###### Abstract

For $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ of degree $k\geq
2$, we show that every subset $A\subseteq\\{1,2,\dots,N\\}$ lacking nonzero
differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_{h}Ne^{-c(\log
N)^{\mu}}$, where $c=c(h)>0$, $\mu=[(k-1)^{2}+1]^{-1}$ if $\ell=2$, and
$\mu=1/2$ if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple
of every natural number and $h$ satisfies certain nonsingularity conditions.
We also explore these conditions in detail, drawing on a variety of tools from
algebraic geometry.

title = Multivariate Polynomial Values
in Difference Sets, author = John R. Doyle and Alex Rice, plaintextauthor =
John R. Doyle, Alex Rice, plaintexttitle = Multivariate Polynomial Values in
Difference Sets, runningtitle = Multivariate Polynomial Values in Difference
Sets, year=2021, number=11, received=14 September 2020, published=8 September
2021, doi=10.19086/da.27995,

[classification=text]

##  1 Introduction

Originating with conjectures of Erdős and Lovász, an extensive literature has
developed over the past several decades concerning the existence of particular
differences within dense sets of integers. For sets $A,B\subseteq\mathbb{Z}$,
we define the sum and difference sets, respectively, as usual by $A\pm
B=\\{a\pm b:a\in A,b\in B\\}$, and we also define the following threshold.

######  Definition 1.1.

For $X\subseteq\mathbb{Z}$ and $N\in\mathbb{N}$, we define
$D(X,N)=\max\left\\{|A|:A\subseteq[1,N],\ (A-A)\cap
X\subseteq\\{0\\}\right\\}.$

We use $[1,N]$ to denote $\\{1,2,\dots,N\\}$ and $|A|$ to denote the size of a
finite set $A$. To clarify, $D(X,N)$ is the threshold such that any subset of
$\\{1,2,\dots,N\\}$ with more than $D(X,N)$ elements necessarily contains two
distinct elements that differ by an element of $X$. As an introductory
offering prior to extensive discussions of history, motivation, notation, and
terminology, a very special case of our results in this paper is the
following:

######  Theorem 1.2.

Suppose $h\in\mathbb{Z}[x,y]$ is a homogeneous polynomial of degree $k\geq 2$.
If $\Delta(h)\neq 0$, then

$D(h(\mathbb{Z}^{2}),N)\ll_{h}Ne^{-c(\log N)^{\mu}},$ (1)

where $c=c(h)>0$ and $\mu=[(k-1)^{2}+1]^{-1}$.

Here $\Delta$ denotes the usual homogeneous discriminant, and we use $\ll$ to
denote “less than a constant times”, with subscripts indicating on what
parameters, if any, the implied constant depends. We take the same convention
with subscripts on Big O notation. Theorem 1.2 follows from Corollary 2.6 and
our main result, Theorem 2.4, of which we discuss various improvements and
important special cases throughout Section 2.

###  1.1 Background

Lovász asked whether a set of positive upper density must contain two distinct
elements that differ by a perfect square, or equivalently whether
$D(S,N)=o(N)$, where $S=\\{n^{2}:n\in\mathbb{N}\\}$. Similarly, Erdős
conjectured that $D(\mathcal{P}-1,N)=o(N)$, where
$\mathcal{P}-1=\\{p-1:p\text{ prime}\\}$. Furstenberg [10] verified the former
using ergodic methods, specifically his correspondence principle, in the same
paper in which he provided the second known proof of Szemerédi’s Theorem on
arithmetic progressions. Independently and concurrently, Sárközy ([35], [36])
verified both conjectures with a Fourier analytic density increment argument
driven by the Hardy-Littlewood circle method. Further, Sárközy’s results
included quantitative information, showing $D(S,N)\ll_{\epsilon}N(\log
N)^{-1/3+\epsilon}$ and $D(\mathcal{P}-1,N)\ll_{\epsilon}N(\log\log
N)^{-2+\epsilon}$ for every $\epsilon>0.$

These results have been incrementally improved and generalized in multiple
ways, both through tightening of the quantitative bounds and expansion of the
possibilities for the set $X$ of prohibited differences. Regarding the former,
Pintz, Steiger and Szemerédi [26] utilized a more elaborate Fourier analytic
strategy to show

$D(S,N)\ll N(\log N)^{-c\log\log\log\log N}$ (2)

for a constant $c>0$.

Dramatically improving Sárközy’s original bound, Ruzsa and Sanders [32] showed

$D(\mathcal{P}-1,N)\ll Ne^{-c(\log N)^{\mu}}$ (3)

with $\mu=1/4$, recently improved to $\mu=1/3$ by Wang [39]. Regarding
alternative choices for the set of prohibited differences, one must first
consider obvious local obstructions. For example, we consider $\mathcal{P}-1$,
rather than $\mathcal{P}$, because $\mathcal{P}\cap 4\mathbb{Z}=\emptyset$
implies $D(\mathcal{P},N)\geq\lceil N/4\rceil$ by taking $A$ to be a
congruence class modulo $4$. Analogously, if $h\in\mathbb{Z}[x]$ and
$h(\mathbb{Z})$ contains no multiples of $q\in\mathbb{N}$, then
$D(h(\mathbb{Z}),N)\geq\lceil N/q\rceil$. Therefore, for even a qualitative
$o(N)$ result, it is clearly necessary that $h(\mathbb{Z})$ contains a nonzero
multiple of every $q\in\mathbb{N}$, in which case we say that $h$ is an
intersective polynomial. Examples of intersective polynomials include any
nonzero polynomial with an integer root or a collection of rational roots with
coprime denominators. However, there are also intersective polynomials with no
rational roots, such as $(x^{3}-19)(x^{2}+x+1)$.

Balog, Pelikán, Pintz, and Szemerédi [1] extended (2) with $S$ replaced by
$\\{n^{k}:n\in\mathbb{N}\\}$ for any fixed $k\in\mathbb{N}$. For a general
univariate intersective polynomial, Kamae and Mendes-France [18] established
the qualitative $o(N)$ result, the first quantitative bounds were due to
Lucier [23], and the second author [28] fully extended (2). In a recent
preprint, Bloom and Maynard [3] both simplified and improved the ideas of
[26], using a more traditional density increment to establish

$D(S,N)\ll N(\log N)^{-c\log\log\log N}$ (4)

for a constant $c>0$, which is currently the best-known bound for the original
square difference question. Further, the methods of [3] are completely
compatible with those of [28], so in fact (4) should hold for the full class
of intersective polynomials. For other intermediate and related results, as
well as alternative proofs, the reader may refer to (in chronological order)
[11], [37], [24], [22], [21], [25], [14], [30], and [12].

Also in [28], the second author showed that if $g,h\in\mathbb{Z}[x]$ are
intersective polynomials, then

$D(g(\mathbb{Z})+h(\mathbb{Z}),N)\ll_{g,h}Ne^{-c(\log N)^{\mu}},$ (5)

where $c=c(g,h)>0$ and $\mu=\mu(\deg(g),\deg(h))>0$. Further, the second
author [31] considered the simplest nontrivial case of a non-diagonal
multivariate polynomial, showing that for a binary quadratic form
$h(x,y)=ax^{2}+bxy+cy^{2}\in\mathbb{Z}[x,y]$ with $b^{2}-4ac\neq 0$, we have

$D(h(\mathbb{Z}^{2}),N)\ll_{h}Ne^{-c\sqrt{\log N}}.$ (6)

###  1.2 Motivation

As outlined in Section 2.4 of [28], the quoted upper bounds in the previous
section, all of which result from adaptations of the two aforementioned
Fourier analytic arguments developed in [35] and [26], respectively, are
partially determined by the degree of decay in local exponential averages
similar to

$q^{-1}\sum_{s=0}^{q-1}e^{2\pi ih(s)a/q}.$ (7)

The best general upper bound for (7) is of the order $q^{-1/k}$ where
$k=\deg(h)$, but the elaborate double iteration method developed in [26], and
the simplified improvement developed in [3], which lead to upper bounds like
(2) and (4), require decay at or near $q^{-1/2}$, which we refer to as square-
root cancellation. Inspired by [1], the second author [28] eliminated this
discrepancy for $k>2$ in the general case by employing a polynomial-specific
sieve to the set of considered inputs that, roughly speaking, reduced the
issue to estimating (7) at prime moduli, for which the desired square-root
cancellation is a well-known result of Weil. This sieve technique can be
thought of as a bridge from the integer setting to the best available
exponential sum estimates over finite fields.

Ruzsa and Sanders [32], and later Wang [39], were able to adapt the more
traditional density increment method to establish (3), which is a stronger
type of upper bound as compared with (2) or (4), based on two key factors: the
high degree of decay in the relevant exponential averages, which are
modifications of

$\phi(q)^{-1}\sum_{\begin{subarray}{c}s=0\\\
(s,q)=1\end{subarray}}^{q-1}e^{2\pi is/q}=\frac{\mu(q)}{\phi(q)},$

and the careful analysis of the distribution of primes in arithmetic
progressions, including the consideration of exceptional zeros of Dirichlet
$L$-functions. In the polynomial setting, the distribution of inputs in
arithmetic progressions is not as delicate of an issue, though it does rear
its head when employing a sieve, but this level of local decay is out of reach
with a single variable. Specifically, bounds like (1) from the density
increment require decay at or near $q^{-1}$ (more specifically, $q^{-1}$ times
a function of average value at most polylogarithmic in $q$, and the exponent
$\mu$ depends on the power of the logarithm), which we refer to as
q-cancellation.

While the image of a multivariate intersective polynomial does not necessarily
contain the image of a univariate intersective polynomial, it is the case
that, by only exploiting cancellation in one variable, the methods of [28] and
[3] can be adapted to show that (4) holds for such an image, so upper bounds
in the multivariate setting are only novel if they are stronger than (4). The
observation made in [28] to establish (5) was a rather simple one: if we
consider differences of the form $g(m)+h(n)$, then the relevant exponential
sum factors into a product, our sieve gives square-root cancellation in each
variable, and these combine to give $q$-cancellation. However, this
observation does not fully generalize to the case of a single polynomial in
several variables with nonzero cross-terms. In particular, simple examples
like $h(x,y)=(x+y)^{2}$ make it clear that one cannot always exploit
cancellation in each variable, so some sort of nonsingularity assumption is
required.

In the setting of binary quadratic forms, the natural assumption is nonzero
discriminant, and since sieving is not required to get square-root
cancellation from each variable in degree $2$, the adaptation of the usual
density increment is relatively straightforward, as done in [31] to establish
(6). Section 2 of [31] provides a helpful description of the density increment
method in a simpler, sieve-free context.

For higher degrees, the sieve technique can indeed be adapted to the
multivariate setting, which leads us toward the best available estimates on
exponential sums for multivariate polynomials over finite fields, due to
Deligne [8] in his proof of the Weil conjectures, and their associated
nonsingularity assumptions. Recall that $\mathbb{A}^{n}$ and $\mathbb{P}^{n}$
denote $n$-dimensional affine and projective space, respectively.

######  Definition 1.3.

Suppose $F$ is a field, $\ell\in\mathbb{N}$, and $g\in
F[x_{1},\dots,x_{\ell}]$ is a homogeneous polynomial. We say that $g$ is
smooth if the vanishing of $g$ defines a smooth hypersurface in
$\mathbb{P}^{\ell-1}$ (as opposed to $\mathbb{A}^{\ell}$). In other words, $g$
is smooth if the system $g(\bm{x})=\frac{\partial g}{\partial
x_{1}}(\bm{x})=\cdots=\frac{\partial g}{\partial x_{\ell}}(\bm{x})=0$ has no
solution besides $x_{1}=\cdots=x_{\ell}=0$ in $\overline{F}^{\ell}$, where the
bar indicates the algebraic closure. For a general polynomial $h\in
F[x_{1},\dots,x_{\ell}]$ with $h=\sum_{i=0}^{k}h^{i}$, where $h^{i}$ is
homogeneous of degree $i$ and $h^{k}\neq 0$, we say that $h$ is Deligne if the
characteristic of $F$ does not divide $k$ and $h^{k}$ is smooth.

######  Remark on notation.

For the remainder of the paper, we take the notational convention that, for a
polynomial $h$, $h^{i}$ denotes the degree-$i$ homogeneous part of $h$, as
opposed to $h$ raised to the $i$-th power.

######  Theorem 1.4 (Theorem 8.4, [8]).

Suppose $\ell\in\mathbb{N}$ and $p\in\mathcal{P}$. If
$h\in\mathbb{F}_{p}[x_{1},\dots,x_{\ell}]$ is Deligne, then

$\left|\sum_{\bm{x}\in\mathbb{F}_{p}^{\ell}}e^{2\pi
ih(\bm{x})/p}\right|\leq(\deg(h)-1)^{\ell}p^{\ell/2}.$

This estimate provides a guide, but additional consideration is required to
develop sufficient conditions on a multivariate polynomial for an application
of Theorem 1.4 that is compatible enough with the density increment procedure
to establish an upper bound like (1). We explore these details in Section 2.

###  1.3 Lower bounds and a special case

In all the nontrivial cases we have explored, there is a large gap in the
best-known upper and lower bounds for $D(X,N)$. For an intersective polynomial
$h\in\mathbb{Z}[x]$, all known lower bounds with $X=h(\mathbb{Z})$ are of
order $N^{c}$ for some $c<1$. The greedy algorithm gives $c=1-1/\deg(h)$, and
higher values of $c$ are known for monomials (see [33] and [20]) and certain
other polynomials divisible by $x^{2}$ (due to Younis [41], and explored from
an algebraic number theory perspective by Wessel [40]). For $X=\mathcal{P}-1$,
the gap is even larger, and the best-known lower bound is of the form
$N^{o(1)}$ (see [34]). Younis [41] established lower bounds for certain
homogeneous multivariate polynomials, including $D(S+S,N)\gg\sqrt{N}$, where
$S$ is the set of squares. All of these results are descended from methods of
Ruzsa that transfer examples from the modular setting to the integer setting.
In the absence of stronger lower bounds, the greedy algorithm gives
$D(X,N)\geq(N-1)/(|X\cap[-N,N]|+1)$ for any set $X\subseteq\mathbb{Z}$ (see
[25]).

As an aside, one very special case where stronger upper bounds on $D(X,N)$ are
available, and where the upper and lower bounds can be relatively close, is
the case when $X$ is itself, or at least contains, a difference set.
Specifically, if $Y\subseteq\\{1,\dots,N\\}$ and $X=Y-Y$, then for a set
$A\subseteq\\{1,\dots,N\\}$ satisfying $(A-A)\cap X\subseteq\\{0\\}$, we have
$a+y\neq a^{\prime}+y^{\prime}$ for all $a,a^{\prime}\in A$ and
$y,y^{\prime}\in Y$ with $(a,y)\neq(a^{\prime},y^{\prime})$. In particular,
the map $(a,y)\mapsto a+y$ into $\\{1,\dots,2N\\}$ is an injection, so
$|A||Y|\leq 2N$, and hence $D(X,N)\leq 2N/|Y|$, while the greedy algorithm
gives $D(X,N)\gg N/|X|\geq N/|Y|^{2}$. For an example relating to our
discussion of multivariate polynomials, if $X$ is the set of differences of
$k$-th powers for a fixed $k\in\mathbb{N}$, then we have $D(X,N)\ll
N^{1-1/k}$, but this observation does not immediately generalize beyond the
case where $X\supseteq Y-Y$.

##  2 Main definitions and results

The density increment procedure takes as input a set
$A\subseteq\\{1,2,\dots,N\\}$ lacking nonzero differences in the image of a
polynomial $h$, and produces a new, denser subset of a slightly smaller
interval lacking nonzero differences in the image of a potentially modified
polynomial. The following definition keeps track of the changes in the
polynomial over the course of the iteration.

######  Definition 2.1.

Fix $\ell\in\mathbb{N}$. As in the univariate setting, we say that
$h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is intersective if
$h(\mathbb{Z}^{\ell})$ contains a nonzero multiple of every $q\in\mathbb{N}$.
Equivalently, $h$ is intersective if it is not identically zero and has a root
in $\mathbb{Z}_{p}^{\ell}$ for every prime $p$, where $\mathbb{Z}_{p}$ denotes
the $p$-adic integers.

Suppose $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is an intersective polynomial
and fix, for each prime $p$, $\bm{z}_{p}\in\mathbb{Z}_{p}^{\ell}$ with
$h(\bm{z}_{p})=0$. All objects defined below certainly depend on this choice
of $p$-adic integer roots, but we suppress that dependence in the subsequent
notation.

By reducing modulo prime powers and applying the Chinese Remainder Theorem,
the choice of roots determines, for each $d\in\mathbb{N}$, a unique
$\bm{r}_{d}\in(-d,0]^{\ell}$ with $\bm{r}_{d}\equiv\bm{z}_{p}\ \text{mod
}p^{j}$ for all prime powers $p^{j}\mid d$.

We define a completely multiplicative function $\lambda$ (depending on $h$ and
$\\{\bm{z}_{p}\\}$) on $\mathbb{N}$ by letting $\lambda(p)=p^{m_{p}}$ for each
prime $p$, where $m_{p}$ is the multiplicity of $\bm{z}_{p}$ as a root of $h$,
that is,

$m_{p}=\min\left\\{i_{1}+\cdots+i_{\ell}:\frac{\partial^{i_{1}+\cdots+i_{\ell}}h}{\partial
x_{1}^{i_{1}}\cdots\partial x_{\ell}^{i_{\ell}}}(\bm{z}_{p})\neq 0\right\\}.$

Roughly speaking, $\lambda(d)$ is the largest guaranteed factor of $h(\bm{n})$
for $\bm{n}\equiv\bm{r}_{d}\ (\text{mod }d)$.

######  Definition 2.2.

With notation as described above, for each $d\in\mathbb{N}$ we define the
auxiliary polynomial $h_{d}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ by

$h_{d}(\bm{x})=h(\bm{r}_{d}+d\bm{x})/\lambda(d).$

Combining the hypotheses of Theorem 1.4 with the technical details of the
density increment iteration, the following definition captures a sufficient
condition for the success of the method.

######  Definition 2.3.

When considering polynomials with integer coefficients, we use the terms
smooth and Deligne as previously defined by embedding the coefficients in the
field of rational numbers. In particular,
$h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ of degree $k\geq 1$ is Deligne if the
system $h^{k}(\bm{x})=\frac{\partial h^{k}}{\partial
x_{1}}(\bm{x})=\cdots=\frac{\partial h^{k}}{\partial x_{\ell}}(\bm{x})=0$ has
no solution besides $x_{1}=\cdots=x_{\ell}=0$ in
$\overline{\mathbb{Q}}^{\ell}$. In this case, there exists a finite set of
primes $X=X(h)$ such that the reduction of $h$ modulo $p$ is Deligne for all
$p\notin X$: Indeed, one can take $X(h)$ to be the set of primes dividing the
Macaulay resultant $\operatorname{Res}\left(h^{k},\frac{\partial
h^{k}}{\partial x_{1}},\cdots,\frac{\partial h^{k}}{\partial
x_{\ell}}\right)$, which is nonzero precisely when $h$ is Deligne. (See also
Prop. A.9.1.6 of [16].)

Further, we say that $h$ is strongly Deligne if there exists a finite set of
primes $X=X(h)$ and a choice $\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$ of $p$-adic
integer roots of $h$ such that the reduction of $h_{d}$ modulo $p$ is Deligne
for all $p\notin X$ and all $d\in\mathbb{N}$. We note that strongly Deligne
polynomials are necessarily both Deligne and intersective.

To highlight some of the subtlety of this definition, we first note that
$h_{d}^{k}=\frac{d^{k}}{\lambda(d)}h^{k}$, so for a prime $p\nmid d$, we have
that if $h$ is Deligne modulo $p$, then $h_{d}$ is Deligne modulo $p$.
However, complications arise when $p\mid d$, because $h_{d}^{i}$ has a factor
of $d^{i}/\lambda(d)$, and hence vanishes modulo $p$ for all $i>m_{p}$. For an
example of a polynomial that is Deligne and intersective but not strongly
Deligne, see “the ugly” in Section 2.4.

For $k,\ell\geq 2$, we let
$\mu(k,\ell)=\begin{cases}[(k-1)^{2}+1]^{-1}&\text{if }\ell=2\\\ 1/2&\text{if
}\ell\geq 3\end{cases}.$ The central result of this paper is the following:

######  Theorem 2.4.

If $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is a strongly
Deligne polynomial of degree $k\geq 2$, then

$D(h(\mathbb{Z}^{\ell}),N)\ll_{h}Ne^{-c(\log N)^{\mu(k,\ell)}},$ (8)

where $c=c(h)>0$.

######  Remark.

In Theorem 2.4, the full image $h(\mathbb{Z}^{\ell})$ is considered for ease
of exposition, and to make the conclusion invariant under input translation.
However, by inspection of the proof, the same upper bound can be seen to hold
for $D(h([1,N^{\epsilon}]^{\ell}),N)$ for any $\epsilon>0$, with $c$ and the
implied constant depending on $\epsilon$. Also, in several of our results and
definitions, we exclude the case $k=1$ only out of convenience due to its
triviality in this context. Specifically, if
$h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(h)=1$, then
$D(h(\mathbb{Z}^{\ell}),N)\ll_{h}1$ if $0\in h(\mathbb{Z}^{\ell})$ and
$D(h(\mathbb{Z}^{\ell}),N)\gg_{h}N$ otherwise.

After setting the stage with preliminary definitions and observations in
Section 3, we prove Theorem 2.4 in Section 4, and then establish the needed
exponential sum estimates, which we state separately as Theorem 3.9, in
Section 7. More imminently, in Sections 2.1 and 2.2, we describe sufficient
conditions under which $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ is strongly
Deligne, and hence (8) holds. Then, in Section 2.3, we explain that in many
cases we may still get a bound similar to (8) even when the strongly Deligne
condition is significantly relaxed.

###  2.1 The integer root case

The simplest sufficient condition for the intersectivity of a nonzero
polynomial is the existence of an integer root. In this case, all $p$-adic
integer roots can be taken to equal said integer root, which simplifies the
auxiliary polynomial definition, giving rise to a pleasantly tangible
sufficient condition for the strongly Deligne property, as captured with the
following proposition.

######  Proposition 2.5.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with
$h(\bm{0})=0$. If the highest and lowest degree homogeneous parts of $h$ are
smooth, then $h$ is strongly Deligne.

###### Proof.

Suppose $h$ satisfies the hypotheses, let $k=\deg(h)$, let $j$ denote the
lowest degree of the nonzero terms of $h$, and let $X$ denote the finite set
of primes $p$ such that $p\mid jk$ or either $h^{k}$ or $h^{j}$ is not smooth
modulo $p$. Making the natural choice of $p$-adic integer roots
$\bm{z}_{p}=\bm{0}$ for all $p$, we then have
$h_{d}(\bm{x})=h(d\bm{x})/d^{j}$, hence
$h^{i}_{d}(\bm{x})=d^{i-j}h^{i}(\bm{x})$. Fix $p\notin X$. If $p\nmid d$, then
the highest degree part of $h_{d}$ modulo $p$ is a nonzero multiple of
$h^{k}$, which is smooth modulo $p$, hence $h_{d}$ is Deligne modulo $p$. If
$p\mid d$, then the only nonvanishing homogeneous part of $h_{d}$ is precisely
$h^{j}$, which is smooth modulo $p$, hence $h_{d}$ is Deligne modulo $p$.∎

######  Remark.

We note that $h(\mathbb{Z}^{\ell})$, hence the threshold
$D(h(\mathbb{Z}^{\ell}),N)$, as well as the Deligne and strongly Deligne
properties, are all invariant under translations of the form
$h(\bm{x}+\bm{n})$ for a fixed $\bm{n}\in\mathbb{Z}^{\ell}$. In particular,
Proposition 2.5 applies provided there exists $\bm{n}\in\mathbb{Z}^{\ell}$
such that $h(\bm{n})=0$ and the highest and lowest degree parts of
$h(\bm{x}+\bm{n})$ are smooth. More generally, all of our results that hold
for a polynomial $h$ also hold for the full translation equivalence class of
$h$.

For homogeneous bivariate polynomials, smoothness of the corresponding
($0$-dimensional) variety is equivalent to non-vanishing of the discriminant.
Therefore, for $\ell=2$, we have the following, which in particular combines
with Theorem 2.4 to yield Theorem 1.2 as a special case.

######  Corollary 2.6.

Suppose $h\in\mathbb{Z}[x,y]$ with $h(0,0)=0$. If the highest and lowest
degree homogeneous parts of $h$ have nonzero homogeneous discriminant, then
$h$ is strongly Deligne.

###  2.2 The Deligne case

Taking the next step in complexity, here we consider the case of a polynomial
that is Deligne and intersective, but may not have an integer root. Recalling
that if $p\mid d$, then $h^{i}_{d}$ vanishes modulo $p$ for all $i>m_{p}$, we
make the following definition with the hopes of exploiting the fact that a
nonzero homogeneous linear polynomial is guaranteed to be smooth.

######  Definition 2.7.

For $\ell\in\mathbb{N}$ and $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ we say
that $h$ is smoothly intersective if there exists a choice
$\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$ of $p$-adic integer roots of $h$ such
that $m_{p}=1$ for all but finitely many $p$. In other words, the variety
defined by $h=0$ has at least one point over $\mathbb{Z}_{p}$ for all $p$, and
at least one nonsingular point over $\mathbb{Z}_{p}$ for all but finitely many
$p$.

For low-hanging examples of polynomials that are intersective but not smoothly
intersective, one could consider the square of any intersective polynomial,
but such polynomials do not pass even our coarsest of nonsingularity filters.
For an example of a polynomial that is intersective but not smoothly
intersective in a more subtle and problematic way, see our discussion of “the
ugly” in Section 2.4. Combining the motivation for the smoothly intersective
definition with the fact that the highest degree part of a Deligne polynomial
is assumed to be smooth, the following proposition provides a sufficient
condition for the strongly Deligne property, and includes two notable special
cases.

######  Proposition 2.8.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ is Deligne
and intersective with $\deg(h)=k\geq 2$. If there exists a choice
$\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$ of $p$-adic integer roots of $h$
satisfying $m_{p}\in\\{1,k\\}$ for all but finitely many $p$, then $h$ is
strongly Deligne. In particular, if $k=2$ or $h$ is smoothly intersective,
then $h$ is strongly Deligne.

Using estimates on the number of nonsingular points on irreducible varieties
over finite fields, we obtain the following convenient criterion for smooth
intersectivity.

######  Proposition 2.9.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ is Deligne
and intersective, and let $h=g_{1}\cdots g_{n}$ be an irreducible
factorization of $h$ in $\mathbb{Z}[x_{1},\dots,x_{\ell}]$. If $g_{i}$ is
geometrically irreducible for some $1\leq i\leq n$, then $h$ is smoothly
intersective, hence strongly Deligne.

######  Remark.

The conclusion of Proposition 2.9 remains true under weaker assumptions on the
factorization of $h$. We give this cleaner statement here, but prove the more
general statement in Corollary 5.4.

For $\ell\geq 3$, the Deligne condition actually implies geometric
irreducibility, yielding the following:

######  Corollary 2.10.

Suppose $\ell\geq 3$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$. If $h$ is
Deligne and intersective, then $h$ is smoothly intersective, hence strongly
Deligne.

###### Proof.

By Proposition 2.9, it suffices to show that if $h$ is a Deligne polynomial in
$\ell\geq 3$ variables, then $h$ is geometrically irreducible. Suppose to the
contrary that $h=g_{1}g_{2}$ with
$g_{1},g_{2}\in\overline{\mathbb{Q}}[x_{1},\ldots,x_{\ell}]$ nonconstant of
degrees $d$ and $k-d$, respectively. In particular, we have
$h^{k}=g_{1}^{d}g_{2}^{k-d}$. Each of $\\{g_{1}^{d}=0\\}$ and
$\\{g_{2}^{k-d}=0\\}$ has codimension $1$ in $\mathbb{P}^{\ell-1}$ (since they
are hypersurfaces) and dimension at least $1$ (since we assumed $\ell\geq 3$).
In particular, $\\{g_{1}^{d}=0\\}$ and $\\{g_{2}^{k-d}=0\\}$ have nontrivial
intersection, and any intersection point must be a singular point of the union
$\\{h^{k}=0\\}$, contradicting the fact that $h$ is Deligne. ∎

In Section 5, we collect some crucial tools from algebraic geometry, which are
followed by the proofs of both Proposition 2.8 and the aforementioned
generalization of Proposition 2.9.

###  2.3 The singular case

While the Deligne condition is required to apply Theorem 1.4 to get the
desired cancellation in our exponential sums, brief consideration reveals that
the condition is not strictly necessary for a bound like (8) to hold, provided
the failure of the Deligne condition is in balance with the freedom of extra
variables. For a particularly simple example, consider
$h(x,y,z)=(x+z)^{4}+(x+z)y^{3}+y^{4}.$ This is a homogeneous degree-$4$
polynomial, and the variety $\widehat{V}\subseteq\mathbb{P}^{2}$ defined by
its vanishing has a unique singular point, namely $(1:0:-1)$. In particular,
$h$ is not Deligne. However, by fixing $z=0$, we can define
$g(x,y)=h(x,y,0)=x^{4}+xy^{3}+y^{4}$, which is a bivariate homogeneous
polynomial of nonzero discriminant. In particular, $g$ is strongly Deligne, so
Theorem 2.4 applies, and moreover $g(\mathbb{Z}^{2})=h(\mathbb{Z}^{3})$, so
(8) holds for $h$ as well, applied as if $\ell=2$ as opposed to $\ell=3$.

This example hints at a less black-and-white consideration of the singularity
of a projective variety. For $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with
$\deg(h)=k\geq 1$, $h$ is Deligne precisely when the variety
$\widehat{V}\subseteq\mathbb{P}^{\ell-1}$ defined by $h^{k}=0$ is nonsingular.
The example above indicates that we should really only need to avoid this
variety being “too singular”, which leads to the following definition.

######  Definition 2.11.

For $\ell\in\mathbb{N}$ and a nonconstant homogeneous polynomial
$g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$, let
$\widehat{V}\subseteq\mathbb{P}^{\ell-1}$ be the variety defined by $g=0$, and
let $\widehat{V}^{\mathrm{s}}$ be the singular locus of $\widehat{V}$. We
define the rank of $g$ to be the codimension of $\widehat{V}^{\mathrm{s}}$ in
$\mathbb{P}^{\ell-1}$, with the convention that the empty set has dimension
$-1$, hence the codimension of the empty set in $\mathbb{P}^{\ell-1}$ is
$\ell$. This is a notion of rank developed by Birch in [2] and utilized, for
example, in [6].

For $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(h)=k\geq 1$, the rank of
$h^{k}$ should, roughly speaking, encode the number of variables $r$ such that
$g(\mathbb{Z}^{r})\subseteq h(\mathbb{Z}^{\ell})$ for some Deligne polynomial
$g\in\mathbb{Z}[x_{1},\dots,x_{r}]$. In particular, $h$ is Deligne if and only
if the rank of $h^{k}$ is $\ell$. In Section 6, using careful dimension-
lowering arguments, we successfully expand the class of polynomials for which
a result analogous to Theorem 2.4 holds, generalizing our efforts from
Sections 2.1 and 2.2 as follows.

######  Theorem 2.12.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with
$h(\bm{0})=0$ and $\deg(h)=k\geq 2$. Let $r$ be the minimum rank of the
highest and lowest degree homogeneous parts of $h$. If $r\geq 2$, then

$D(h(\mathbb{Z}^{\ell}),N)\ll_{h}Ne^{-c(\log N)^{\mu(k,r)}},$ (9)

where $c=c(h)>0$.

######  Theorem 2.13.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is
intersective of degree $k\geq 2$. Let $r$ be the rank of $h^{k}$. If $r\geq
3$, OR if $r=2$ and there exists a choice $\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$
of $p$-adic integer roots of $h$ satisfying $m_{p}\in\\{1,k\\}$ for all but
finitely many $p$, then (9) holds.

######  Remark.

To shed light on the hypotheses of Theorems 2.12 and 2.13, we note that, for
$\ell\geq 2$ and a nonconstant homogeneous polynomial
$g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ of rank $r$, we have $r\geq 2$ if and
only if $g$ is squarefree—in other words, if and only if $g=0$ defines a
reduced variety.

###  2.4 Summary of results

For this section, we suppose $k,\ell\geq 2$ and
$h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(h)=k$, and we let $r$ denote
the rank of $h^{k}$. We assume $h$ is intersective, as otherwise
$D(h(\mathbb{Z}^{\ell}),N)\gg_{h}N$. The following bullet points summarize the
reach and limitations of our results.

  * •

The good: In addition to previously known results on sums of univariate
intersective polynomials (Theorems 1.2 and 5.7 of [28]), we now have that (9)
holds provided $h$, or in the case of (iii) any translation of $h$, meets any
of the following criteria:

    1. (i)

$r\geq 3$ (including Deligne with $\ell\geq 3$)

    2. (ii)

$r=k=2$ (including Deligne with $\ell=k=2$)

    3. (iii)

$r=2$ (including Deligne with $\ell=2$), $h(\bm{0})=0$, and the lowest degree
homogeneous part of $h$ has rank at least $2$. This includes as a special case
bivariate homogeneous polynomials with nonzero discriminant, which is Theorem
1.2 from the introduction.

    4. (iv)

$r=2$ (including Deligne with $\ell=2$) and $h$ is smoothly intersective, the
latter of which in particular holds if any irreducible (over $\mathbb{Z}$)
factor of $h$ is geometrically irreducible. Parts of this item can be made
slightly more general, as seen in the hypotheses of Proposition 2.9 and
Corollary 5.4.

An interesting example of (iv) that does not fit into any other category is
$h(x,y)=x^{3}+y^{3}-q$, where $q$ is a prime congruent to $1$ modulo $90090$
that is not expressable as the sum of two integer cubes, of which there are
plenty. This polynomial has no rational root, and it cannot be decomposed into
a sum of two univariate intersective polynomials, but it is Deligne and it has
simple roots in $\mathbb{Z}_{p}^{2}$ for all primes $p$. This example was
discussed in a remark following Theorem 1.2 in [28] to illustrate a limitation
of that result.

  * •

The bad: The methods utilized here fail to improve on univariate results in
the case that $r=1$, or equivalently the case that $h^{k}$ has a repeated
factor. It should be noted that we can only definitively say that it is
impossible to reach beyond the cutting edge of the univariate setting if
$h=f\circ g$ for some $g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ and
$f\in\mathbb{Z}[x]$ with $\deg(f)\geq 2$, because in this case
$h(\mathbb{Z}^{\ell})\subseteq f(\mathbb{Z})$. This was hinted at in the
introduction with the example $h(x,y)=(x+y)^{2}$. In this situation, $h^{k}$
is a proper power of the highest-degree part of $g$, so we definitely have
$r=1$. While it is certainly possible to have $r=1$ without $h$ being given as
a composition of this sort, our current methods cannot distinguish between the
two.

  * •

The ugly: A more subtle remaining hurdle is the case where $r=2$ (including
Deligne with $\ell=2$), $k\geq 3$, and $h$ does not meet either of the
criteria described in items (iii) or (iv). Focusing on the $\ell=2$ Deligne
case, such a polynomial must satisfy $\Delta(h^{k})\neq 0$, must be
intersective and hence have roots in $\mathbb{Z}_{p}^{2}$ for all primes $p$,
but by Proposition 2.8, for infinitely many $p$, all roots in
$\mathbb{Z}_{p}^{2}$ must satisfy $2\leq m_{p}\leq k-1$. In particular, by
Proposition 2.9, at least one coefficient in every geometrically irreducible
factor of $h$ must fail to be an integer. Finally, by Corollary 2.6, if $h$
satisfies $h(0,0)=0$, then the lowest degree part of $h$ must have
discriminant $0$.

One example is

$h(x,y)=x^{4}-2y^{4}+2x^{2}(x+y)+(x+y)^{2}=\left(x^{2}-\sqrt{2}y^{2}+(x+y)\right)\left(x^{2}+\sqrt{2}y^{2}+(x+y)\right).$

For any prime $p$ such that $2$ is not a square in $\mathbb{Q}_{p}$, the only
$\mathbb{Q}_{p}$ roots of $h$ are $(0,0)$ and $(-1,0)$. With these choices for
$\bm{z}_{p}$, the highest degree nonvanishing part of $h_{p}$ modulo $p$ is
either $(x+y)^{2}$ or $(x-y)^{2}$, respectively. In both cases
$\Delta(h_{p}^{2})=0$, and hence $h_{p}$ is not Deligne at this infinite
collection of primes. In other words, $h$ is not strongly Deligne, and we
cannot claim that (9) holds.

  * •

The future: The issue in the previous bullet point may represent an avoidable
artifact of the method, in which case the upper bound (9) could be shown to
hold for all intersective polynomials satisfying $r\geq 2$. Regarding improved
bounds, as noted in Section 2.3 of [31], and as implicitly referenced in [32]
when noting that the exponent $\mu$ in (3) could be increased to $1/2$
conditioned on the Generalized Riemann Hypothesis, an upper bound of order
$Ne^{-c\sqrt{\log N}}$ appears to be the limit of a Fourier analytic $L^{2}$
density increment. More specifically, if
$(\delta,N)\mapsto(\delta^{\prime},N^{\prime})$ represents the change in
density and interval size at each step of the iteration, then any further
improvement would require either $N^{\prime}/N$ to decay more slowly than any
power of $\delta$, or $\delta^{\prime}/\delta$ to tend to infinity, as
$\delta\to 0$, both of which appear incompatible with the method. To be clear,
this is not at all to say that much stronger upper bounds do not hold, even in
the univariate polynomial setting. As discussed in Section 1.3, this question
is rather murky. However, to achieve such a goal would likely require a
fundamentally different proof strategy.

##  3 Preliminaries

In this section we make some preliminary definitions and observations required
to execute the sieve-powered $L^{2}$ density increment strategy utilized to
prove Theorem 2.4.

###  3.1 Fourier analysis and the circle method on $\mathbb{Z}$

We embed our finite sets in $\mathbb{Z}$, on which we utilize an unnormalized
discrete Fourier transform. Specifically, for a function
$F:\mathbb{Z}\to\mathbb{C}$ with finite support, we define
$\widehat{F}:\mathbb{T}\to\mathbb{C}$, where $\mathbb{T}$ denotes the circle
parameterized by the interval $[0,1]$ with $0$ and $1$ identified, by

$\widehat{F}(\alpha)=\sum_{x\in\mathbb{Z}}F(x)e^{-2\pi ix\alpha}.$

Given $N\in\mathbb{N}$ and a set $A\subseteq[1,N]$ with $|A|=\delta N$, we
examine the Fourier analytic behavior of $A$ by considering the balanced
function, $f_{A}$, defined by $f_{A}=1_{A}-\delta 1_{[1,N]}.$

As is standard, we decompose the frequency space into two pieces: the points
of $\mathbb{T}$ that are close to rational numbers with small denominator, and
the complement.

######  Definition 3.1.

Given $\gamma>0$ and $Q\geq 1$, we define, for each $a,q\in\mathbb{N}$ with
$0\leq a\leq q-1$,

$\mathbf{M}_{a/q}(\gamma)=\left\\{\alpha\in\mathbb{T}:\Big{|}\alpha-\frac{a}{q}\Big{|}<\gamma\right\\},\
\mathbf{M}_{q}(\gamma)=\bigcup_{(a,q)=1}\mathbf{M}_{a/q}(\gamma),\text{ and
}\mathbf{M}^{\prime}_{q}(\gamma)=\bigcup_{r\mid
q}\mathbf{M}_{r}(\gamma)=\bigcup_{a=0}^{q-1}\mathbf{M}_{a/q}(\gamma).$

We then define the major arcs by

$\mathfrak{M}(\gamma,Q)=\bigcup_{q=1}^{Q}\mathbf{M}_{q}(\gamma),$

and the minor arcs by
$\mathfrak{m}(\gamma,Q)=\mathbb{T}\setminus\mathfrak{M}(\gamma,Q).$ We note
that if $2\gamma Q^{2}<1$, then

$\mathbf{M}_{a/q}(\gamma)\cap\mathbf{M}_{b/r}(\gamma)=\emptyset$ (10)

whenever $a/q\neq b/r$ and $q,r\leq Q$.

###  3.2 Inheritance proposition

As previously noted, we defined auxiliary polynomials to keep track of an
inherited lack of prescribed differences at each step of a density increment
iteration. The following proposition makes this inheritance precise.

######  Proposition 3.2.

Suppose $\ell\in\mathbb{N}$, $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is
intersective, $d,q\in\mathbb{N}$, and $A\subseteq\mathbb{N}$.

If $(A-A)\cap h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$ and
$A^{\prime}\subseteq\\{a:x+\lambda(q)a\in A\\}$ for some $x\in\mathbb{Z}$,
then $(A^{\prime}-A^{\prime})\cap h_{qd}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$.

###### Proof.

Suppose that $A\subseteq\mathbb{N}$, $A^{\prime}\subseteq\\{a:x+\lambda(q)a\in
A\\}$, and

$a-a^{\prime}=h_{qd}(\bm{n})=h(\bm{r}_{qd}+qd\bm{n})/\lambda(qd)\neq 0$

for some $\bm{n}\in\mathbb{Z}^{\ell}$, $a,a^{\prime}\in A^{\prime}$. By
construction we have that $\bm{r}_{qd}\equiv\bm{r}_{d}$ mod $d$, so there
exists $\bm{s}\in\mathbb{Z}^{\ell}$ such that
$\bm{r}_{qd}=\bm{r}_{d}+d\bm{s}$. Further, $\lambda$ is completely
multiplicative, and therefore

$0\neq
h_{d}(\bm{s}+q\bm{n})=h(\bm{r}_{d}+d(\bm{s}+q\bm{n}))/\lambda(d)=\lambda(q)h_{qd}(\bm{n})=\lambda(q)a-\lambda(q)a^{\prime}\in
A-A.$

Since $a-a^{\prime}\neq 0$, we have $(A-A)\cap
h_{d}(\mathbb{Z}^{\ell})\not\subseteq\\{0\\}$, and the contrapositive is
established. ∎

###  3.3 Sieve definitions and observations

As in [28], we apply a polynomial-specific sieve to our set of considered
inputs in order to, roughly speaking, reduce our analysis of local exponential
averages to the case of prime moduli, which in the multivariate setting allows
for the application of Theorem 1.4. To this end, for $\ell\in\mathbb{N}$, an
intersective polynomial $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$, and each prime
$p$ and $d\in\mathbb{N}$, we define $\gamma_{d}(p)$ to be the smallest power
such that $\nabla h_{d}$ modulo $p^{\gamma_{d}(p)}$ does not vanish
identically as a function on
$(\mathbb{Z}/p^{\gamma_{d}(p)}\mathbb{Z})^{\ell}$, and we let $j_{d}(p)$
denote the number of solutions to $\nabla h_{d}=\bm{0}$ in
$(\mathbb{Z}/p^{\gamma_{d}(p)}\mathbb{Z})^{\ell}$. Then, for $d\in\mathbb{N}$
and $Y>0$ we define

$W_{d}(Y)=\left\\{\bm{n}\in\mathbb{N}^{\ell}:\nabla
h_{d}(\bm{n})\not\equiv\bm{0}\text{ mod }p^{\gamma_{d}(p)}\text{ for all
}p\leq Y\right\\}.$

In the absence of a subscript $d$ in the usage of $\gamma(p),j(p),$ and
$W(Y)$, we assume $d=1$, in which case the definitions make sense even for
non-intersective polynomials. Further, for any
$g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ and $q\in\mathbb{N}$, we define

$W^{q}(Y)=\left\\{\bm{n}\in\mathbb{N}^{\ell}:\nabla
g(\bm{n})\not\equiv\bm{0}\text{ mod }p^{\gamma(p)}\text{ for all }p\leq Y,\
p^{\gamma(p)}\mid q\right\\}.$

Unlike in the univariate case, the size of $W(Y)$ here can be estimated with a
straightforward application of the inclusion-exclusion principle, as opposed
to a Brun sieve truncation thereof (see Proposition 2.4 in [28]). To achieve
this goal, however, we must first look forward and invoke an estimate
established in Section 7.1. For the following two statements, we assume
$\ell\geq 2$ and $g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(g)=k\geq
1$.

######  Lemma 3.3.

If $p$ is prime and $g$ is Deligne modulo $p$, then $j(p)\ll_{k,\ell}1$.

######  Proposition 3.4.

For any $x_{1},\dots,x_{\ell},Y>0$ we have

$\left|B\cap W(Y)\right|=x_{1}x_{2}\cdots x_{\ell}\prod_{p\leq
Y}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)+E,$ (11)

where $B=[1,x_{1}]\times\cdots\times[1,x_{\ell}]$,

$E=\begin{cases}O(X^{\ell-1}\log^{C}(Y))&\text{if }\ell=2\\\
O(X^{\ell-1})&\text{if }\ell\geq 3\end{cases},$

$X=\max\\{x_{1},\dots,x_{\ell}\\}$, $C=C(k,\ell)$, and the implied constants
depend only on $k$, $\ell$, the moduli at which $\nabla g$ identically
vanishes, and the primes $p\leq Y$ modulo which $g$ is not Deligne.

###### Proof.

Fix $x_{1},\dots,x_{\ell},Y>0$ and let $X=\max\\{x_{1},\dots,x_{\ell}\\}$. For
primes $p_{1}<p_{2}<\cdots<p_{s}$, we let

$\mathcal{A}_{p_{1}\cdots p_{s}}=\mathcal{A}_{p_{1}\cdots
p_{s}}(x_{1},\dots,x_{\ell})=\left|\left\\{\bm{n}\in B:\nabla
g(\bm{n})\equiv\bm{0}\text{ mod }p_{i}^{\gamma(p_{i})}\ \text{for all }1\leq
i\leq s\right\\}\right|.$

Fixing $Y>0$ and letting $r$ denote the number of primes that are at most $Y$,
we have by the Chinese Remainder Theorem and the inclusion-exclusion principle
that

$\left|B\cap W(Y)\right|=\sum_{s=0}^{r}(-1)^{s}\sum_{p_{1}<\dots<p_{s}\leq
Y}\mathcal{A}_{p_{1}\cdots p_{s}}.$ (12)

Further,

$\mathcal{A}_{p}=\frac{j(p)}{p^{\gamma(p)\ell}}x_{1}\cdots x_{\ell}+R_{p},$
(13)

where $|R_{p}|\ll_{\ell}j(p)(X/p^{\gamma(p)})^{\ell-1}$. We trivially have
$j(p)\leq p^{\gamma(p)\ell}$, while if $g$ is Deligne modulo $p$, then
$\gamma(p)=1$ and, by Lemma 3.3, $j(p)\leq C=C(k,\ell)$. In particular, we can
apply the Chinese Remainder Theorem again and extend (13) to

$\mathcal{A}_{p_{1}\cdots p_{s}}=x_{1}\cdots
x_{\ell}\prod_{i=1}^{s}\frac{j(p_{i})}{p_{i}^{\gamma(p_{i})\ell}}+R_{p_{1}\cdots
p_{s}},$ (14)

where $|R_{p_{1}\cdots p_{s}}|\leq KC^{s}(X/p_{1}\cdots p_{s})^{\ell-1}$,
where $K$ depends only on the moduli at which $\nabla g$ identically vanishes
and the primes $p\leq Y$ modulo which $g$ is not Deligne. Now, by (12) and
(14) we have

$\displaystyle\left|B\cap W(Y)\right|$
$\displaystyle=\sum_{s=0}^{r}(-1)^{s}\sum_{p_{1}<\dots<p_{s}\leq
Y}\mathcal{A}_{p_{1}\cdots p_{s}}$
$\displaystyle=\sum_{s=0}^{r}(-1)^{s}\sum_{p_{1}<\dots<p_{s}\leq
Y}\left(x_{1}\cdots
x_{\ell}\prod_{i=1}^{s}\frac{j(p_{i})}{p_{i}^{\gamma(p_{i})\ell}}+R_{p_{1}\cdots
p_{s}}\right)$ $\displaystyle=x_{1}x_{2}\cdots x_{\ell}\prod_{p\leq
Y}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)+E,$

where

$\displaystyle|E|$ $\displaystyle\leq
KX^{\ell-1}\sum_{s=0}^{r}\sum_{p_{1}<\dots<p_{s}\leq
Y}\frac{C^{s}}{(p_{1}\cdots p_{s})^{\ell-1}}=KX^{\ell-1}\prod_{p\leq
Y}\left(1+\frac{C}{p^{\ell-1}}\right),$

and the estimate follows. ∎

###  3.4 Control over gradient vanishing: Part I

A potential hazard of the density increment method is the possibility that, as
$d$ grows, $\nabla h_{d}$ could identically vanish at a larger and larger
collection of moduli. This section is dedicated to establishing that, for
strongly Deligne polynomials, this does not occur. We begin by noting that the
collection of moduli at which a polynomial identically vanishes is firmly
controlled in terms of its degree and the gcd of its coefficients. Throughout
this section we assume $k,\ell\in\mathbb{N}$.

######  Definition 3.5.

We define a multi-index to be an $\ell$-tuple $\bm{i}=(i_{1},\dots,i_{\ell})$
of nonnegative integers. We let $|\bm{i}|=i_{1}+\dots+i_{\ell}$, we let
$\bm{i}!=i_{1}!\cdots i_{\ell}!$, and for $\bm{x}=(x_{1},\dots,x_{\ell})$, we
let $\bm{x}^{\bm{i}}=x_{1}^{i_{1}}\cdots x_{\ell}^{i_{\ell}}$. Finally, for a
polynomial $g(\bm{x})$, we let
$\partial^{\bm{i}}g=\frac{\partial^{|\bm{i}|}g}{\partial
x_{1}^{i_{1}}\cdots\partial x_{\ell}^{i_{\ell}}}$.

######  Proposition 3.6.

If $g(\bm{x})=\sum_{|\bm{i}|\leq
k}a_{\bm{i}}\bm{x}^{\bm{i}}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is identically
zero modulo $q\in\mathbb{N}$, then

$q\mid k!\gcd(\\{a_{\bm{i}}\\}).$

###### Proof.

We first note that $g$ is identically zero as a function on
$\mathbb{Z}/q\mathbb{Z}$ if and only if the polynomial $g/q$ is integer-
valued. In this case, since products of binomial coefficients

$\binom{\bm{x}}{\bm{i}}=\binom{x_{1}}{i_{1}}\cdots\binom{x_{\ell}}{i_{\ell}}=\frac{x(x-1)\dots(x-i_{1}+1)}{i_{1}!}\cdots\frac{x(x-1)\dots(x-i_{\ell}+1)}{i_{\ell}!}$

form a $\mathbb{Z}$-basis for integer-valued polynomials in
$\mathbb{Q}[x_{1},\dots,x_{\ell}]$, we can write $g(x)=\sum_{|\bm{i}|\leq
k}qb_{\bm{i}}\binom{\bm{x}}{\bm{i}}$ for $b_{\bm{i}}\in\mathbb{Z}$. In
particular, by clearing denominators we see that the coefficients of $k!g$ are
all divisible by $q$, and the proposition follows. ∎

Further, we note that the gcd of the coefficients of each partial derivative
of a polynomial $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ divides $k!$ times the
gcd of the nonconstant coefficients of $h$. With this in mind, the following
definition and proposition complete the task at hand.

######  Definition 3.7.

For $h(\bm{x})=\sum_{|\bm{i}|\leq
k}a_{\bm{i}}\bm{x}^{\bm{i}}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$, we define

$\textnormal{cont}(h)=\gcd(\\{a_{\bm{i}}:|\bm{i}|>0\\}).$

We note that our use of $\textnormal{cont}(h)$ does not precisely align with
the standard notion of the content of a polynomial, as we exclude the constant
coefficient.

######  Proposition 3.8.

If $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is a strongly Deligne polynomial of
degree $k$, then

$\textnormal{cont}(h_{d})\ll_{h}1.$

###### Proof.

Suppose $d\in\mathbb{N}$ and $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is a
strongly Deligne polynomial of degree $k$. Let
$\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$ and $X$ denote the choice of $p$-adic
integer roots and the finite set of primes, respectively, guaranteed by the
strongly Deligne condition. In particular, $h_{d}$ is Deligne modulo $p$ for
all $p\notin X$. Because constant polynomials are not Deligne,
$\textnormal{cont}(h_{d})$ can only be divisible by primes in $X$.

Recalling that $h_{d}(\bm{x})=h(\bm{r}_{d}+d\bm{x})/\lambda(d)$, we make the
trivial note that for any multi-index $\bm{i}$ with $|\bm{i}|=k$, the
$\bm{x}^{\bm{i}}$ coefficient of $h_{d}$ is precisely $d^{k}/\lambda(d)$ times
the corresponding coefficient $a_{\bm{i}}$ of $h$. In particular,

$\textnormal{cont}(h_{d})\mid\frac{d^{k}}{\lambda(d)}a_{\bm{i}}\
\text{whenever}\ |\bm{i}|=k.$ (15)

Now fix $p\in X$. By definition of the multiplicity $m_{p}$, there exists a
multi-index $\bm{i}$ with $|\bm{i}|=m_{p}$ and
$\partial^{\bm{i}}h(\bm{z}_{p})\neq 0$, so in particular
$\partial^{\bm{i}}h(\bm{z}_{p})$ has some finite $p$-adic valuation
$v_{1}(p)$.

If $p^{v_{1}(p)+1}\nmid d$, then by (15), we have that
$p^{kv_{1}(p)+v_{2}(p)+1}\nmid\textnormal{cont}(h_{d})$, where $v_{2}(p)$ is
the minimum $p$-adic valuation amongst the degree-$k$ coefficients of $h$. Now
suppose that $p^{v_{1}(p)+1}\mid d$.

Let $b_{\bm{i}}$ denote the $\bm{x}^{\bm{i}}$ coefficient of $h_{d}$. By
Taylor’s formula, we have that

$b_{\bm{i}}=\frac{d^{m_{p}}}{\lambda(d)}\frac{\partial^{\bm{i}}h(\bm{r}_{d})}{\bm{i}!}.$

By definition of $\lambda$ we have $p\nmid(d^{m_{p}}/\lambda(d))$, and since
$\bm{r}_{d}\equiv\bm{z}_{p}\ \text{mod }p^{v_{1}(p)+1}$ and
$p^{v_{1}(p)+1}\nmid\partial^{\bm{i}}h(\bm{z}_{p})$, we have that
$p^{v_{1}(p)+1}\nmid b_{\bm{i}}$. In either case, we have that
$p^{kv_{1}(p)+v_{2}(p)+1}\nmid\textnormal{cont}(h_{d})$, and hence

$\textnormal{cont}(h_{d})\leq\prod_{p\in X}p^{kv_{1}(p)+v_{2}(p)+1}\ll_{h}1,$

as required.∎

For strongly Deligne $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(h)=k$,
we have now established control over not only the error term in the size of
$W_{d}(Y)$, but also the main term, since Lemma 3.3, Proposition 3.6, and
Proposition 3.8 give

$\prod_{p\leq
Y}\left(1-\frac{j_{d}(p)}{p^{\gamma_{d}(p)\ell}}\right)\gg_{h}\prod_{C=C(h)\leq
p\leq Y}\left(1-\frac{C}{p^{\ell}}\right)\gg_{h}1$ (16)

for all $d\in\mathbb{N}$ and $Y\geq 2$.

###  3.5 Summary of new exponential sum estimates

In Section 7, we combine new and old techniques to establish the sieved
multivariate exponential sum estimates necessary to prove Theorem 2.4. These
estimates are obtained through a sequence of lemmas presented in the context
of the larger proof, so we separately present a summary here in case the
estimates are of independent interest to the reader.

For the following theorem, a multivariate generalization of Theorem 2.7 in
[28], we utilize all the sieve-related notation and definitions from Section
3.3. Further, we use $\tau$ and $\omega$ to denote the divisor and distinct
prime divisor counting functions, respectively, as well as $\phi$ to denote
the Euler totient function.

######  Theorem 3.9.

For $k,\ell\geq 2$, $g(\bm{x})=\sum_{|\bm{i}|\leq
k}a_{\bm{i}}\bm{x}^{\bm{i}}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$,
$J=\sum_{|\bm{i}|\leq k}|a_{\bm{i}}|$, and $a,q\in\mathbb{N}$, the following
estimates hold:

  1. (i)

Major arc estimate: If $X,Y>0$ and $\alpha=a/q+\beta$, then

$\displaystyle\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi ig(\bm{n})\alpha}$
$\displaystyle=q^{-\ell}\prod_{\begin{subarray}{c}p\leq Y\\\
p^{\gamma(p)}\nmid
q\end{subarray}}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap
W^{q}(Y)}e^{2\pi ig(\bm{s})a/q}\int_{[0,X]^{\ell}}e^{2\pi
ig(\bm{x})\beta}d\bm{x}$
$\displaystyle\qquad+O_{k,\ell}\left(qE(1+JX^{k}|\beta|)^{\ell}\right),$

where $E$ is as in Proposition 3.4.

  2. (ii)

Local cancellation: If $(a,q)=1$ and $Y>0$, then

$\left|\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap W^{q}(Y)}e^{2\pi
ig(\bm{s})a/q}\right|\leq
C_{1}\begin{cases}(k-1)^{\ell\omega(q)}\Phi(q,\ell)q^{\ell/2}&\text{if }q\leq
Y\\\ C_{2}^{\omega(q)}\tau(q)^{\ell}q^{\ell-1/k}&\text{for all }q\end{cases},$

where $C_{2}=C_{2}(k)$, $\Phi(q,2)=(q/\phi(q))^{C_{2}}$,
$\Phi(q,\ell)\ll_{k,\ell}1$ for $\ell\geq 3$, and $C_{1}$ depends only on the
moduli at which $\nabla g$ identically vanishes and the primes $p\leq Y$
dividing $q$ modulo which $g$ is not Deligne.

  3. (iii)

Minor arc estimate: If $X,Y,Z\geq 2$, $YZ\leq X$, $(a,q)=1$, and
$|\alpha-a/q|<q^{-2}$, then

$\left|\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi
ig(\bm{n})\alpha}\right|\ll_{k,\ell}\textnormal{cont}(g)^{6}(\log
Y)^{ek}X^{\ell}\left(e^{-\frac{\log Z}{\log
Y}}+\left(J\log^{k^{2}}(JqX)\left(q^{-1}+\frac{Z}{X}+\frac{qZ^{k}}{X^{k}}\right)\right)^{2^{-k}}\right).$

##  4 Proof of Theorem 2.4

In this section, we exploit the estimates enumerated in Theorem 3.9 and apply
a Fourier analytic $L^{2}$ density increment, essentially an improved,
streamlined version of Sárközy’s [35] original method, in order to prove
Theorem 2.4. The core of this method has been utilized in [23], [22], [32],
and [30], among others. Most specifically, this section very closely follows
Section 5 of [28].

###  4.1 Main iteration lemma and proof of Theorem 2.4

For the remainder of Section 4 we fix $k,\ell\geq 2$, a strongly Deligne
polynomial $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(h)=k$, and
positive constants $C_{0}=C_{0}(h)$ and $c_{0}=c_{0}(h)$ that are
appropriately large and small, respectively. For $N\in\mathbb{N}$ we let

$\mathcal{Q}=\mathcal{Q}(N,h)=e^{c_{0}\sqrt{\log N}}.$

For a density $\delta\in(0,1]$, we define $\theta(k,\ell,\delta)$ by
$\theta(k,\ell,\delta)=1$ if $\ell\geq 3$ and
$\theta(k,2,\delta)=\log^{-k(k-2)}((c_{0}\delta)^{-1})$.

We deduce Theorem 2.4 from the following iteration lemma, which makes precise
the aforementioned passage from a set lacking nonzero differences in the image
of a polynomial to a new, denser subset of a slightly smaller interval lacking
nonzero differences in the image of an appropriate auxiliary polynomial.

######  Lemma 4.1.

Suppose $A\subseteq[1,N]$ with $|A|=\delta N$. If $(A-A)\cap
h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$, $C_{0},\delta^{-1}\leq\mathcal{Q}$,
and $d\leq N^{c_{0}}$, then there exists $q\ll_{h}\delta^{-2}$ and
$A^{\prime}\subseteq[1,N^{\prime}]$ such that $N^{\prime}\gg_{h}\delta^{4k}N$,

$|A^{\prime}|\geq(1+c\theta(k,\ell,\delta))\delta N^{\prime},$

where $c=c(h)>0$, and

$(A^{\prime}-A^{\prime})\cap h_{qd}(\mathbb{Z}^{\ell})\subseteq\\{0\\}.$

###### Proof of Theorem 2.4.

Throughout this proof, we let $C$ and $c$ denote sufficiently large or small
positive constants, respectively, which we allow to change from line to line,
but which depend only on $h$.

Suppose $A\subseteq[1,N]$ with $|A|=\delta N$ and

$(A-A)\cap h(\mathbb{Z}^{\ell})\subseteq\\{0\\}.$

Setting $A_{0}=A$, $N_{0}=N$, $d_{0}=1$, and $\delta_{0}=\delta$, Lemma 4.1
yields, for each $m$, a set $A_{m}\subseteq[1,N_{m}]$ with
$|A_{m}|=\delta_{m}N_{m}$ and $(A_{m}-A_{m})\cap
h_{d_{m}}(\mathbb{Z}^{\ell})\subseteq\\{0\\}.$ Further, we have that

$N_{m}\geq c\delta^{4k}N_{m-1}\geq(c\delta)^{4km}N,$ (17)
$\delta_{m}\geq(1+c\theta(k,\ell,\delta))\delta_{m-1},$ (18)

and

$d_{m}\leq(c\delta)^{-2}d_{m-1}\leq(c\delta)^{-2m},$ (19)

as long as

$C,\delta_{m}^{-1}\leq e^{c\sqrt{\log N_{m}}},\quad d\leq N_{m}^{c}.$ (20)

By (18), the density $\delta_{m}$ will exceed $1$, and hence (20) must fail,
for $m=M=M(h,\delta)$, where

$M(h,\delta)=\begin{cases}C\log(C\delta^{-1})&\text{if }\ell\geq 3\\\
C\log^{(k-1)^{2}}(C\delta^{-1})&\text{if }\ell=2\end{cases}.$

However, by (17), (18), and (19), (20) holds for $m=M$ if

$(c\delta)^{4kM}=e^{C\log^{\mu(k,\ell)^{-1}}(C\delta^{-1})}\leq N^{c}.$ (21)

Therefore, (21) must fail, or in other words $\delta\ll_{h}e^{-c(\log
N)^{\mu(k,\ell)}},$ as claimed. ∎

###  4.2 $L^{2}$ Fourier concentration and proof of Lemma 4.1

The philosophy behind the proof of Lemma 4.1 is that the condition $(A-A)\cap
h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$ represents highly nonrandom
behavior, which should be detectable in the Fourier analytic behavior of $A$.
Specifically, we locate one small denominator $q$ such that $\widehat{f_{A}}$
has $L^{2}$ concentration around rationals with denominator $q$, then invoke a
standard lemma stating that $L^{2}$ concentration of $\widehat{f_{A}}$ implies
the existence a long arithmetic progression on which $A$ has increased
density.

######  Lemma 4.2.

Suppose $A\subseteq[1,N]$ with $|A|=\delta N$, $\eta=c_{0}\delta$, and
$\gamma=\eta^{-2k}/N$. If $(A-A)\cap
h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$, $C_{0},\delta^{-1}\leq\mathcal{Q}$,
$d\leq N^{c_{0}}$, and $|A\cap(N/9,8N/9)|\geq 3\delta N/4$, then there exists
$q\leq\eta^{-2}$ such that

$\int_{\mathbf{M}^{\prime}_{q}(\gamma)}|\widehat{f_{A}}(\alpha)|^{2}d\alpha\gg_{h}\theta(k,\ell,\delta)\delta^{2}N.$

Lemma 4.1 follows from Lemma 4.2 and the following standard $L^{2}$ density
increment lemma.

######  Lemma 4.3 (Lemma 2.3 in [29], see also [23], [32]).

Suppose $A\subseteq[1,N]$ with $|A|=\delta N$. If $0<\theta\leq 1$,
$q\in\mathbb{N}$, $\gamma>0$, and

$\int_{\mathbf{M}^{\prime}_{q}(\gamma)}|\widehat{f_{A}}(\alpha)|^{2}d\alpha\geq\theta\delta^{2}N,$

then there exists an arithmetic progression

$P=\\{x+\ell q:1\leq\ell\leq L\\}$

with $qL\gg\min\\{\theta N,\gamma^{-1}\\}$ and $|A\cap
P|\geq(1+\theta/32)\delta L$.

###### Proof of Lemma 4.1.

Suppose $A\subseteq[1,N]$, $|A|=\delta N$, $(A-A)\cap
h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$, $C_{0},\delta^{-1}\leq\mathcal{Q}$,
and $d\leq N^{c_{0}}$. If $|A\cap(N/9,8N/9)|<3\delta N/4$, then
$\max\\{|A\cap[1,N/9]|,|A\cap[8N/9,N]|\\}>\delta N/8$. In other words, $A$ has
density at least $9\delta/8$ on one of these intervals.

Otherwise, Lemmas 4.2 and 4.3 apply, so in either case, letting
$\eta=c_{0}\delta$, there exists $q\leq\eta^{-2}$ and an arithmetic
progression

$P=\\{x+\ell q:1\leq\ell\leq L\\}$

with $qL\gg_{h}\delta^{2k}N$ and

$|A\cap P|\geq(1+c\theta(k,\ell,\delta))\delta L.$

Partitioning $P$ into subprogressions of step size $\lambda(q)$, the
pigeonhole principle yields a progression

$P^{\prime}=\\{y+a\lambda(q):1\leq a\leq N^{\prime}\\}\subseteq P$

with $N^{\prime}\geq qL/2\lambda(q)$ and $|A\cap
P^{\prime}|/N^{\prime}\geq|A\cap P|/L$. This allows us to define a set
$A^{\prime}\subseteq[1,N^{\prime}]$ by

$A^{\prime}=\\{a\in[1,N^{\prime}]:y+a\lambda(q)\in A\\},$

which satisfies $|A^{\prime}|=|A\cap P^{\prime}|$ and
$N^{\prime}\gg_{h}\delta^{2k}N/\lambda(q)\gg_{h}\delta^{4k}N$. Moreover,
$(A-A)\cap h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$ implies
$(A^{\prime}-A^{\prime})\cap h_{qd}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$ by
Proposition 3.2. ∎

Our task for this section is now completely reduced to a proof of Lemma 4.2.

###  4.3 Preliminary notation for proof of Lemma 4.2

Before delving into the proof of Lemma 4.2, we take the opportunity to define
some relevant sets and quantities, depending on our strongly Deligne
polynomial $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$, scaling parameter $d$, a
parameter $Y>0$, and the size $N$ of the ambient interval. In all the notation
defined below, we suppress all of the aforementioned dependence, as the
relevant objects will be fixed in context.

We define $W_{d}$, $\gamma_{d}$, and $j_{d}$ in terms of $h$ as in Section
3.3. We then let $M=\left(\frac{N}{9J}\right)^{1/k}$, where $J$ is the sum of
the absolute value of all the coefficients of $h_{d}$, and hence
$h_{d}([1,M]^{\ell})\subseteq[-N/9,N/9]$. We let

$w=\prod_{p\leq Y}\left(1-\frac{j_{d}(p)}{p^{\gamma_{d}(p)\ell}}\right),$

and we let $T=wM^{\ell}$.

We let $Z=\\{\bm{n}\in\mathbb{Z}^{\ell}:h_{d}(\bm{n})=0\\}$, and we let
$H=\left([1,M]^{\ell}\cap W_{d}(Y)\right)\setminus Z$. We note that the
hypothesis $\mathcal{Q}\geq C_{0}$ allows us to assume at any point that
$\mathcal{Q}$, and hence also $N$, are sufficiently large with respect to $h$,
which we take as a perpetual assumption moving forward. Under this assumption,
it follows from (11), (16), and the estimate

$|Z\cap[1,M]^{\ell}|\ll_{h}M^{\ell-1}$ (22)

that

$\left|H\right|\geq T/2.$ (23)

###  4.4 Proof of Lemma 4.2

Suppose $A\subseteq[1,N]$ with $|A|=\delta N$, $(A-A)\cap
h_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\}$, $C_{0},\delta^{-1}\leq\mathcal{Q}$,
and $d\leq N^{c_{0}}$. Further, let $\eta=c_{0}\delta$, let $Q=\eta^{-2}$, and
let $Y=\eta^{-2k}$. Since $h_{d}(H)\subseteq[-N/9,N/9]\setminus\\{0\\}$, we
have

$\displaystyle\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\ \bm{n}\in
H\end{subarray}}f_{A}(x)f_{A}(x+h_{d}(\bm{n}))$
$\displaystyle=\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\ \bm{n}\in
H\end{subarray}}1_{A}(x)1_{A}(x+h_{d}(\bm{n}))-\delta\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\
\bm{n}\in H\end{subarray}}1_{A}(x)1_{[1,N]}(x+h_{d}(\bm{n}))$
$\displaystyle\qquad-\delta\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\
\bm{n}\in
H\end{subarray}}1_{A}(x+h_{d}(\bm{n}))1_{[1,N]}(x)+\delta^{2}\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\
\bm{n}\in H\end{subarray}}1_{[1,N]}(x)1_{[1,N]}(x+h_{d}(\bm{n}))$
$\displaystyle\leq\Big{(}\delta^{2}N-2\delta|A\cap(N/9,8N/9)|\Big{)}|H|.$

Therefore, if $|A\cap(N/9,8N/9)|\geq 3\delta N/4$, then by (23) we have

$\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\ \bm{n}\in
H\end{subarray}}f_{A}(x)f_{A}(x+h_{d}(\bm{n}))\leq-\delta^{2}NT/4.$ (24)

We see from (22) and orthogonality of characters that

$\sum_{\begin{subarray}{c}x\in\mathbb{Z}\\\ \bm{n}\in
H\end{subarray}}f_{A}(x)f_{A}(x+h_{d}(\bm{n}))=\int_{0}^{1}|\widehat{f_{A}}(\alpha)|^{2}S(\alpha)d\alpha+O_{h}(NM^{\ell-1}),$
(25)

where

$S(\alpha)=\sum_{\bm{n}\in[1,M]^{\ell}\cap W_{d}(Y)}e^{2\pi
ih_{d}(\bm{n})\alpha}.$

Combining (24) and (25), we have

$\int_{0}^{1}|\widehat{f_{A}}(\alpha)|^{2}|S(\alpha)|d\alpha\geq\delta^{2}NT/8.$
(26)

Letting $\gamma=\eta^{-2k}/N$, Theorem 3.9 yields that for
$\alpha\in\mathbf{M}_{q}(\gamma),\ q\leq Q$, we have

$|S(\alpha)|\ll_{h}(k-1)^{\ell\omega(q)}\Phi(q,\ell)q^{-\ell/2}T,$ (27)

where $\Phi(q,2)=(q/\phi(q))^{C}$ for $C=C(k)$ and $\Phi(q,\ell)\ll_{k,\ell}1$
for $\ell\geq 3$. Further, for $\alpha\in\mathfrak{m}(\gamma,Q)$ we have

$|S(\alpha)|\leq\delta T/16.$ (28)

The proof of the estimates in Theorem 3.9 and the subsequent deduction of (27)
and (28) can be found in Section 7. From (28) and Plancherel’s Identity, we
have

$\int_{\mathfrak{m}(\gamma,Q)}|\widehat{f_{A}}(\alpha)|^{2}|S(\alpha)|d\alpha\leq\delta^{2}NT/16,$

which together with (26) yields

$\int_{\mathfrak{M}(\gamma,Q)}|\widehat{f_{A}}(\alpha)|^{2}|S(\alpha)|d\alpha\geq\delta^{2}NT/16.$
(29)

From (27) and (29) , we have

$\sum_{q=1}^{Q}(k-1)^{\ell\omega(q)}\Phi(q)q^{-\ell/2}\int_{\mathbf{M}_{q}(\gamma)}|\widehat{f_{A}}(\alpha)|^{2}{d}\alpha\gg_{h}\delta^{2}N.$
(30)

For $\ell=2$, the function $b(q)=(k-1)^{2\omega(q)}(q/\phi(q))^{C}$ satisfies
$b(qr)\geq b(r)$, and we make use of the following proposition, which is based
on a trick that originated in [32].

######  Proposition 4.4 (Proposition 5.6, [28]).

For any $\gamma,Q>0$ satisfying $2\gamma Q^{2}<1$, and for any function
$b:\mathbb{N}\to[0,\infty)$ satisfying $b(qr)\geq b(r)$ for all
$q,r\in\mathbb{N}$, we have

$\max_{q\leq
Q}\int_{\mathbf{M}^{\prime}_{q}(\gamma)}|\widehat{f_{A}}(\alpha)|^{2}{d}\alpha\geq
Q\Big{(}2\sum_{q=1}^{Q}b(q)\Big{)}^{-1}\sum_{r=1}^{Q}\frac{b(r)}{r}\int_{\mathbf{M}_{r}(\gamma)}|\widehat{f_{A}}(\alpha)|^{2}{d}\alpha.$

Because $b$ is a multiplicative function,
$b(p^{v})=(k-1)^{2}(1+1/(p-1))^{C}\ll_{k}1$ for all prime powers $p^{v}$, and

$\sum_{q=1}^{Q}\frac{b(q)}{q}\leq\prod_{p\leq
Q}\left(1+\frac{b(p)}{p}+\frac{b(p)}{p^{2}}+\cdots\right)=\prod_{p\leq
Q}\left(1+\frac{(k-1)^{2}}{p}+O_{k}(1/p^{2})\right)\ll_{k}\log^{(k-1)^{2}}Q,$

it follows from Theorem 01 of [13] that

$\sum_{q=1}^{Q}b(q)\ll_{k}Q\log^{(k-1)^{2}-1}Q,$

and the lemma for $\ell=2$ follows from (30) and Proposition 4.4. For
$\ell\geq 3$, since $(k-1)^{\ell\omega(q)}\ll_{k,\ell,\epsilon}q^{\epsilon}$
for any $\epsilon>0$, the sum
$\sum_{q=1}^{\infty}(k-1)^{\ell\omega(q)}q^{-\ell/2}$ is convergent, and hence
(30) immediately yields

$\max_{q\leq
Q}\int_{\mathbf{M}_{q}(\gamma)}|\widehat{f_{A}}(\alpha)|^{2}{d}\alpha\gg_{h}\delta^{2}N.$

Since $\mathbf{M}_{q}(\gamma)\subseteq\mathbf{M}^{\prime}_{q}(\gamma),$ this
establishes the lemma for $\ell\geq 3$. ∎

##  5 Criteria for strongly Deligne polynomials

In this section, we prove Proposition 2.8 and a stronger version of
Proposition 2.9. We begin, though, by collecting a few facts from algebraic
geometry that will be useful in subsequent sections. Throughout this section,
for a variety $V$, we let $V^{\mathrm{s}}$ denote the singular locus of $V$,
and we let $V^{\operatorname{ns}}=V\setminus V^{\mathrm{s}}$.

###  5.1 Results from algebraic geometry

We first state a classical version of Bézout’s Theorem; see [9, Example
8.4.6].

######  Lemma 5.1 (Bézout’s Theorem).

Let $V_{1},\ldots,V_{k}$ be subvarieties of $\mathbb{P}^{\ell}$. Then
$\deg\bigcap_{i=1}^{k}V_{i}\leq\prod_{i=1}^{k}\deg V_{i}.$ In particular, if
the intersection is finite, then
$\left|\bigcap_{i=1}^{k}V_{i}\right|\leq\prod_{i=1}^{k}\deg V_{i}.$

We now record estimates due to Lang and Weil [19] on the number of points on
varieties over finite fields. The following is a well known consequence of
Theorem 1 of [19] (see, for example, Theorem 5.1 of [27]), but we give the
short proof for completeness.

######  Lemma 5.2.

Let $k$, $\ell$, $m$, and $r$ be positive integers, and let $q$ be a prime
power. Let $V$ be a (reduced) closed subvariety of $\mathbb{P}^{\ell}$,
defined over $\mathbb{F}_{q}$ (the field with $q$ elements), of degree $k$ and
dimension $r$. Let $m\geq 1$ be the number of geometrically irreducible
components of $V$ which are defined over $\mathbb{F}_{q}$. Then

$|V(\mathbb{F}_{q})|,\
|V^{\operatorname{ns}}(\mathbb{F}_{q})|=mq^{r}+O_{k,\ell,r}(q^{r-1/2}).$ (31)

Moreover, the same is true if we replace $V$ with a closed subvariety
$W\subseteq\mathbb{A}^{\ell}$.

###### Proof.

The proof is by induction on $r$, noting that the case $r=0$ is elementary,
and amounts to considering the following observations.

  1. 1.

If $P\in Z(\mathbb{F}_{q})$ for a component $Z\subset V$ not defined over
$\mathbb{F}_{q}$, then $P=P^{\sigma}\in Z^{\sigma}\neq Z$ for nontrivial
$\sigma\in\operatorname{Gal}(\overline{\mathbb{F}_{q}}/\mathbb{F}_{q})$, hence
$P\in Z\cap Z^{\sigma}$, which has dimension strictly less than $r$. Thus, the
number of points on components not defined over $\mathbb{F}_{q}$ is absorbed
into the error term.

  2. 2.

Each component of $V$ defined over $\mathbb{F}_{q}$ has
$q^{r}+O_{k,\ell,r}(q^{r-1/2})$ by Theorem 1 of [19]. Summing the number of
points on each component is an overcount, but the surplus is due to points on
pairwise intersections of components, which again is absorbed into the error
term. (Note that $m\leq k$, so even after multiplying the error by $m$, the
implied constant still depends only on $k$, $\ell$, and $r$.) Thus
$|V(\mathbb{F}_{q})|$ has the claimed magnitude.

  3. 3.

We have $V^{\operatorname{ns}}:=V\setminus V^{\mathrm{s}}$; since
$V^{\mathrm{s}}$ has dimension at most $r-1$ and degree controlled by $k$,
$r$, and $\ell$ (by Bézout’s Theorem), the size of
$V^{\mathrm{s}}(\mathbb{F}_{q})$ is included in the error term. Thus,
$|V^{\operatorname{ns}}(\mathbb{F}_{q})|$ also has the desired magnitude.

  4. 4.

Finally, if we let $V$ be the projective closure of $W$, then
$W=V\setminus(V\cap H)$, where $H$ is the hyperplane at infinity. Since $V\cap
H$ has lower dimension and degree $k$, we are once again removing a set whose
cardinality is subsumed by the error term, so $W(\mathbb{F}_{q})$ (and,
similarly, $W^{\operatorname{ns}}(\mathbb{F}_{q})$) has the appropriate
cardinality. ∎

###  5.2 A key equivalence

The following equivalence observation yields a strengthening of Proposition
2.9 as a corollary, and is also instrumental in subsequent proofs.

######  Lemma 5.3.

Let $V$ be a variety (reduced, but not necessarily irreducible) of dimension
$d\geq 1$ defined over $\mathbb{Z}$. For a sufficiently large (with respect to
$V$) prime $p$, the following are equivalent:

  1. (a)

$V^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$.

  2. (b)

$V^{\operatorname{ns}}(\mathbb{Z}_{p})\neq\emptyset$.

  3. (c)

At least one of the geometric components of $V$ is defined over
$\mathbb{Z}_{p}$.

###### Proof.

((a) $\Longrightarrow$ (b)) Suppose
$V^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$, and let $Q\in
V^{\operatorname{ns}}(\mathbb{F}_{p})$. By Hensel’s lemma, there exists $P\in
V(\mathbb{Z}_{p})$ such that $\widetilde{P}=Q$. Since $\widetilde{P}$ is
nonsingular, so must be $P$.

((b) $\Longrightarrow$ (c)) Let $P\in V^{\operatorname{ns}}(\mathbb{Z}_{p})$,
and let $Z$ be a geometric component of $V$ containing $P$. As in part 1 of
the proof of Lemma 5.2, if $Z$ were not defined over $\mathbb{F}_{q}$, then
$P$ would lie in the intersection of two components, hence would be a singular
point on $V$, contradicting our assumption on $P$.

((c) $\Longrightarrow$ (a)) Let $Z_{1},\ldots,Z_{m}$ be the irreducible
components of $V$. By Lemma 5.2, for each $1\leq i\leq m$ there exists a bound
$B_{i}$ such that for all $p\geq B_{i}$ with $Z_{i}$ defined over
$\mathbb{Z}_{p}$, $Z_{i}^{\operatorname{ns}}(\mathbb{F}_{p})$ contains a point
that does not lie on $Z_{j}$ for any $j\neq i$. Letting
$B=\max\\{B_{1},\ldots,B_{m}\\}$, we have that for $p\geq B$, the existence of
$Z_{i}$ defined over $\mathbb{Z}_{p}$ implies the existence of $Q\in
Z_{i}^{\operatorname{ns}}(\mathbb{F}_{p})\setminus\bigcup_{j\neq
i}Z_{j}(\mathbb{F}_{p})$. Since $Q$ is nonsingular on $Z_{i}$ and is not a
point of intersection with any other component $Z_{j}$, we have $Q\in
V^{\operatorname{ns}}(\mathbb{F}_{p})$. ∎

As previously noted, if $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is Deligne,
then $h=0$ defines a reduced variety. Further, a nonsingular point over
$\mathbb{Z}_{p}$ on this variety corresponds precisely to a root
$\bm{z}_{p}\in\mathbb{Z}_{p}^{\ell}$ of $h$ satisfying $m_{p}=1$, hence Lemma
5.3 establishes the following sufficient condition for smooth intersectivity.
Here we let $\overline{\mathbb{Z}}$ denote the ring of algebraic integers.

######  Corollary 5.4.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ is Deligne
and intersective, and let $h=g_{1}\cdots g_{n}$ be an irreducible
factorization of $h$ in $\overline{\mathbb{Z}}[x_{1},\dots,x_{\ell}]$. If, for
all but finitely many $p\in\mathcal{P}$, $g_{i}$ has coefficients in
$\mathbb{Z}_{p}$ for some $1\leq i\leq n$, then $h$ is smoothly intersective,
hence strongly Deligne.

Note that Proposition 2.9 is an immediate consequence of Corollary 5.4, since
the hypotheses of the proposition imply that one of the factors over
$\overline{\mathbb{Z}}$ is defined over $\mathbb{Z}$, hence over
$\mathbb{Z}_{p}$ for all $p$. We now complete this section by using Lemma 5.3
to prove Proposition 2.8.

###### Proof of Proposition 2.8.

Let $\ell\geq 2$, and suppose $h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ is
Deligne and intersective with $\deg(h)=k\geq 2$. Let
$\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$ be a choice of $p$-adic integer roots of
$h$ satisfying $m_{p}\in\\{1,k\\}$ for all but finitely many $p$. Let $X$
denote the finite set of primes such that

  * •

$m_{p}\notin\\{1,k\\}$, or

  * •

$p\mid k$, or

  * •

$h^{k}$ is not smooth modulo $p$, or

  * •

the equivalence in Lemma 5.3 fails.

We note that the first item is assumed to be finite, the second item is
clearly finite, the fourth item is proven to be finite in Lemma 5.3, and the
third item is finite because $h$ is Deligne (see Definition 2.3).

Fix $d\in\mathbb{N}$ and $p\notin X$. If $p\nmid d$ or $m_{p}=k$, then
$p\nmid\frac{d^{k}}{\lambda(d)}$, so $h_{d}^{k}=\frac{d^{k}}{\lambda(d)}h^{k}$
is a nonzero scalar multiple of $h^{k}$, hence remains smooth modulo $p$.
Therefore, $h_{d}$ is Deligne modulo $p$.

The remaining case is $p\mid d$ and $m_{p}=1$. In this case, since $h^{i}_{d}$
has a factor of $\frac{d^{i}}{\lambda(d)}$, the definition of $\lambda$
assures that the polynomial $h^{i}_{d}$ identically vanishes modulo $p$ for
all $i>1$. Since nonzero homogeneous linear polynomials are automatically
smooth, we need only argue that

$h_{d}^{1}(\bm{x})=\frac{d}{\lambda(d)}\sum_{i=1}^{\ell}\frac{\partial
h}{\partial x_{i}}(\bm{r}_{d})x^{i}$

does not identically vanish modulo $p$. We know that
$p\nmid\frac{d}{\lambda(d)}$ by definition of $\lambda$. Further, the fact
that $h$ is Deligne ensures that $h=0$ defines a reduced variety, so by Lemma
5.3, we can choose $\bm{z}_{p}$ to reduce to a nonsingular point over
$\mathbb{F}_{p}$. Since $\bm{r}_{d}\equiv\bm{z}_{p}\ (\text{mod }p)$, we have
that $\frac{\partial h}{\partial x_{i}}(\bm{r}_{d})\equiv\frac{\partial
h}{\partial x_{i}}(\bm{z}_{p})\not\equiv 0\ (\text{mod }p)$ for some $1\leq
i\leq\ell$, as required. Therefore, $h_{d}$ is Deligne modulo $p$ for all
$p\notin X$, hence $h$ is strongly Deligne. ∎

##  6 Dimension lowering argument

In this section, we generalize the phenomenon exemplified at the beginning of
Section 2.3, establishing Theorems 2.12 and 2.13 by reducing to the case
covered in Theorem 2.4. In the integer root setting, this reduction is very
direct, as Theorem 2.12 follows immediately from Theorem 2.4 and the following
proposition.

######  Proposition 6.1.

Suppose $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ with
$h(\bm{0})=0$. Let $r$ be the minimum rank of the highest and lowest degree
homogeneous parts of $h$. If $r\geq 2$, then there exists a strongly Deligne
polynomial $g\in\mathbb{Z}[x_{1},\ldots,x_{r}]$ such that
$g(\mathbb{Z}^{r})\subseteq h(\mathbb{Z}^{\ell})$.

Before delving into the proof of this proposition, we state a version of
Bertini’s theorem that will allow us to eliminate the singularity in the top-
degree parts of our polynomials, one dimension at a time. Throughout this
section we let $(\mathbb{P}^{n})^{*}$ denote the dual space of
$\mathbb{P}^{n}$, that is, the space of hyperplanes in $\mathbb{P}^{n}$. Note
that $(\mathbb{P}^{n})^{*}$ is isomorphic to $\mathbb{P}^{n}$, with the
hyperplane $\\{a_{0}x_{0}+\cdots+a_{n}x_{n}=0\\}\in(\mathbb{P}^{n})^{*}$
corresponding to the point $(a_{0}:\cdots:a_{n})\in\mathbb{P}^{n}$. A linear
system of hyperplanes in $\mathbb{P}^{n}$ is a linear subspace of
$(\mathbb{P}^{n})^{*}$.

######  Theorem 6.2 (Bertini’s Theorem).

Let $V$ be a (quasi-projective) subvariety of $\mathbb{P}^{n}$ with
irreducible components $V_{1},\ldots,V_{m}$ of equal dimension $d\geq 1$, and
let ${\mathcal{L}}\subseteq(\mathbb{P}^{n})^{*}$ be a linear system. After a
change of coordinates if necessary, we may assume that there exists
$k\in\\{0,\ldots,n\\}$ such that ${\mathcal{L}}$ is the space of all
hyperplanes of the form $\\{a_{k}x_{k}+\cdots+a_{n}x_{n}=0\\}$. Assume that
the coordinates $x_{k},\ldots,x_{n}$ do not simultaneously vanish at any point
on $V$ (i.e., the linear system ${\mathcal{L}}$ has no base-points in $V$), so
that

$\displaystyle\Phi_{\mathcal{L}}:V$
$\displaystyle\longrightarrow\mathbb{P}^{n-k}$
$\displaystyle(z_{0}:\cdots:z_{n})$
$\displaystyle\longmapsto(z_{k}:\cdots:z_{n})$

defines a morphism. Then there exists a nonempty open subset
$U\subseteq{\mathcal{L}}$ such that for all hyperplanes $H\in U$,

  1. (a)

$V^{\operatorname{ns}}\cap H$ is nonsingular, and

  2. (b)

$\dim\left(V^{\mathrm{s}}\cap H\right)<\dim V^{\mathrm{s}}$ (if
$V^{\mathrm{s}}\neq\emptyset$).

Moreover, if $\dim\Phi_{\mathcal{L}}(V)\geq 2$, then $U$ may be chosen so that
for all $H\in U$ we have

  1. (c)

for all $1\leq i\leq m$, the intersection $V_{i}\cap H$ is either empty or
geometrically irreducible.

######  Remark.

Theorem 6.2 is stated somewhat more generally than we need, so we specify the
two situations for which we will actually need the result:

  1. 1.

Let $V$ be a closed hypersurface in $\mathbb{P}^{n}$ and let
${\mathcal{L}}=(\mathbb{P}^{n})^{*}$. Then $\Phi_{\mathcal{L}}$ is just the
inclusion map of $V$ into $\mathbb{P}^{n}$, and the hypotheses of Theorem 6.2
are satisfied. Moreover, since each component $V_{i}$ is a closed subvariety
of $\mathbb{P}^{n}$ of positive dimension, each intersection $V_{i}\cap H$ is
nonempty; thus, if $d=\dim V=\dim\Phi_{\mathcal{L}}(V)\geq 2$, then $V_{i}\cap
H$ is irreducible for all $1\leq i\leq m$ and all $H\in U$.

  2. 2.

Identify $\mathbb{A}^{n}$ with the Zariski open subset $\\{x_{0}\neq
0\\}\subset\mathbb{P}^{n}$. Let $V$ be a closed hypersurface in
$\mathbb{A}^{n}$ not containing the origin $\bm{0}=(0,\ldots,0)$, and let
${\mathcal{L}}$ be the space of all hypersurfaces of the form
$\\{a_{1}x_{1}+\cdots+a_{n}x_{n}=0\\}$. Then the conditions of Theorem 6.2 are
satisfied once again. A fiber of $\Phi_{\mathcal{L}}$ is precisely the
intersection of $V$ with a line in $\mathbb{A}^{n}$ passing through $\bm{0}$.
Since $V$ is closed in $\mathbb{A}^{n}$ and does not contain $\bm{0}$, $V$
cannot contain a line through $\bm{0}$, hence each such intersection is
finite. In particular, this means the map $\Phi_{\mathcal{L}}$ has finite
fibers, so $\dim\Phi_{\mathcal{L}}(V)=\dim V=d$. Moreover, the failure of a
hyperplane $H\in{\mathcal{L}}$ to intersect every $V_{i}$ is a proper Zariski
closed condition. Therefore, removing such hyperplanes from $U$ if necessary,
we again have that $V_{i}\cap H$ is nonempty for all $1\leq i\leq n$ and $H\in
U$, thus $V_{i}\cap H$ is irreducible for all $1\leq i\leq n$ and $H\in U$ as
long as $d\geq 2$.

###### Proof of Theorem 6.2.

Consider the set ${\mathcal{X}}$ of hyperplanes $H\in{\mathcal{L}}$ satisfying
the following conditions:

  1. (a′)

$V_{i}^{\operatorname{ns}}\cap H$ is nonsingular for all $1\leq i\leq m$;

  2. (b′)

$H$ does not contain any components of $V_{i}^{\mathrm{s}}$ nor $(V_{i}\cap
V_{j})$ for $1\leq i,j\leq m$ with $i\neq j$; and

  3. (c′)

for all $1\leq i\leq m$, the intersection $V_{i}\cap H$ is either empty or
geometrically irreducible (if $d\geq 2)$.

We begin by showing that if $H\in{\mathcal{X}}$, then $H$ satisfies properties
(a), (b), and (c). Indeed, condition (c′) is exactly condition (c), so we need
only show that $H$ also satisfies (a) and (b).

For (a), note that a point $P\in V$ is nonsingular if and only if $P$ is a
nonsingular point on $V_{i}$ for some $1\leq i\leq m$ and $P\notin V_{j}$ for
all $j\neq i$. Thus $V^{\operatorname{ns}}$ is a disjoint union
$V^{\operatorname{ns}}=\bigsqcup_{i=1}^{m}W_{i}$, where each $W_{i}$ is a
subset of $V_{i}^{\operatorname{ns}}$. Then (a) follows from (a′) since
$V^{\operatorname{ns}}\cap H=\bigsqcup_{i=1}^{m}(W_{i}\cap H)$ and each
$W_{i}\cap H\subseteq V_{i}^{\operatorname{ns}}\cap H$ is nonsingular.
Finally, (b′) implies that $H$ intersects each component of $V^{\mathrm{s}}$
properly (assuming $V^{\mathrm{s}}\neq\emptyset$), so (b) follows.

It remains to show that ${\mathcal{X}}$ contains a Zariski open subset of
${\mathcal{L}}$. By the standard form of Bertini’s Theorem (see Corollaire
6.11 of [17], or Corollary 10.9 and Remark 10.9.1 of [15]), since
${\mathcal{L}}$ has no base-points in $V$, the set of hyperplanes
$H\in{\mathcal{L}}$ satisfying (a′) and (c′) contains a nonempty open subset
of ${\mathcal{L}}$. Moreover, $H$ containing any of a finite collection of
nonempty subvarieties of $\mathbb{P}^{n}$ is a proper closed condition on $H$,
so condition (b′) is a nonempty open condition; therefore, ${\mathcal{X}}$
contains a nonempty open subset of ${\mathcal{L}}$. ∎

Armed with Theorem 6.2, the proof of Proposition 6.1 is pleasingly
straightforward.

###### Proof of Proposition 6.1.

Suppose that $\ell\geq 2$ and $h\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$ with
$h(\bm{0})=0$. Let $k$ and $j$ denote the highest and lowest degrees,
respectively, of the terms appearing in $h$, and let $r$ denote the minimum
rank of $h^{k}$ and $h^{j}$. Let
${\widehat{V}}_{k},{\widehat{V}}_{j}\subseteq\mathbb{P}^{\ell-1}$ denote the
varieties defined by $h^{k}=0$ and $h^{j}=0$, respectively. By Theorem 6.2
(see also case 1 of the remark that follows) applied to the linear system
${\mathcal{L}}=(\mathbb{P}^{\ell-1})^{*}$ and the varieties
${\widehat{V}}_{k}$ and ${\widehat{V}}_{j}$, respectively, the set of
hyperplanes $H$ in $\mathbb{P}^{\ell-1}$ satisfying

  * •

$H\cap{\widehat{V}}_{k}^{\operatorname{ns}}$ and
$H\cap{\widehat{V}}_{j}^{\operatorname{ns}}$ are nonsingular, and

  * •

$\dim(H\cap{\widehat{V}}_{k}^{\mathrm{s}})<\dim{\widehat{V}}_{k}^{\mathrm{s}}$,
if $V^{\mathrm{s}}\neq\emptyset$, and
$\dim(H\cap{\widehat{V}}_{j}^{\mathrm{s}})<\dim{\widehat{V}}_{j}^{\mathrm{s}}$,
if ${\widehat{V}}_{j}^{\mathrm{s}}\neq\emptyset$,

contains a nonempty open subset $U\subseteq(\mathbb{P}^{\ell-1})^{*}$. Thus,
we can choose $H\in U$ defined by the vanishing of
$l(x_{1},\ldots,x_{\ell})=a_{1}x_{1}+\cdots+a_{\ell-1}x_{\ell-1}-x_{\ell}$
with $a_{1},\ldots,a_{\ell-1}\in\mathbb{Z}$. Here, we’re using the fact that
the set of integer points is Zariski dense in the affine space
$\mathbb{A}^{\ell-1}\subset\mathbb{P}^{\ell-1}\cong(\mathbb{P}^{\ell-1})^{*}$.

Let $\mu(x_{1},\ldots,x_{\ell-1}):=a_{1}x_{1}+\cdots+a_{n-1}x_{\ell-1}$, and
set

$g_{1}(x_{1},\ldots,x_{\ell-1}):=h(x_{1},\ldots,x_{\ell-1},\mu(x_{1},\ldots,x_{\ell-1})).$

Note that, by construction, $g_{1}(\mathbb{Z}^{\ell-1})\subseteq
h(\mathbb{Z}^{\ell})$, $g_{1}(\bm{0})=0$, and the highest and lowest degrees
of the nonzero terms of $g_{1}$ are still $k$ and $j$, respectively.

Now, the subvariety ${\widehat{W}}_{k}$ (resp., ${\widehat{W}}_{j}$) of
$\mathbb{P}^{\ell-2}$ defined by $g_{1}^{k}=0$ (resp., $g_{1}^{j}=0$) is
isomorphic to $H\cap{\widehat{V}}_{k}$ (resp., $H\cap{\widehat{V}}_{j}$). In
particular, the minimum rank of $g_{1}^{k}$ and $g_{1}^{j}$ can only drop
below $r$ if both singular loci were originally empty, which would imply
$r=\ell$. Thus, repeating this process $(\ell-r)$ times yields a sequence of
polynomials $\left(g_{i}(x_{1},\ldots,x_{\ell-i})\right)_{i=0}^{\ell-r}$, with
$g_{0}:=h$, satisfying

  * •

$g_{i}(\mathbb{Z}^{\ell-i})\subseteq g_{i-1}(\mathbb{Z}^{\ell-i+1})$ for all
$1\leq i\leq\ell-r$,

  * •

$g_{i}(\bm{0})=0$ for all $0\leq i\leq\ell-r$,

  * •

the highest and lowest degrees of the nonzero terms of each $g_{i}$ are $k$
and $j$, respectively, and

  * •

the rank of each $g_{i}^{k}$ (resp., $g_{i}^{j}$) is at least $r$.

Finally, let $g:=g_{\ell-r}\in\mathbb{Z}[x_{1},\dots,x_{r}]$, so the rank for
each of $g^{k}$ and $g^{j}$ is $r$. In other words, $g^{k}$ and $g^{j}$ are
smooth, and thus by Proposition 2.5, $g$ is strongly Deligne. ∎

######  Remark.

The conclusion of Proposition 6.1 technically holds for $r=1$ as well, since
nonconstant univariate polynomials are necessarily Deligne; however, this case
is not useful for our purposes.

###  6.1 Proof of Theorem 2.13

We now proceed with the more elaborate of our two dimension-lowering
arguments, in which we cannot exploit the existence of an integer root.
Throughout this section, we fix $\ell\geq 2$ and a polynomial
$h\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ satisfying all hypotheses of Theorem
2.13, and we recall that $k=\deg(h)$ and $r$ denotes the rank of $h^{k}$. Note
that the hypotheses of Theorem 2.13 imply that $r\geq 2$. We assume without
loss of generality that $h(\bm{0})\neq 0$, which is permissible because
$h(\mathbb{Z}^{\ell})$ is invariant under input translation. We let
$V\subseteq\mathbb{A}^{\ell}$ and ${\widehat{V}}\subseteq\mathbb{P}^{\ell-1}$
denote the varieties defined by $h=0$ and $h^{k}=0$, respectively. The
following crucial lemma says that we can eliminate the singularity in the top-
degree part of $h$, one dimension at a time, while maintaining the existence
of nonsingular $\mathbb{F}_{p}$-points.

######  Lemma 6.3.

Suppose $\ell\geq 3$. Then there exists a homogeneous linear polynomial
$l\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$, monic in $x_{\ell}$, for which the
following holds: Letting ${\widehat{L}}$ and $L$ denote the hyperplanes in
$\mathbb{P}^{\ell-1}$ and $\mathbb{A}^{\ell}$, respectively, defined by $l=0$,
we have

  1. (i)

$\dim({\widehat{V}}\cap{\widehat{L}})^{\mathrm{s}}<\dim{\widehat{V}}^{\mathrm{s}}$,
if ${\widehat{V}}^{\mathrm{s}}\neq\emptyset$; and

  2. (ii)

For sufficiently large $p$,
$V^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$ implies $(V\cap
L)^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$.

###### Proof.

Let $\widehat{{\mathcal{L}}}$ and ${\mathcal{L}}$ denote the linear systems of
hyperplanes in $\mathbb{P}^{\ell-1}$ and hyperplanes in $\mathbb{A}^{\ell}$
passing through $\bm{0}$, respectively. We identify each of
$\widehat{{\mathcal{L}}}$ and ${\mathcal{L}}$ with $\mathbb{P}^{\ell-1}$, with
the point ${\bm{a}}=(a_{1}:\cdots:a_{\ell})\in\mathbb{P}^{\ell-1}$
corresponding to the hyperplanes $\\{a_{1}x_{1}+\cdots+a_{\ell}x_{\ell}=0\\}$
in $\mathbb{P}^{\ell-1}$ and $\mathbb{A}^{\ell}$, respectively.

The hypotheses of Theorem 6.2 are satisfied by ${\widehat{V}}$ and
$\widehat{{\mathcal{L}}}$ (resp., $V$ and ${\mathcal{L}}$), as explained in
case 1 (resp., case 2) of the remark immediately following the theorem. Thus,
there is a nonempty open set $U\subseteq\mathbb{P}^{\ell-1}$ such that for all
${\bm{a}}=(a_{1}:\cdots:a_{\ell})\in U$, the hyperplanes
${\widehat{L}}_{\bm{a}}\subset\mathbb{P}^{\ell-1}$ and
$L_{\bm{a}}\subset\mathbb{A}^{\ell}$ defined by
$l_{\bm{a}}:=a_{1}x_{1}+\cdots+a_{\ell}x_{\ell}=0$ satisfy the conclusion of
Theorem 6.2 (intersected with ${\widehat{V}}\subset\mathbb{P}^{\ell-1}$ and
$V\subset\mathbb{A}^{\ell}$, respectively). Similar to the proof of
Proposition 6.1, we may choose ${\bm{a}}\in U$ of the form
${\bm{a}}=(a_{1}:\cdots:a_{\ell-1}:1)$ with
$a_{1},\ldots,a_{\ell-1}\in\mathbb{Z}$. Set $l:=l_{\bm{a}}$ for such a choice
of ${\bm{a}}\in U$, hence also ${\widehat{L}}={\widehat{L}}_{\bm{a}}$ and
$L=L_{\bm{a}}$. By construction, we immediately have that
$l\in\mathbb{Z}[x_{1},\ldots,x_{\ell}]$, $l$ is monic in $x_{\ell}$, and
property (i) holds, so it remains only to show that (ii) holds.

Let $V_{1},\ldots,V_{m}$ be the geometrically irreducible components of $V$.
Since $\dim V=\ell-1\geq 2$, our choice of ${\bm{a}}$ guarantees that the
geometrically irreducible components of $V\cap L$ are $V_{i}\cap L$ with
$1\leq i\leq m$. (We are again using Theorem 6.2 and case 2 of the remark that
follows.) By Lemma 5.3, if $p$ is sufficiently large, then
$V^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$ implies that $V_{i}$ is
defined over $\mathbb{Z}_{p}$ for some $1\leq i\leq m$. Since $L$ is defined
over $\mathbb{Z}$, hence over $\mathbb{Z}_{p}$, the intersection $V_{i}\cap L$
is also defined over $\mathbb{Z}_{p}$. Appealing to Lemma 5.3 once more
implies that $(V\cap L)^{\operatorname{ns}}(\mathbb{F}_{p})$ is nonempty. ∎

The hyperplane produced by Lemma 6.3 quickly yields a suitable polynomial with
one fewer variable.

######  Corollary 6.4.

Suppose $\ell\geq 3$. Then there exists
$g^{\prime}\in\mathbb{Z}[x_{1},\ldots,x_{\ell-1}]$ with $\deg(g^{\prime})=k$
such that

  1. (i)

$g^{\prime}(\bm{0})\neq 0$;

  2. (ii)

$g^{\prime}(\mathbb{Z}^{\ell-1})\subseteq h(\mathbb{Z}^{\ell})$;

  3. (iii)

$\dim({\widehat{W}}^{\prime})^{\mathrm{s}}<\dim{\widehat{V}}^{\mathrm{s}}$, if
${\widehat{V}}^{\mathrm{s}}\neq\emptyset$; and

  4. (iv)

for sufficiently large $p$,
$V^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$ implies
$(W^{\prime})^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$;

where ${\widehat{W}}^{\prime}\subset\mathbb{P}^{\ell-2}$ and
$W^{\prime}\subset\mathbb{A}^{\ell-1}$ are the varieties defined by
$(g^{\prime})^{k}=0$ and $g^{\prime}=0$, respectively.

###### Proof.

Let $L=L_{\bm{a}}$ be as in Lemma 6.3, and write
$l=l_{\bm{a}}=a_{1}x_{1}+\cdots+a_{\ell-1}x_{\ell-1}+x_{\ell}$. To ease
notation, we also set
$\mu=\mu_{\bm{a}}:=-(a_{1}x_{1}+\cdots+a_{\ell-1}x_{\ell-1})$. Now, define

$g^{\prime}(x_{1},\ldots,x_{\ell-1}):=h(x_{1},\ldots,x_{\ell-1},\mu(x_{1},\ldots,x_{\ell-1})).$

Clearly $g^{\prime}(\mathbb{Z}^{\ell-1})\subseteq h(\mathbb{Z}^{\ell})$ and,
since $\mu$ is homogeneous, $g^{\prime}(\bm{0})=h(\bm{0})\neq 0$. Finally,
since $V\cap L\cong W^{\prime}$ and
${\widehat{V}}\cap{\widehat{L}}\cong{\widehat{W}}^{\prime}$, properties (iii)
and (iv) follow immediately from Lemma 6.3. ∎

Recall our assumption that the rank satisfies $r\geq 2$. Repeated application
of Corollary 6.4 yields the following:

######  Corollary 6.5.

There exists $g\in\mathbb{Z}[x_{1},\ldots,x_{r}]$ with $\deg(g)=k$ such that

  1. (i)

$g(\mathbb{Z}^{r})\subseteq h(\mathbb{Z}^{\ell})$;

  2. (ii)

$g$ is Deligne; and

  3. (iii)

for sufficiently large $p$,
$V^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$ implies
$W^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$;

where $W\subseteq\mathbb{A}^{r}$ is the variety defined by $g=0$.

###### Proof.

When $\ell\geq 3$, this follows immediately by applying Corollary 6.4
recursively $(\ell-r)$ times. The fact that $r\geq 2$ ensures that at each
step we are applying Corollary 6.4 to a polynomial in at least $3$ variables.

When $\ell=2$, the statement is trivial, since $r=2$ implies that $h$ is
already Deligne, so we can take $g=h$. ∎

######  Remark.

Using the construction from the proof of Corollary 6.4, the polynomial $g$ of
Corollary 6.5 may be written in the form

$g(x_{1},\ldots,x_{r})=h(x_{1},\ldots,x_{r},\mu_{r+1}(x_{1},\ldots,x_{r}),\ldots,\mu_{\ell}(x_{1},\ldots,x_{r})),$

where each $\mu_{j}$ is a homogeneous linear polynomial. We will use this
precise form in our proof of Theorem 2.13, which we are now ready to begin.

###### Proof of Theorem 2.13.

Let $g\in\mathbb{Z}[x_{1},\ldots,x_{r}]$ be as in Corollary 6.5, and let
$W\subseteq\mathbb{A}^{r}$ be the variety defined by $g=0$. Throughout this
proof we use the notation $\tilde{\bm{x}}=(x_{1},\dots,x_{r})$ and
$\bm{x}=(x_{1},\dots,x_{\ell})$.

As mentioned in the remark above, $g$ may be given by
$g(\tilde{\bm{x}})=h(M\tilde{\bm{x}})$, where

$M(x_{1},\dots,x_{r})=(x_{1},\dots,x_{r},\mu_{r+1}(x_{1},\dots,x_{r}),\dots,\mu_{\ell}(x_{1},\dots,x_{r}))$

for linear forms $\mu_{r+1},\dots,\mu_{\ell}$. Note that $g$ and the linear
forms have been constructed once and for all from $h$, so any quantities
depending on them implicitly depend only on $h$.

Let $X=X(h)$ be the set of primes $p$ for which

  * •

$p\mid k$;

  * •

$g^{k}$ is not smooth modulo $p$; or

  * •

$W^{\operatorname{ns}}(\mathbb{F}_{p})=\emptyset$ and $m_{p}\neq k$ for all
$\bm{z}_{p}\in V(\mathbb{Z}_{p})$.

The first item clearly defines a finite set, the second item defines a finite
set because $g$ is Deligne (see Definition 2.3). If $r\geq 3$, then the third
item defines a finite set by Lemma 5.3 and the fact that Deligne polynomials
in $r\geq 3$ variables are geometrically irreducible, as seen in the proof of
Corollary 2.10. If $r=2$, then item (iii) of Corollary 6.5, Lemma 5.3, and the
hypotheses of Theorem 2.13 ensure that the third item defines a finite set.
Thus, $X$ is finite.

In order to construct auxiliary polynomials $h_{d}$ for $d\in N$, we first
choose $\mathbb{Z}_{p}$-roots of $h$ as follows: If $p\in X$, then choose a
point $\bm{z}_{p}\in V(\mathbb{Z}_{p})$ arbitrarily; such points exist because
$h$ is intersective. For $p\notin X$ with
$W^{\operatorname{ns}}(\mathbb{F}_{p})\neq\emptyset$, choose
$\tilde{\bm{z}}_{p}\in W(\mathbb{Z}_{p})$ to be a Hensel lift of a nonsingular
point on $W(\mathbb{F}_{p})$, then set $\bm{z}_{p}=M\tilde{\bm{z}}_{p}\in
V(\mathbb{Z}_{p}).$ Finally, for all remaining $p\notin X$, fix $\bm{z}_{p}\in
V(\mathbb{Z}_{p})$ with $m_{p}=k$.

For each prime $p$, by definition of multiplicity, we have a decomposition of
the form

$h(\bm{x}+\bm{z}_{p})=\sum_{m_{p}\leq|\bm{i}|\leq k}b_{\bm{i}}\bm{x}^{\bm{i}}$
(32)

for $b_{\bm{i}}\in\mathbb{Z}_{p}$. However, the substitution
$\bm{x}=M\tilde{\bm{x}}$ could cause some homogeneous parts to identically
vanish, so we define $\tilde{m}_{p}$ to be the multiplicity of $\bm{0}$ as a
root of $h(M\tilde{\bm{x}}+\bm{z}_{p})$, so in particular

$h(M\tilde{\bm{x}}+\bm{z}_{p})=\sum_{m_{p}\leq|\bm{i}|\leq
k}b_{\bm{i}}(M\tilde{\bm{x}})^{\bm{i}}=\sum_{\tilde{m}_{p}\leq|\bm{i}|\leq
k}a_{\bm{i}}\tilde{\bm{x}}^{\bm{i}},$ (33)

where $a_{\bm{i}}\neq 0$ for some $\bm{i}$ with $|\bm{i}|=\tilde{m}_{p}$. We
quickly note that $\tilde{m}_{p}=m_{p}$ for all $p\notin X$. If $p\notin X$
with $m_{p}=k$, the degree-$k$ part of $h(M\tilde{\bm{x}}+\bm{z}_{p})$ is the
same as the degree-$k$ part of $g$. If $p\notin X$ and
$\bm{z}_{p}=M\tilde{\bm{z}}_{p}$ as above, then
$h(M\tilde{\bm{x}}+\bm{z}_{p})$ is precisely
$g(\tilde{\bm{x}}+\tilde{\bm{z}}_{p})$, and in particular the linear part does
not vanish modulo $p$.

To account for this possible increase in multiplicity for primes $p\in X$, we
define a completely multiplicative function $\tilde{\lambda}(d)$ by setting
$\lambda(p)=p^{\tilde{m}_{p}}$ for all primes $p$. We define
$\\{\bm{r}_{d}\\}_{d\in\mathbb{N}}$ from $\\{\bm{z}_{p}\\}_{p\in\mathcal{P}}$
as usual from the Chinese remainder theorem, then define the slightly modified
auxiliary polynomials $\\{\tilde{h}_{d}\\}_{d\in\mathbb{N}}$ by

$\tilde{h}_{d}(\bm{x})=h(\bm{r}_{d}+d\bm{x})/\tilde{\lambda}(d).$

We note that $\tilde{h}_{d}$ can potentially have non-integer coefficients,
with denominators divisible by primes in $X$. However, the analog of
Proposition 3.2, and the deduction of Lemma 4.1 from Lemma 4.2 and Proposition
3.2, still hold because $d\mid\tilde{\lambda}(d)$ and $\tilde{\lambda}$ is
completely multiplicative.

We now let $d^{\prime}=\prod_{p\mid
d}p^{(\tilde{m}_{p}-m_{p}+1)\text{ord}_{p}(d)}\leq d^{k},$ and we define

$g_{d}(\tilde{\bm{x}})=\tilde{h}_{d}(\bm{s}_{d}+M\tilde{\bm{x}})=h(\bm{r}_{d^{\prime}}+dM\tilde{\bm{x}})/\tilde{\lambda}(d),$

where $\bm{s}_{d}$ satisfies $\bm{r}_{d^{\prime}}=\bm{r}_{d}+d\bm{s}_{d}$. We
will establish the following properties of $g_{d}$:

  1. (i)

$g_{d}(\mathbb{Z}^{r})\subseteq\tilde{h}_{d}(\mathbb{Z}^{\ell})$,

  2. (ii)

$g_{d}$ has integer coefficients,

  3. (iii)

$g_{d}$ is Deligne modulo $p$ for all $p\notin X$,

  4. (iv)

The coefficients of $g_{d}$ are of size $O_{h}(d^{k^{2}})$,

  5. (v)

$\text{cont}(g_{d})\ll_{h}1$.

Unlike Proposition 6.1, these efforts cannot be applied “externally” to
immediately yield Theorem 2.13 because the family
$\\{g_{d}\\}_{d\in\mathbb{N}}$ is not necessarily the set of auxiliary
polynomials of a single intersective polynomial. However, the enumerated
properties of this family make it perfectly suited for us to apply our efforts
“internally”, using the estimates enumerated in Theorem 3.9, as follows:

  * (1)

Replace all occurrences of $h_{d}$ in the proof of Theorem 2.4 with
$\tilde{h}_{d}$. The fact that $\tilde{h}_{d}$ potentially has non-integer
coefficients is not a problem, as the analog of Proposition 3.2 still holds,
and as explained in the next step.

  * (2)

When proving Lemma 4.2 (the only piece of the proof of Theorem 2.4 that
requires integer coefficients or a nonsingularity condition), use that
$(A-A)\cap
g_{d}(\mathbb{Z}^{r})\subseteq(A-A)\cap\tilde{h}_{d}(\mathbb{Z}^{\ell})\subseteq\\{0\\},$
then do the remainder of the proof with $h_{d}$ replaced by $g_{d}$. For this
purpose, properties (ii)-(v) above assure that $g_{d}$ functions as if it were
the auxiliary polynomial of a strongly Deligne polynomial in $r$ variables. In
particular, the conclusion of Lemma 4.2 holds with $\theta(k,\ell,\delta)$
replaced by $\theta(k,r,\delta)$.

  * (3)

The remainder of the argument is identical, and Theorem 2.13 follows.

Our task is now reduced to verifying properties (i)-(v). Properties (i) and
(iv) are immediate from the definition of $g_{d}$ and $\tilde{h}_{d}$. We next
simultaneously establish (ii) and the property

$\text{ord}_{p}(\text{cont}(g_{d}))\ll_{h,p}1\text{ for all }p\in\mathcal{P}.$
(34)

When we later establish (iii), it immediately combines with (34) to yield (v),
because $p\nmid\text{cont}(g_{d})$ if $g_{d}$ is Deligne modulo $p$. We fix
$p\in\mathcal{P}$ and set $j=\text{ord}_{p}(d)$. By (32), we have

$g_{d}(\tilde{\bm{x}})=\tilde{h}_{d}(\bm{s}_{d}+M\tilde{\bm{x}})=\frac{1}{\tilde{\lambda}(d)}h(\bm{r}_{d^{\prime}}+dM\tilde{\bm{x}})=\frac{1}{\tilde{\lambda}(d)}\sum_{m_{p}\leq|i|\leq
k}b_{\bm{i}}(dM\tilde{\bm{x}}+\bm{r}_{d^{\prime}}-\bm{z}_{p})^{\bm{i}}.$ (35)

Since $p^{j}\mid d$ and $p^{(\tilde{m}_{p}-m_{p}+1)j}$ divides all coordinates
of $\bm{r}_{d^{\prime}}-\bm{z}_{p}$, all terms in the summation apart from

$\sum_{m_{p}\leq|\bm{i}|\leq\tilde{m}_{p}-1}b_{\bm{i}}(dM\tilde{\bm{x}})^{\bm{i}}$
(36)

have coefficients divisible by $p^{j\tilde{m}_{p}}$, and the polynomial (36)
identically vanishes by definition of $\tilde{m}_{p}$. Since
$\text{ord}_{p}(\tilde{\lambda}(d))=j\tilde{m}_{p}$, all coefficients of
$g_{d}$ have nonnegative $p$-adic valuation. Since $p\in\mathcal{P}$ was
arbitrary, it follows that $g_{d}$ has integer coefficients.

Further, we see in (35) that all degree-$\tilde{m}_{p}$ terms have a factor of
$p^{j}$ apart from those arising from

$\frac{d^{\tilde{m}_{p}}}{\tilde{\lambda}(d)}\sum_{|\bm{i}|=\tilde{m}_{p}}b_{\bm{i}}(M\tilde{\bm{x}})^{\bm{i}}=\frac{d^{\tilde{m}_{p}}}{\tilde{\lambda}(d)}\sum_{|\bm{i}|=\tilde{m}_{p}}a_{\bm{i}}\tilde{\bm{x}}^{\bm{i}},$

where $a_{\bm{i}}\neq 0$ for some $\bm{i}$ with $|\bm{i}|=\tilde{m}_{p}$.

Since $p\nmid(d^{\tilde{m}_{p}}/\tilde{\lambda}(d))$, we have that

$\text{ord}_{p}(\text{cont}(g_{d}))\leq
v:=\min_{|\bm{i}|=\tilde{m}_{p}}\text{ord}_{p}(a_{\bm{i}}),$

provided $j>v$. Alternatively, if $j\leq v$, then
$\text{ord}_{p}(\text{cont}(g_{d}))$ is at most $kv$ plus the minimum $p$-adic
valuation of the degree-$k$ coefficients of $g$, which establishes (34).

Our task is now reduced to verifying property (iii), for which we fix $p\notin
X$, and proceed similarly to the proof of Proposition 2.8. Since $g_{d}^{k}$
is precisely $\frac{d^{k}}{\tilde{\lambda}(d)}g^{k}$, we know that if $p\nmid
d$ or $m_{p}=k$, then $g_{d}^{k}$ modulo $p$ is a nonzero multiple of $g^{k}$,
hence remains smooth. Therefore, $g_{d}$ is Deligne modulo $p$.

The remaining case is when $p\mid d$ and $\bm{z}_{p}=M\tilde{z}_{p}$, where
$\tilde{z}_{p}\in W(\mathbb{Z}_{p})$ is a Hensel lift of a nonsingular point
of $W(\mathbb{F}_{p})$, so in particular the linear part of
$g(\tilde{\bm{x}}+\tilde{z}_{p})=h(M\tilde{\bm{x}}+\bm{z}_{p})$ does not
identically vanish modulo $p$.

Using (32), letting $j=\text{ord}_{p}(d)$, we note that
$\text{ord}_{p}(\tilde{\lambda}(d))=j$ and $p^{j}$ divides all coordinates of
$\bm{r}_{d^{\prime}}-\bm{z}_{p}$, and we have

$g_{d}(\tilde{\bm{x}})=\frac{1}{\tilde{\lambda}(d)}\sum_{1\leq|\bm{i}|\leq
k}b_{\bm{i}}(dM\tilde{\bm{x}}+\bm{r}_{d^{\prime}}-\bm{z}_{p})^{\bm{i}}=p^{j}f(\tilde{\bm{x}})+\frac{d}{\tilde{\lambda}(d)}\sum_{|\bm{i}|=1}b_{\bm{i}}(M\tilde{\bm{x}})^{\bm{i}}+C$

for some $f\in\mathbb{Z}_{p}[x_{1},\dots,x_{r}]$ and constant $C$. In
particular, modulo $p$, the highest-degree part of $g_{d}$ is a nonzero
multiple of the nonvanishing linear part of
$g(\tilde{\bm{x}}+\tilde{\bm{z}}_{p})$, hence $g_{d}$ is Deligne modulo $p$.
All five properties of $g_{d}$ are now verified and the proof of Theorem 2.13
is complete. ∎

##  7 Exponential sum estimates

In this final section, we establish the exponential sum estimates claimed in
Theorem 3.9, which we then use to deduce (27) and (28). This effort consists
primarily of careful multivariate adaptations of the tools used to prove
Theorem 2.7 in [28], but we begin with another foray into varieties over
finite fields.

###  7.1 Control over gradient vanishing: Part II

Since we are sieving away inputs at which the gradient of our polynomial
vanishes, but then appealing to Theorem 1.4, which is a complete exponential
sum estimate, it is important for us to have an upper bound on the number of
points our sieve might be throwing away. With this in mind, we make the
following definition.

######  Definition 7.1.

For a field $F$ and $g\in F[x_{1},\dots,x_{\ell}]$ we define the gradient
locus of $g$ to be the variety

$\mathcal{G}_{g}=\\{\bm{x}\in\mathbb{A}^{\ell}:\nabla
g(\bm{x})=\bm{0}\\}\subseteq\mathbb{A}^{\ell}.$

The following proposition establishes firm control over the gradient locus of
a Deligne polynomial.

######  Proposition 7.2.

Suppose $F$ is a field, $\ell\in\mathbb{N}$, and $g\in
F[x_{1},\ldots,x_{\ell}]$ with $\deg(g)=k\geq 1$. If $g$ is Deligne, then
$\mathcal{G}_{g}=\emptyset$ or $\dim\mathcal{G}_{g}=0$.

###### Proof.

First, assume $g$ is not homogeneous. Let $G(x_{0},x_{1},\ldots,x_{\ell})$ be
the homogenization of $g$. Thus, we have

$g(x_{1},\ldots,x_{\ell})=G(1,x_{1},\ldots,x_{\ell})\quad\text{and}\quad
g^{k}(x_{1},\ldots,x_{\ell})=G(0,x_{1},\ldots,x_{\ell}).$

The variety

${\widehat{W}}:=\\{G=0\\}\cap\\{x_{0}=0\\}\subset\mathbb{P}^{\ell}$

is isomorphic to $\\{g^{k}=0\\}$, hence is nonsingular since $g$ is Deligne.
Thus, the Jacobian matrix

$\begin{pmatrix}\dfrac{\partial G}{\partial x_{0}}&\dfrac{\partial G}{\partial
x_{1}}&\cdots&\dfrac{\partial G}{\partial x_{\ell}}\\\ \\\
1&0&\cdots&0\end{pmatrix}$

has rank $2$ at every point on ${\widehat{W}}$. In other words, the system

$\displaystyle G=x_{0}=\frac{\partial G}{\partial x_{1}}=\cdots=\frac{\partial
G}{\partial x_{\ell}}$ $\displaystyle=0$

has no solutions in $\mathbb{P}^{\ell}$. The equation $G=0$ is actually
superfluous here; by Euler’s theorem on homogeneous functions, we have

$kG(x_{0},x_{1},\ldots,x_{\ell})=x_{0}\dfrac{\partial G}{\partial
x_{0}}+x_{1}\dfrac{\partial G}{\partial x_{1}}+\cdots+x_{\ell}\dfrac{\partial
G}{\partial x_{\ell}},$

so the vanishing of $x_{0}$ and the $x_{1}$\- through $x_{\ell}$-partials
would guarantee the vanishing of $G$. Here we use the fact that the
characteristic of $F$ does not divide $k$, as included in the definition of
the Deligne property. It follows that the system

$\displaystyle x_{0}=\frac{\partial G}{\partial x_{1}}=\cdots=\frac{\partial
G}{\partial x_{\ell}}$ $\displaystyle=0$

has no solutions in $\mathbb{P}^{\ell}$, so the subvariety of
$\mathbb{P}^{\ell}$ defined by

$\frac{\partial G}{\partial x_{1}}=\cdots=\frac{\partial G}{\partial
x_{\ell}}=0$ (37)

is contained in $\\{\bm{x}\in\mathbb{P}^{\ell}\mid x_{0}\neq
0\\}\cong\mathbb{A}^{\ell}$ and has dimension $0$. But, for
$\bm{\alpha}=(\alpha_{1},\ldots,\alpha_{\ell})\in\mathbb{A}^{\ell}$, we have

$\frac{\partial g}{\partial x_{i}}(\bm{\alpha})=\frac{\partial G}{\partial
x_{i}}(1,\bm{\alpha})$

for all $1\leq i\leq\ell$. Thus, $\mathcal{G}_{g}$ is (isomorphic to) the
zero-dimensional subvariety of $\mathbb{P}^{\ell}$ given by (37), concluding
the proof in the case that $g$ is not homogeneous.

Finally, suppose $g$ is homogeneous. Again using Euler’s theorem on
homogeneous functions, we write

$kg(x_{1},\ldots,x_{\ell})=x_{1}\frac{\partial g}{\partial
x_{1}}+\cdots+x_{\ell}\frac{\partial g}{\partial x_{\ell}}.$

Thus, if all partials of $g$ vanish at $\bm{x}$, then $g(\bm{x})=0$ as well.
By hypothesis, $g=g^{k}$ is smooth, so there are no common zeroes of
$g,\frac{\partial g}{\partial x_{1}},\ldots,\frac{\partial g}{\partial
x_{\ell}}$ in $\mathbb{P}^{\ell}$, so in $\mathbb{A}^{\ell}$ the only possible
common zero is the origin. Therefore, $\mathcal{G}_{g}$ contains at most one
point. ∎

Proposition 7.2 combines with Bézout’s Theorem (Lemma 5.1) to yield the
following estimate on the size of the gradient vanishing locus for a Deligne
polynomial over a finite field, which yields Lemma 3.3 as a special case.

######  Corollary 7.3.

If $\ell\geq 1$ and $g\in\mathbb{F}_{q}[x_{1},\dots,x_{\ell}]$ is a Deligne
polynomial of degree $k\geq 1$, then $|\mathcal{G}_{g}|$ is bounded by a
constant depending only $k$ and $\ell$.

###  7.2 Major arc estimates

In this section we establish item (i) of Theorem 3.9. Derivations of
asymptotic formulas of this type typically rely on partial summation, so we
begin with a multivariate version thereof, proven by induction from the usual
formula.

######  Lemma 7.4 (Multivariable Partial Summation).

Suppose $\ell\in\mathbb{N}$ and $a:\mathbb{N}^{\ell}\to\mathbb{C}$. Suppose
further that $\psi:\mathbb{R}^{\ell}\to\mathbb{C}$ is $C^{\ell}$. For any
$X>0$, we have

$\displaystyle\sum_{\bm{n}\in[1,X]^{\ell}}a(\bm{n})\psi(\bm{n})$
$\displaystyle=A(X,\dots,X)\psi(X,\dots,X)$
$\displaystyle\qquad+\sum_{i=1}^{\ell}(-1)^{i}\sum_{1\leq
j_{1}<\cdots<j_{i}\leq\ell}\int_{[0,X]^{i}}A(\star)\frac{\partial^{i}\psi}{\partial
x_{j_{1}}\cdots\partial x_{j_{i}}}(\star)\ dx_{j_{1}}\cdots dx_{j_{i}},$

where

$A(x_{1},\dots,x_{\ell})=\sum_{\bm{n}\in[1,x_{1}]\times\cdots\times[1,x_{\ell}]}a(\bm{n})$

and $\star=(X,\dots,x_{j_{1}},\dots,x_{j_{i}},\dots,X),$ with
$x_{j_{1}},\dots,x_{j_{i}}$ plugged into coordinate positions
$j_{1},\dots,j_{i}$ and all other coordinates evaluated at $X$.

###### Proof.

We induct on $\ell$. The base case $\ell=1$ is the usual partial summation
formula

$\sum_{1\leq n\leq X}a(n)\psi(n)=A(X)\psi(X)-\int_{0}^{X}A(x)\psi^{\prime}(x)\
dx.$

Fix $\ell\geq 2$ and assume the formula holds for $\ell-1$. Defining some
notation before proceeding, let

$\tilde{A}(x_{1},\dots,x_{\ell-1},n_{\ell})=\sum_{\bm{n}\in[1,x_{1}]\times\cdots\times[1,x_{\ell-1}]}a(\bm{n},n_{\ell}),$

let

$\tilde{I}(j_{1},\dots,j_{i},n_{\ell})=\int_{[0,X]^{i}}\tilde{A}(\star,n_{\ell})\frac{\partial^{i}\psi}{\partial
x_{j_{1}}\cdots\partial x_{j_{i}}}(\star,n_{\ell})\ dx_{j_{1}}\cdots
dx_{j_{i}},$

and let

$I(j_{1},\dots,j_{i})=\int_{[0,X]^{i}}A(\star)\frac{\partial^{i}\psi}{\partial
x_{j_{1}}\cdots\partial x_{j_{i}}}(\star)\ dx_{j_{1}}\cdots dx_{j_{i}},$

where $A$ and $\star$ are as defined in the statement of the lemma. By our
inductive hypothesis, we have

$\displaystyle\sum_{\bm{n}\in[1,X]^{\ell}}a(\bm{n})\psi(\bm{n})$
$\displaystyle=\sum_{1\leq n_{\ell}\leq
X}\sum_{\bm{n}\in[1,X]^{\ell-1}}a(\bm{n},n_{\ell})\psi(\bm{n},n_{\ell})$
$\displaystyle=\sum_{1\leq n_{\ell}\leq
X}\Big{(}\tilde{A}(X,\dots,X,n_{\ell})\psi(X,\dots,X,n_{\ell})+\sum_{i=1}^{\ell-1}(-1)^{i}\sum_{1\leq
j_{1}<\cdots<j_{i}\leq\ell-1}\tilde{I}(j_{1},\dots,j_{i},n_{\ell})\Big{)}.$

We now apply the standard single-variable formula to the first term and each
individual integral, yielding

$\sum_{1\leq n_{\ell}\leq
X}\tilde{A}(X,\dots,n_{\ell})\psi(X,\dots,n_{\ell})=A(X,\dots,X)\psi(X,\dots,X)-\int_{0}^{X}A(X,\dots,x_{\ell})\frac{\partial\psi}{\partial
x_{\ell}}(X,\dots,x_{\ell})dx_{\ell},$ (38)

and

$\displaystyle\sum_{1\leq n_{\ell}\leq X}$
$\displaystyle(-1)^{i}\tilde{I}(j_{1},\dots,j_{i},n_{\ell})$
$\displaystyle=(-1)^{i}\int_{[0,X]^{i}}\left(A(\star,X)\frac{\partial^{i}\psi}{\partial
x_{j_{1}}\cdots\partial
x_{j_{i}}}(\star,X)-\int_{0}^{X}A(\star,x_{\ell})\frac{\partial^{i+1}\psi}{\partial
x_{j_{1}}\cdots\partial x_{j_{i}}\partial
x_{\ell}}(\star,x_{\ell})dx_{\ell}\right)\ dx_{j_{1}}\cdots dx_{j_{i}}$
$\displaystyle=(-1)^{i}I(j_{1},\dots,j_{i})+(-1)^{i+1}I(j_{1},\dots,j_{i},\ell).$

Summing this final expression over $1\leq i\leq\ell-1$ and over all choices of
$1\leq j_{1}<\cdots<j_{i}\leq\ell-1$ accounts for all required terms with
$1\leq i\leq\ell$ and $1\leq j_{1}<\cdots<j_{i}\leq\ell$, with the single
exception of $i=1$ and $j_{1}=\ell$, which is precisely the integral present
in (38), and the induction is complete. ∎

We use Lemma 7.4 and the same calculation as in Proposition 3.4 to establish
our asymptotic formula for sieved multivariate exponential sums near rationals
with small denominator.

######  Lemma 7.5.

Suppose $\ell,k\in\mathbb{N}$, $g(\bm{x})=\sum_{|\bm{i}|\leq
k}a_{\bm{i}}\bm{x}^{\bm{i}}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$, and let
$J=\sum_{|\bm{i}|\leq k}|a_{\bm{i}}|$. If $X,Y>0$, $a,q\in\mathbb{N}$, and
$\alpha=a/q+\beta$, then

$\displaystyle\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi ig(\bm{n})\alpha}$
$\displaystyle=q^{-\ell}\prod_{\begin{subarray}{c}p\leq Y\\\
p^{\gamma(p)}\nmid
q\end{subarray}}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap
W^{q}(Y)}e^{2\pi ig(\bm{s})a/q}\int_{[0,X]^{\ell}}e^{2\pi
ig(\bm{x})\beta}d\bm{x}$
$\displaystyle\qquad+O_{k,\ell}\left(qE(1+JX^{k}|\beta|)^{\ell}\right),$

where $E$ is as in Proposition 3.4.

###### Proof.

We begin by noting that for any $a,q\in\mathbb{N}$ and $0\leq
x_{1},\dots,x_{\ell}\leq X$, letting

$B=[1,x_{1}]\times\cdots\times[1,x_{\ell}],$

we have

$\displaystyle T(x_{1},\dots,x_{\ell})$ $\displaystyle:=\sum_{\bm{n}\in B\cap
W(Y)}e^{2\pi ig(\bm{n})a/q}$
$\displaystyle\phantom{:}=\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}}e^{2\pi
ig(\bm{s})a/q}\left|\left\\{\bm{n}\in B\cap W(Y):\bm{n}\equiv\bm{s}\
(\text{mod }q)\right\\}\right|.$

For $s\in W^{q}(Y)$ we have by the same calculation as Proposition 3.4 that

$\left|\left\\{\bm{n}\in B\cap W(Y):\bm{n}\equiv\bm{s}\ (\text{mod
}q)\right\\}\right|=\frac{x_{1}\cdots
x_{\ell}}{q^{\ell}}\prod_{\begin{subarray}{c}p\leq Y\\\ p^{\gamma(p)}\nmid
q\end{subarray}}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)+E/q^{\ell-1},$

where $E$ is as in Proposition 3.4, whereas for $s\notin W^{q}(Y)$ the set is
empty.

Therefore,

$T(x_{1},\dots,x_{\ell})=\frac{x_{1}\cdots
x_{\ell}}{q^{\ell}}\prod_{\begin{subarray}{c}p\leq Y\\\ p^{\gamma(p)}\nmid
q\end{subarray}}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap
W^{q}(Y)}e^{2\pi ig(\bm{s})a/q}+O\left(qE\right).$ (39)

Letting $\psi(\bm{n})=e^{2\pi ig(\bm{n})\beta}$, we now decompose our sum as

$\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi
ig(\bm{n})\alpha}=\sum_{\bm{n}\in[1,X]^{\ell}}\left(1_{W(Y)}(\bm{n})e^{2\pi
ig(\bm{n})a/q}\right)\psi(\bm{n})$

and apply Lemma 7.4, yielding

$\displaystyle\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi ig(\bm{n})\alpha}$
$\displaystyle=T(X,\dots,X)\psi(X,\dots,X)$
$\displaystyle\qquad+\sum_{m=1}^{\ell}(-1)^{m}\sum_{1\leq
j_{1}<\cdots<j_{m}\leq\ell}\int_{[0,X]^{m}}T(\star)\frac{\partial^{m}\psi}{\partial
x_{j_{1}}\cdots\partial x_{j_{m}}}(\star)\ dx_{j_{1}}\cdots dx_{j_{m}},$

where $\star$ is as in Lemma 7.4. Substituting (39) gives the main term

$\displaystyle q^{-\ell}\prod_{\begin{subarray}{c}p\leq Y\\\
p^{\gamma(p)}\nmid
q\end{subarray}}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap
W^{q}(Y)}e^{2\pi ig(\bm{s})a/q}\Big{(}X^{\ell}\psi(X,\dots,X)$
$\displaystyle\qquad+\sum_{m=1}^{\ell}(-1)^{m}\sum_{1\leq
j_{1}<\cdots<j_{m}\leq\ell}X^{\ell-m}\int_{[0,X]^{m}}x_{j_{1}}\cdots
x_{j_{m}}\frac{\partial^{m}\psi}{\partial x_{j_{1}}\cdots\partial
x_{j_{m}}}(\star)\ dx_{j_{1}}\cdots dx_{j_{m}}\Big{)}.$

By iteratively applying integration by parts, this equals

$q^{-\ell}\prod_{\begin{subarray}{c}p\leq Y\\\ p^{\gamma(p)}\nmid
q\end{subarray}}\left(1-\frac{j(p)}{p^{\gamma(p)\ell}}\right)\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap
W^{q}(Y)}e^{2\pi ig(\bm{s})a/q}\int_{[0,X]^{\ell}}\psi(\bm{x})d\bm{x},$

as desired. It remains to bound the error term that results from our
substitution of (39). This error term is the sum of a first term of order $qE$
and $2^{\ell}-1$ terms of the form

$qE\left(\int_{[0,X]^{m}}\frac{\partial^{m}\psi}{\partial
x_{j_{1}}\cdots\partial x_{j_{m}}}(\star)\ dx_{j_{1}}\cdots
dx_{j_{m}}\right).$

Iteratively applying the product rule, we see that
$\frac{\partial^{m}\psi}{\partial x_{j_{1}}\cdots\partial x_{j_{m}}}$ is the
sum of less than $m!$ terms bounded in absolute value by $(2\pi
k^{m}J|\beta|)^{j}X^{jk-m}$ for some $1\leq j\leq m$. In particular, each
integral is bounded by

$\ell!\max_{1\leq j\leq\ell}(2\pi k^{\ell}JX^{k}|\beta|)^{j}\leq\ell!(1+2\pi
k^{\ell}JX^{k}|\beta|)^{\ell},$

and the error bound follows.∎

###  7.3 Local cancellation

In this section, we apply Theorem 1.4 to establish the necessary cancellation
in our sieved local exponential sums, yielding item (ii) in Theorem 3.9. We
begin by invoking a multivariate version of Hensel’s Lemma that allows us to
reduce to the case of prime moduli. This statement in particular follows from
Theorem 1.1 of [5].

######  Lemma 7.6 (Multivariable Hensel’s Lemma).

Suppose $\ell\in\mathbb{N}$, $g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$, $p$ is
prime, $\bm{n}\in\mathbb{Z}^{\ell}$, and $\gamma,v\in\mathbb{N}$ with $v\geq
2\gamma-1$. If

$g(\bm{n})\equiv 0\ \textnormal{mod}\ p^{2\gamma-1}$

and $\nabla g(\bm{n})\not\equiv\bm{0}$ mod $p^{\gamma}$, then there exists
$\bm{m}\in\mathbb{Z}^{\ell}$ with $g(\bm{m})\equiv 0$ mod $p^{v}$.

We now prove the following multivariate generalization of Lemma 4.3 in [28].

######  Lemma 7.7.

Suppose $\ell\in\mathbb{N}$, $g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with
$\deg(g)=k\geq 2$, and $Y>0$. If $q\in\mathbb{N}$ has prime factorization
$q=p_{1}^{v_{1}}\cdots p_{r}^{v_{r}}$ with $p_{1}<\cdots<p_{t}\leq
Y<p_{t+1}<\cdots<p_{r}$, and $(a,q)=1$, then

$\left|\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap W^{q}(Y)}e^{2\pi
ig(\bm{s})a/q}\right|\leq
C_{1}\prod_{i=1}^{t}\left((k-1)^{\ell}p_{i}^{\ell/2}+j(p_{i})\right)\prod_{i=t+1}^{r}C_{2}(v_{i}+1)^{\ell}p_{i}^{v_{i}(\ell-1/k)},$

where $C_{2}=C_{2}(k)$ and $C_{1}$ depends only on the moduli at which $\nabla
g$ identically vanishes and the primes $p\leq Y$ dividing $q$ modulo which $g$
is not Deligne. Further, the sum is $0$ if $v_{i}\geq 2\gamma(p_{i})$ for some
$1\leq i\leq t$.

###### Proof.

Factor $q=p_{1}^{v_{1}}\cdots p_{r}^{v_{r}}$ as in the lemma. By the Chinese
Remainder Theorem, we have

$\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap W^{q}(Y)}e^{2\pi
ig(\bm{s})a/q}=\prod_{m=1}^{r}\sum_{\bm{s}\in\\{0,\dots,p_{m}^{v_{m}}-1\\}^{\ell}\cap
W^{p_{m}^{v_{m}}}(Y)}e^{2\pi ig(\bm{s})a_{m}/p_{m}^{v_{m}}},$

where $a_{1},\dots,a_{r}$ are the unique residues satisfying $a/q\equiv
a_{1}/p_{1}^{v_{1}}+\cdots+a_{r}/p_{r}^{v_{r}}\ \text{mod }1.$

Suppose $p^{v}=p_{m}^{v_{m}}$ with $\gamma(p)>1$ and $v<2\gamma(p)$. By
definition of $\gamma$, $\nabla g$ identically vanishes modulo
$p^{\gamma(p)-1}$. Since $p^{2\gamma(p)-1}\leq p^{3(\gamma(p)-1)}$, we can
bound $p^{v}$ by the cube of a modulus at which $\nabla g$ identically
vanishes, trivially bound the corresponding sum, and absorb it into the
constant $C_{1}$ in the conclusion of the lemma.

Next suppose $p^{v}=p_{m}^{v_{m}}$ with $p\leq Y$ and $v=\gamma(p)=1$.
Recalling that $j(p)$ is the number of zeros of $\nabla g$ modulo $p$ and
applying Theorem 1.4, we have for $p\nmid b$ that

$\left|\sum_{\bm{s}\in\\{0,\dots,p-1\\}^{\ell}\cap W^{p}(Y)}e^{2\pi
ig(\bm{s})b/p}\right|\leq(k-1)^{\ell}p^{\ell/2}+j(p),$

provided $g$ is Deligne modulo $p$, and the remaining such primes are absorbed
into $C_{1}$.

Now suppose that $p^{v}=p_{m}^{v_{m}}$ with $p\leq Y$ and $v\geq 2\gamma(p)$,
and let $w=2\gamma(p)-1$. If $\bm{s}\in\\{0,\dots,p^{v}-1\\}^{\ell}$ and
$\tilde{\bm{s}}$ is the reduced residue class of $\bm{s}$ modulo $p^{w}$, then
we have that $g(\bm{s})\equiv p^{w}t+g(\tilde{\bm{s}})\ (\text{mod }p^{v})$
for some $0\leq t\leq p^{v-w}-1$. Conversely, if
$\tilde{\bm{s}}\in\\{0,\dots,p^{w}-1\\}^{\ell}$ with $\nabla
g(\tilde{\bm{s}})\not\equiv\bm{0}\ (\text{mod }p^{\gamma(p)})$, then for every
$0\leq t\leq p^{v-w}-1$, Lemma 7.6 applied to the polynomial
$g(\bm{x})-(p^{w}t+g(\tilde{\bm{s}}))$ yields
$\bm{s}\in\\{0,\dots,p^{v}-1\\}^{\ell}$ with $g(\bm{s})\equiv
p^{w}t+g(\tilde{\bm{s}})\ (\text{mod }p^{v})$.

In other words, the map $F$ on $\mathbb{Z}/p^{v-w}\mathbb{Z}$ defined by
$g(p^{w}t+\tilde{\bm{s}})\equiv p^{w}F(t)+g(\tilde{\bm{s}})\ (\text{mod
}p^{v})$ is a bijection. In particular, if $p\nmid b$, then

$\displaystyle\sum_{\bm{s}\in\\{0,\dots,p^{v}-1\\}^{\ell}\cap
W^{p^{v}}(Y)}e^{2\pi ig(\bm{s})b/p^{v}}$
$\displaystyle=\sum_{\begin{subarray}{c}\tilde{\bm{s}}\in\\{0,\dots,p^{w}-1\\}^{\ell}\\\
\nabla g(\tilde{\bm{s}})\not\equiv\bm{0}\ (\text{mod
}p^{\gamma(p)})\end{subarray}}\sum_{t=0}^{p^{v-w}-1}e^{2\pi
ig(p^{w}t+\tilde{\bm{s}})b/p^{v}}$
$\displaystyle=\sum_{\begin{subarray}{c}\tilde{\bm{s}}\in\\{0,\dots,p^{w}-1\\}^{\ell}\\\
\nabla g(\tilde{\bm{s}})\not\equiv\bm{0}\ (\text{mod
}p^{\gamma(p)})\end{subarray}}\sum_{t=0}^{p^{v-w}-1}e^{2\pi
i\left(p^{w}t+g(\tilde{\bm{s}})\right)b/p^{v}}$ $\displaystyle=0,$

where the last equality is the fact that the sum in $t$ runs over the full
collection of $p^{v-w}$-th roots of unity.

Finally, suppose $p^{v}=p_{m}^{v_{m}}$ with $p>Y$. We note that
$W^{p^{v}}(Y)=\mathbb{N}$ and we only exploit cancellation in a single
variable. To this end, for each
$\tilde{\bm{s}}=(s_{2},\dots,s_{\ell})\in\\{0,\dots,p^{v}-1\\}^{\ell-1}$, we
define $\tilde{g}$ by $\tilde{g}(x)=g(x,\tilde{\bm{s}})$. Utilizing the
standard single-variable complete sum estimate (see [4] for example), we have
for $b\nmid p$ that

$\displaystyle\left|\sum_{\bm{s}\in\\{0,\dots,p^{v}-1\\}^{\ell}}e^{2\pi
ig(\bm{s})b/p^{v}}\right|$
$\displaystyle\leq\sum_{\tilde{\bm{s}}\in\\{0,\dots,p^{v}-1\\}^{\ell-1}}\left|\sum_{s=0}^{p^{v}-1}e^{2\pi
i\tilde{g}(s)b/p^{v}}\right|$
$\displaystyle\ll_{k}p^{v(1-1/k)}\sum_{\tilde{\bm{s}}\in\\{0,\dots,p^{v}-1\\}^{\ell-1}}\gcd(\text{cont}(\tilde{g}),p^{v})^{1/k}.$

To analyze the remaining sum, we note that at the expense of the term
$\gcd(\text{cont}(g),p^{v})^{1/k}$ in our final estimate, we can cancel
factors of $p$ from the coefficients of $g$ and assume that
$p\nmid\text{cont}(g)$. In this case, suppose
$a_{\bm{i}}=a_{i_{1},\dots,i_{\ell}}$ with $0<|\bm{i}|\leq k$ is a coefficient
of $g$, corresponding to $x_{1}^{i_{1}}\cdots x_{\ell}^{i_{\ell}}$, that is
not divisible by $p$. Further, assume that $i_{1}>0$, as if $i_{1}=0$ then we
could just relabel our coordinates. In this case, for each $0\leq w\leq v$,
$\gcd(\text{cont}(\tilde{g}),p^{v})=p^{w}$ only if $p^{w}\mid
s_{2}^{i_{2}}\cdots s_{\ell}^{i_{\ell}}$, so in particular $p^{\lceil
w/k\rceil}\mid s_{2}\cdots s_{\ell}$, which occurs for fewer than
$(w+1)^{\ell-1}p^{v(\ell-1)-w/k}$ choices of $\tilde{\bm{s}}$. In particular,

$\displaystyle\sum_{\tilde{s}\in\\{0,\dots,p^{v}-1\\}^{\ell-1}}\gcd(\text{cont}(\tilde{g}),p^{v})^{1/k}$
$\displaystyle\leq\gcd(\text{cont}(g),p^{v})^{1/k}\sum_{w=0}^{v}(w+1)^{\ell-1}p^{v(\ell-1)-w/k}p^{w/k}$
$\displaystyle\leq(v+1)^{\ell}\gcd(\text{cont}(g),p^{v})^{1/k}p^{v(\ell-1)}.$

The $\gcd(\text{cont}(g),p^{v})^{1/k}$ term can be absorbed into $C_{1}$, and
the remaining bound on the exponential sum modulo $p^{v}$ is a constant
depending on $k$ times
$p^{v(1-1/k)}(v+1)^{\ell}p^{\ell(v-1)}=(v+1)^{\ell}p^{v(\ell-1/k)}$, as
required. Having accounted for all prime divisors of $q$, the proof is
complete. ∎

Lemma 7.7 combines with Lemma 3.3 as well as the estimates $\prod_{p\mid
q}\left(1+\frac{C}{p}\right)\leq(q/\phi(q))^{C}$ and $\prod_{p\mid
q}\left(1+\frac{C}{p^{3/2}}\right)\ll_{C}1$ to yield item (ii) of Theorem 3.9,
restated below.

######  Corollary 7.8.

If $\ell\geq 2$, $g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(g)=k\geq
2$, and $(a,q)=1$, then

$\left|\sum_{\bm{s}\in\\{0,\dots,q-1\\}^{\ell}\cap W^{q}(Y)}e^{2\pi
ig(\bm{s})a/q}\right|\leq
C_{1}\begin{cases}(k-1)^{\ell\omega(q)}\Phi(q,\ell)q^{\ell/2}&\text{if }q\leq
Y\\\ C_{2}^{\omega(q)}\tau(q)^{\ell}q^{\ell-1/k}&\text{for all }q\end{cases},$

where $C_{2}=C_{2}(k)$, $\Phi(q,2)=(q/\phi(q))^{C_{2}}$,
$\Phi(q,\ell)\ll_{k,\ell}1$ for $\ell\geq 3$, and $C_{1}$ depends only on the
moduli at which $\nabla g$ identically vanishes and the primes $p\leq Y$
dividing $q$ modulo which $g$ is not Deligne.

###  7.4 Oscillatory integral estimate

In order to establish (28) in the case that $\alpha$ is close, but not too
close, to a rational with very small denominator, we need to control the
oscillatory integral in the asymptotic formula given by Lemma 7.5. To achieve
this, we invoke the following standard estimate, given for example in Lemma
2.8 of [38].

######  Lemma 7.9 (Van der Corput’s Lemma).

If $X>0$, $\beta\neq 0$, $k\in\mathbb{N}$, and $g\in\mathbb{Z}[x]$ with
$\deg(g)=k$, then

$\left|\int_{0}^{X}e^{2\pi ig(x)\beta}dx\right|\ll|\beta|^{-1/k}.$

Utilizing Lemma 7.9 to exploit cancellation in a single variable, then
trivially bounding the integral in the remaining variables, we have the
following bound for the integral in the conclusion of Lemma 7.5.

######  Corollary 7.10.

If $X>0$, $\beta\neq 0$, $k,\ell\in\mathbb{N}$, and
$g\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with $\deg(g)=k$, then

$\left|\int_{[0,X]^{\ell}}e^{2\pi
ig(\bm{x})\beta}d\bm{x}\right|\ll\min\\{X^{\ell},X^{\ell-1}|\beta|^{-1/k}\\}.$

###  7.5 Minor arc estimates

In an effort to establish item (iii) of Theorem 3.9, we begin by invoking a
variation of the most traditional minor arc estimate, Weyl’s Inequality.

######  Lemma 7.11 (Lemma 3, [7]).

Suppose $k\in\mathbb{N}$, $g(x)=a_{0}+a_{1}x+\cdots+a_{k}x^{k}$ with
$a_{0}\dots,a_{k}\in\mathbb{R}$ and $a_{k}\in\mathbb{N}$. If $X>0$,
$a,q\in\mathbb{N}$ with $(a,q)=1$, and $|\alpha-a/q|<q^{-2}$, then

$\left|\sum_{n=1}^{X}e^{2\pi
ig(n)\alpha}\right|\ll_{k}X\left(a_{k}\log^{k^{2}}(a_{k}qX)\left(q^{-1}+X^{-1}+\frac{q}{a_{k}X^{k}}\right)\right)^{2^{-k}}.$

We now carefully adapt Lemma 7.11 to our particular sieve, and to the
multivariate setting, though as in Corollary 7.10, we ultimately only exploit
cancellation in a single variable.

######  Lemma 7.12.

Suppose $k,\ell\in\mathbb{N}$ and $g(\bm{x})=\sum_{|\bm{i}|\leq
k}a_{\bm{i}}\bm{x}^{\bm{i}}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with
$\deg(g)=k$. Suppose further that $X,Y,Z\geq 2$, $YZ\leq X$, and
$a,q\in\mathbb{N}$ with $(a,q)=1$, and let $J=\sum_{|\bm{i}|\leq
k}|a_{\bm{i}}|$. If $|\alpha-a/q|<q^{-2}$, then

$\left|\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi
ig(\bm{n})\alpha}\right|\ll_{k,\ell}\textnormal{cont}(g)^{6}(\log
Y)^{ek}X^{\ell}\left(e^{-\frac{\log Z}{\log
Y}}+\left(J\log^{k^{2}}(JqX)\left(q^{-1}+\frac{Z}{X}+\frac{qZ^{k}}{X^{k}}\right)\right)^{2^{-k}}\right).$

###### Proof.

Suppose $k,\ell\in\mathbb{N}$ and $g(\bm{x})=\sum_{|\bm{i}|\leq
k}a_{\bm{i}}\bm{x}^{\bm{i}}\in\mathbb{Z}[x_{1},\dots,x_{\ell}]$ with
$\deg(g)=k$. We begin by conducting an invertible (over $\mathbb{Z}$) change
of variables to reduce to the case where the $x_{1}^{k}$ coefficient
$a_{(k,0,\dots,0)}$ is nonzero. To this end, consider the polynomial
$\tilde{g}\in\mathbb{Z}[x_{2},\dots,x_{\ell}]$ defined by
$\tilde{g}(x_{2},\dots,x_{\ell})=g^{k}(1,x_{2},\dots,x_{\ell})$, where $g^{k}$
denotes the top degree homogeneous part of $g$, noting that $\tilde{g}$ is not
identically zero. Let $(c_{2},\dots,c_{\ell})\in\\{0,1,\dots,k\\}^{\ell-1}$ be
such that $\tilde{g}(c_{2},\dots,c_{\ell})\neq 0$.

As an aside, the existence of such a “small integer non-root” of a general
nonzero multivariate polynomial $F\in\mathbb{Z}[x_{1},\dots,x_{j}]$ can be
shown via induction, which we sketch here. The base case $j=1$ corresponds to
nonzero univariate polynomials, which have at most $k$ roots, hence at least
one non-root in $\\{0,1,\dots,k\\}$. Then, for higher degrees, fix one
variable that appears at least once in $F$ (without loss of generality, assume
$x_{1}$ appears at least once), let $d$ be the degree of $F$ as a polynomial
in $x_{1}$ only, and let $\tilde{F}(x_{2},\dots,x_{j})$ be the polynomial of
degree at most $k-d$ that forms the $x_{1}^{d}$ coefficient. By the inductive
hypothesis, we can choose $(m_{2},\dots,m_{j})\in\\{0,\dots,k\\}^{j-1}$ such
that $\tilde{F}(m_{2},\dots,m_{j})\neq 0$. Then, $F(x_{1},m_{2},\dots,m_{j})$
is a nonzero degree-$d$ polyomial in $x_{1}$, which has a non-root in
$\\{0,\dots,k\\}$, completing the induction.

Back to the proof at hand, we see that the change of variables $x_{1}=y_{1}$
and $x_{j}=y_{j}+c_{j}y_{1}$ for $2\leq j\leq\ell$ yields a $y_{1}^{k}$
coefficient of $\tilde{g}(c_{2},\dots,c_{\ell})\neq 0$. Let $M$ denote the
$\ell\times\ell$ matrix satisfying $M\bm{x}=\bm{y}$ corresponding to the
described change of variables, and let
$f(y_{1},\dots,y_{\ell})=\sum_{|\bm{i}|\leq k}b_{\bm{i}}y^{\bm{i}}$ be the
polynomial satisfying $f(\bm{y})=g(M^{-1}\bm{y})$. By taking the complex
conjugate of the relevant exponential sum if necessary, we can assume that
$b=b_{(k,0,\dots,0)}>0$. Further, the effect of the transformation on the size
of this coefficient is well-controlled, in that $b\ll_{k,l}J$.

Let $T=M([1,X]^{\ell})$, so

$\sum_{\bm{n}\in[1,X]^{\ell}\cap W(Y)}e^{2\pi
ig(\bm{n})\alpha}=\sum_{\bm{n}\in T\cap W(Y)}e^{2\pi if(\bm{n})\alpha},$

where $W(Y)$ is defined on each side in terms of the corresponding polynomial.

Let $\tilde{T}$ denote the projection of $T$ onto the last $\ell-1$
coordinates, noting that $|\tilde{T}|\leq(2kX)^{\ell-1}$ due to the details of
our change of variables. For each fixed
$\tilde{\bm{n}}=(n_{2},\dots,n_{\ell})\in\mathbb{N}^{\ell-1}$, we let
$I=\\{n\in\mathbb{N}:(n,\tilde{\bm{n}})\in T\\}$, which is an interval of
integers of length at most $X$, we let
$\tilde{W}(Y)=\\{n\in\mathbb{N}:(n,\tilde{\bm{n}})\in W(Y)\\}$, and we let
$\tilde{f}(x)=f(x,\tilde{\bm{n}})$. We see trivially that

$\left|\sum_{\bm{n}\in T\cap W(Y)}e^{2\pi
if(\bm{n})\alpha}\right|\leq(2kX)^{\ell-1}\max_{\tilde{\bm{n}}\in\tilde{T}}\left|\sum_{n\in
I\cap\tilde{W}(Y)}e^{2\pi i\tilde{f}(n)\alpha}\right|.$ (40)

We now proceed with $\tilde{\bm{n}}=(n_{2},\dots,n_{\ell})$ fixed, and we
define $L$ and $m$ so that $I=[m,L+m]$, so in particular $L\leq X$. All
subsequent conclusions will be independent of $\tilde{\bm{n}}$. Let $P$ be the
set of products $p_{1}^{\gamma(p_{1})}\cdots p_{s}^{\gamma(p_{s})}$ for primes
$p_{1}<\cdots<p_{s}\leq Y$, let $P_{1}$ denote the set of elements of $P$ that
are at most $Z$, and let $P_{2}$ denote the set of elements of $P$ that are
greater than $Z$.

By inclusion-exclusion, we have

$\left|\sum_{n\in I\cap\tilde{W}(Y)}e^{2\pi
i\tilde{f}(n)\alpha}\right|=\left|\sum_{D\in
P}(-1)^{\omega(D)}\sum_{\begin{subarray}{c}n\in I\\\ \nabla
f(n,\tilde{\bm{n}})\equiv\bm{0}\ (\text{mod }D)\end{subarray}}e^{2\pi
i\tilde{f}(n)\alpha}\right|,$ (41)

where $\omega(D)$ is the number of distinct prime factors of $D$. For $D\in
P_{1}$, we use the fact that the set of $n$ for which $\nabla
f(n,\tilde{\bm{n}})\equiv\bm{0}\ (\text{mod }D)$ is contained in the set of
$n$ for which $\tilde{f}^{\prime}(n)\equiv 0\ (\text{mod }D)$. Noting that
$\tilde{f}^{\prime}$ can have at most $k$ roots modulo any prime at which it
does not identically vanish, we have

$\displaystyle\left|\sum_{D\in
P_{1}}(-1)^{\omega(D)}\sum_{\begin{subarray}{c}n\in I\\\ \nabla
f(n,\tilde{\bm{n}})\equiv\bm{0}\ (\text{mod }D)\end{subarray}}e^{2\pi
i\tilde{f}(n)\alpha}\right|\ll_{k}(\text{cont}(g))^{2}\sum_{D\in
P_{1}}k^{\omega(D)}\max_{0\leq c\leq D}\left|\sum_{n=0}^{L/D}e^{2\pi
i\tilde{f}(Dn+m+c)\alpha}\right|,$

where the $\text{cont}(g)^{2}$ term accounts for the primes $p$ for which
$\gamma(p)>1$ by Proposition 3.6. Further, we see from Lemma 7.11 and the
estimate $1\leq b\ll_{k,l}J$ that

$\displaystyle\sum_{D\in P_{1}}k^{\omega(D)}\max_{0\leq c\leq
D}\left|\sum_{n=0}^{L/D}e^{2\pi i\tilde{f}(Dn+m+c)\alpha}\right|$
$\displaystyle\ll_{k,l}\sum_{D\in
P_{1}}k^{\omega(D)}\frac{L}{D}\left(b\log^{k^{2}}(bqL)\left(q^{-1}+\frac{D}{L}+\frac{qD^{k}}{bL^{k}}\right)\right)^{2^{-k}}$
$\displaystyle\ll_{k,l}X\left(J\log^{k^{2}}(JqX)\left(q^{-1}+\frac{Z}{X}+\frac{qZ^{k}}{X^{k}}\right)\right)^{2^{-k}}\sum_{D\in
P_{1}}\frac{k^{\omega(D)}}{D}$ $\displaystyle\ll_{k,l}X(\log
Y)^{k}\left(J\log^{k^{2}}(JqX)\left(q^{-1}+\frac{Z}{X}+\frac{qZ^{k}}{X^{k}}\right)\right)^{2^{-k}},$

where the last inequality uses that if $C>0$, then

$\sum_{D\in P}\frac{C^{\omega(D)}}{D}=\prod_{p\leq
Y}\left(1+\frac{C}{p^{\gamma(p)}}\right)\leq\prod_{p\leq
Y}\left(1+\frac{C}{p}\right)\ll(\log Y)^{C}.$ (42)

This combines with (40) to close the book on the contributions to (41) from
$P_{1}$. It remains to account for the contribution to (41) from $P_{2}$.
Because $P_{2}$ has so many elements, it is crucial for us to exploit the
cancellation provided by the term $(-1)^{\omega(D)}$.

To this end, for a fixed $n\in I$, let $P^{n}=\\{D\in P:\nabla
f(n,\tilde{\bm{n}})\equiv\bm{0}\ (\text{mod }D)\\}$, and let
$P^{n}_{2}=P^{n}\cap P_{2}$. The only issue is the possibility that way more
elements of $P^{2}_{n}$ have an even number of prime factors than odd, or vice
versa, which we show below does not happen.

Let $q$ be the largest prime power of the form $p^{\gamma(p)}$ with $p\leq Y$,
and let $q_{n}$ be the largest such prime power lying in $P^{n}$, noting that
$q_{n}\leq q\ll_{k}\textnormal{cont}(g)Y$ by Proposition 3.6. Let $A$ denote
the set of elements of $P^{n}$ that have an even number of prime factors, let
$B$ denote the set of elements of $P^{n}$ that have odd number of prime
factors, and let $A^{\prime}$ and $B^{\prime}$, respectively, denote the same
for elements of $P^{n}_{2}$. The quantity we need control of is
$\left||A^{\prime}|-|B^{\prime}|\right|$.

Let $A_{1}$ be the elements of $A$ that are greater than $Z$ and not divisible
by $q_{n}$, and let $A_{2}$ be the elements of $A$ that are greater than
$q_{n}Z$ and divisible by $q_{n}$. Likewise define $B_{1}$ and $B_{2}$. The
map $D\to q_{n}D$ defines an injection from $A_{1}$ to $B_{2}$, while the map
$D\to D/q_{n}$ defines an injection from $A_{2}$ to $B_{1}$. Letting $A_{3}$
denote all the elements of $A$ greater than $q_{n}Z$, we have

$|A_{3}|\leq|A_{1}|+|A_{2}|\leq|B_{1}|+|B_{2}|\leq|B^{\prime}|.$

Symmetrically, we have $|B_{3}|\leq|A^{\prime}|$. Finally, letting $A_{4}$ and
$B_{4}$ denote the elements of $A^{\prime}$ and $B^{\prime}$ satisfying
$Z<D\leq q_{n}Z$, we have
$|A^{\prime}|=|A_{3}|+|A_{4}|\leq|B^{\prime}|+|A_{4}|$ and similarly
$|B^{\prime}|\leq|A^{\prime}|+|B_{4}|$, so the magnitude of
$|A^{\prime}|-|B^{\prime}|$ is bounded above by $|A_{4}|+|B_{4}|$, which is
the size of the set $\overline{P^{n}}$ of elements of $P^{n}$ satisfying
$Z<D\leq q_{n}Z$.

We now see

$\displaystyle\left|\sum_{D\in