b+4u)R$

holds for $i=1,2$, which is what we wanted to show. ∎

###  7.3. Proof of Theorem 1.2

###### Proof of Theorem 1.2.

If $n_{1}=n_{2}$ then the result follows immediately from Proposition 7.1. We
have two cases to consider and although their strategies are very similar they
are not entirely symmetric. Therefore it is necessary to consider them
individually.

Case 1: $n_{1}>n_{2}$. We consider a new system of equations
$\widetilde{F}_{i}({\bm{x}},\tilde{\bm{y}})$ in the variables
${\bm{x}}=(x_{1},\ldots,x_{n})$ and
$\tilde{\bm{y}}=(y_{1},\ldots,y_{n_{2}},y_{n_{2}+1},\ldots,y_{n_{1}})$ where
the forms $\widetilde{F}_{i}({\bm{x}},\tilde{\bm{y}})$ satisfy

$\widetilde{F}_{i}({\bm{x}},\tilde{\bm{y}})=F(\bm{x},\bm{y}),$

where $\bm{y}=(y_{1},\ldots,y_{n_{2}})$. Write $\widetilde{N}(P_{1},P_{2})$
for the counting function associated to the system $\widetilde{\bm{F}}=\bm{0}$
and the boxes
$\mathcal{B}_{1}\times(\mathcal{B}_{2}\times[0,1]^{n_{1}-n_{2}})$. Note in
particular, that if we replace $F$ by $\widetilde{F}$ in (5.30) and (5.29)
then the expressions for the singular series and the singular integral remain
unchanged. Further denote by $\tilde{s}_{\mathbb{R}}^{(i)}$ the quantities
defined in (1.7) and (1.8) but with $F$ replaced by $\widetilde{F}$. Note that
we have $\tilde{s}_{\mathbb{R}}^{(1)}=s_{\mathbb{R}}^{(1)}$ and
$\tilde{s}_{\mathbb{R}}^{(2)}\leq s_{\mathbb{R}}^{(2)}+\frac{n_{1}-n_{2}}{2}$.
Therefore the assumptions (1.9) imply

$n_{1}-\tilde{s}_{\mathbb{R}}^{(i)}>(8b+4u)R$

for $i=1,2$. Hence we may apply Proposition 7.1 in order to obtain

$\widetilde{N}(P_{1},P_{2})={\mathfrak{I}}{\mathfrak{S}}P_{1}^{n_{1}-2R}P_{2}^{n_{1}-R}+O(P_{1}^{n_{1}-2R}P_{2}^{n_{1}-R}\min\\{P_{1},P_{2}\\}^{-\delta}),$

for some $\delta>0$. Finally it is easy to see that

$\displaystyle\widetilde{N}(P_{1},P_{2})$
$\displaystyle=N(P_{1},P_{2})\\#\left\\{\bm{t}\in\mathbb{Z}^{n_{1}-n_{2}}\cap[0,P_{2}]^{n_{1}-n_{2}}\right\\}$
$\displaystyle=N(P_{1},P_{2})(P_{2}^{n_{1}-n_{2}}+O(P_{2}^{n_{1}-n_{2}-1})),$

and so (1.10) follows.

Case 2: $n_{2}>n_{1}$ We deal with this very similarly as in the first case;
we define a new system of forms $\widetilde{F}_{i}(\tilde{\bm{x}},\bm{y})$ in
the variables $\tilde{\bm{x}}=(x_{1},\ldots,x_{n_{2}})$ and
$\bm{y}=(y_{1},\ldots,y_{n_{2}})$ such that

$\widetilde{F}_{i}({\bm{x}},\tilde{\bm{y}})=F_{i}(\bm{x},\bm{y})$

holds. As before we define a new counting function
$\widetilde{N}(P_{1},P_{2})$ with respect to the new product of boxes
$\left(\mathcal{B}_{1}\times[0,1]^{n_{2}-n_{1}}\right)\times\mathcal{B}_{2}$,
and we define $\tilde{s}_{\mathbb{R}}^{(i)}$ similarly to the previous case.
Note that $\tilde{s}_{\mathbb{R}}^{(1)}=s_{\mathbb{R}}^{(1)}+n_{2}-n_{1}$ and
$\tilde{s}_{\mathbb{R}}^{(2)}\leq s_{\mathbb{R}}^{(2)}+\frac{n_{2}-n_{1}}{2}$
so that (1.9) gives

$n_{2}-\tilde{s}_{\mathbb{R}}^{(i)}>(8b+4u)R,$

for $i=1,2$. Therefore Proposition 7.1 applies and we deduce again that (1.10)
holds as desired.

Finally we turn to the case when $\mathbb{V}(\bm{F})$ defines a smooth
complete intersection. Note first that by Lemma 7.15 we have

$s_{\mathbb{R}}^{(2)}\leq\frac{n_{2}+R}{2},$

and therefore the condition

$\frac{n_{1}+n_{2}}{2}-s_{\mathbb{R}}^{(2)}>(8b+4u)R$

is satisfied if we assume $n_{1}>(16b+8u+1)R$. Further, by Lemma 7.4 we have

$s_{\mathbb{R}}^{(1)}\leq\max\\{0,n_{1}+R-n_{2}\\},$

and so we may replace the condition $n_{1}-s_{\mathbb{R}}^{(1)}>(8b+4u)R$ by

$n_{1}-\max\\{0,n_{1}+R-n_{2}\\}>(8b+4u)R.$

If $n_{2}\geq n_{1}+R$ then this reduces to assuming $n_{1}>(8b+4u+1)R$, which
follows immediately since we assumed $n_{1}>(16b+8u+1)R$. If $n_{2}\leq
n_{1}+R$ on the other hand, then this is equivalent to assuming

$n_{2}>(8b+4u+1)R.$

In any case, the assumptions (1.11) imply the assumptions (1.9) as desired. ∎

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