# A motivic circle method

Margaret Bilu  Institut de Mathématiques de Bordeaux
Université de Bordeaux
351 cours de la Libération
Talence cedex
France<EMAIL_ADDRESS>and  Tim Browning  IST Austria
Am Campus 1
3400 Klosterneuburg
Austria<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

The circle method has been successfully used over the last century to study
rational points on hypersurfaces. More recently, a version of the method over
function fields, combined with spreading out techniques, has led to a range of
results about moduli spaces of rational curves on hypersurfaces. In this paper
a version of the circle method is implemented in the setting of the
Grothendieck ring of varieties. This allows us to approximate the classes of
these moduli spaces directly, without relying on point counting, and leads to
a deeper understanding of their geometry.

###### 2010 Mathematics Subject Classification:

14H10 (11D72, 11P55 , 14E18)

###### Contents

  1. 1 Introduction
  2. 2 The Grothendieck ring of varieties with exponentials
  3. 3 The motivic circle method
  4. 4 The weight function
  5. 5 Weyl differencing and a general exponential sum bound
  6. 6 The motivic major arcs
  7. 7 The motivic minor arcs
  8. 8 The space of morphisms
  9. 9 The variety of lines
  10. A Geometry of numbers over function fields

##  1\. Introduction

Let $k$ be a field and let $f\in k[x_{1},\dots,x_{n}]$ be a non-singular
homogeneous polynomial of degree $d\geq 2$, defining a hypersurface
$X\subset\mathbf{A}^{n}$. For global fields, the density of $k$-points on $X$
has been the object of intense study over the years. When $k=\mathbf{Q}$, the
Hardy–Littlewood circle method can be used to cast light on the limit

$\lim_{B\to\infty}B^{-(n-d)}\\#\\{\mathbf{x}\in\mathbf{Z}^{n}:|\mathbf{x}|\leq
B,\leavevmode\nobreak\ f(\mathbf{x})=0\\},$

where $\mathbf{x}=(x_{1},\dots,x_{n})$ and $|\mathbf{x}|=\max_{1\leq i\leq
n}|x_{i}|.$ Thus, it follows from work of Birch [Bir61] that this limit exists
as a product of local densities, provided that $n>2^{d}(d-1)$. Lee [Lee13] has
worked out the analogous statement when $k=\mathbf{F}_{q}(t)$, for a finite
field $\mathbf{F}_{q}$ of characteristic $>d$. Thus, under the same assumption
$n>2^{d}(d-1)$, a similar statement is proved about the existence and nature
of the limit

$\lim_{e\to\infty}q^{-e(n-d)}\\#\\{\mathbf{g}\in\mathbf{F}_{q}[t]^{n}:\deg(g_{1}),\dots,\deg(g_{n})\leq
e,\leavevmode\nobreak\ f(\mathbf{g})=0\\},$

where $\mathbf{g}=(g_{1},\dots,g_{n})$. This paper is concerned with the field
$k=\mathbf{C}(t)$ and the geometry of the affine variety

$M_{e}=\\{\mathbf{g}\in\mathbf{C}[t]^{n}:\deg(g_{1}),\dots,\deg(g_{n})\leq
e,\leavevmode\nobreak\ f(\mathbf{g})=0\\},$

as $e\to\infty$. We have $(e+1)n$ indeterminates and $de+1$ equations, and so
the expected dimension of $M_{e}$ is

(1.1) $\mu(e):=(e+1)n-(de+1)=e(n-d)+n-1.$

Typically, tools from algebraic geometry only allow one to prove that the
expected dimension is actually the true dimension when $X$ is generic. This
follows from recent work of Riedl and Yang [RY19] if $n\geq d+3$.

It turns out that the circle method can shed light on the geometry of $M_{e}$
when $d$ is small enough compared to $n$, for any hypersurface
$X\subset\mathbf{A}^{n}$ defined by a non-singular form. Thus, under the
assumption $n>(5d-4)2^{d-1}$, it follows from [BV17] that $M_{e}$ is
irreducible and of dimension $\mu(e)$. This was later refined in [BS23, Thm.
1.2], with the outcome that for $n>3\cdot 2^{d-1}(d-1)$, the variety $M_{e}$
is a reduced and irreducible complete intersection of dimension $\mu(e)$,
which is smooth outside a set of codimension at least
$(n/2^{d-2}-6(d-1))(1+\lfloor\frac{e+1}{d-1}\rfloor)$. The arguments in [BV17,
BS23] rely on spreading out, which translates the problem into a counting
problem over a finite field $\mathbf{F}_{q}$. An application of the Lang–Weil
estimates yields the desired conclusion via an application of the circle
method over $\mathbf{F}_{q}(t)$.

This paper aims to tackle deeper questions about the geometry of $M_{e}$
without recourse to spreading out and a reduction to counting arguments. A
step in this direction was taken recently in [BS20], where a sheaf-theoretic
version of the circle method is developed to study a variant of $M_{e}$, using
tools from étale cohomology. Our goal is to rework the circle method in a
motivic setting and to try and understand the class of $M_{e}$ in the
Grothendieck ring of varieties.

The Grothendieck ring of varieties $K_{0}(\mathrm{Var}_{\mathbf{C}})$ over
$\mathbf{C}$ is defined by generators and relations. The generators are
$\mathbf{C}$-varieties $X$. The relations are given by $X-Y$, whenever $X$ and
$Y$ are $\mathbf{C}$-varieties that are isomorphic over $\mathbf{C}$, and the
scissor relation $X-Y-U$, whenever $X$ is a $\mathbf{C}$-variety, $Y$ a closed
subscheme of $X$ and $U=X\setminus Y$ its open complement. We will write $[X]$
for the class in $K_{0}(\mathrm{Var}_{\mathbf{C}})$ of $X$. The product
$[X][Y]=[X\times_{\mathbf{C}}Y]$ endows $K_{0}(\mathrm{Var}_{\mathbf{C}})$
with a ring structure. We denote by $\mathbf{L}$ the class of $\mathbf{A}^{1}$
in $K_{0}(\mathrm{Var}_{\mathbf{C}})$, which is a zero divisor in the ring, by
work of Borisov [Bor18]. Inverting $\mathbf{L}$ gives the ring
$\mathscr{M}_{\mathbf{C}}=K_{0}(\mathrm{Var}_{\mathbf{C}})[\mathbf{L}^{-1}]$.
We will be interested in the class of $M_{e}$ in $\mathscr{M}_{\mathbf{C}}$.
In fact, since the circle method involves working with exponential sums, in
order to be able to set up a motivic version of it we will need to work in the
Grothendieck ring of varieties with exponentials
$K_{0}(\mathrm{ExpVar}_{\mathbf{C}})$ and its localisation
$\mathscr{E}xp\mathscr{M}_{\mathbf{C}}=K_{0}(\mathrm{ExpVar}_{\mathbf{C}})[\mathbf{L}^{-1}]$.
The precise definition of $K_{0}(\mathrm{ExpVar}_{\mathbf{C}})$ will be given
in Section 2, together with some of its properties. In order to talk about
convergence, as $e\to\infty$, we will need to introduce a topology on
$\mathscr{E}xp\mathscr{M}_{\mathbf{C}}$, which refines the obvious filtration
by dimension. In Section 4 we will carry out the construction of an
appropriate weight function

$w:\mathscr{E}xp\mathscr{M}_{\mathbf{C}}\to\mathbf{Z},$

coming from Hodge theory. We’ll denote by $\widehat{\mathscr{M}_{\mathbf{C}}}$
the completion of $\mathscr{M}_{\mathbf{C}}$ with respect to this weight
topology.

Working in $\mathscr{E}xp\mathscr{M}_{\mathbf{C}}$, our main result will show
that the class of $M_{e}$ converges to the classes of local terms that
resemble the main term in the circle method, as $e\to\infty$. Given
$x\in\mathbf{A}^{1}$ and integer $N\geq 0$, our work involves the jet spaces

(1.2)
$\Lambda_{N}(f,x)=\left\\{\mathbf{g}\in\mathbf{C}[t]^{n}:\deg(g_{1}),\dots,\deg(g_{n})<N,\leavevmode\nobreak\
f(\mathbf{g})\in(t-x)^{N}\mathbf{C}[t]\right\\},$

parameterising solutions of $f(\mathbf{g})=0$ modulo $(t-x)^{N}$, together
with

(1.3)
$\Lambda_{N}(f,\infty)=\left\\{\mathbf{g}\in\mathbf{C}[t]^{n}:\begin{array}[]{l}\deg(g_{1}),\dots,\deg(g_{n})<N\\\
f(g_{1}(t^{-1}),\dots,g_{n}(t^{-1}))\in
t^{-N}\mathbf{C}[t^{-1}]\end{array}\right\\}.$

Note that $\Lambda_{N}(f,\infty)$ can naturally be identified with
$\Lambda_{N}(f,0)$. Bearing this notation in mind, we shall prove the
following result.

######  Theorem 1.1.

Let $f\in\mathbf{C}[x_{1},\dots,x_{n}]$ be a non-singular homogeneous
polynomial of degree $d\geq 3$, defining a hypersurface
$X\subset\mathbf{A}^{n}$. Assume $n>2^{d}(d-1)$ and let $e\geq 1$. Then

$[M_{e}]=\mathbf{L}^{\mu(e)}\left(\mathfrak{S}(f)\cdot\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,\infty)]+R_{e}\right)$

in $\widehat{\mathscr{M}_{\mathbf{C}}}$, where

$\mathfrak{S}(f)=\prod_{x\in\mathbf{A}^{1}}\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)]$

is a motivic Euler product and $R_{e}$ is an error term satisfying

$w(R_{e})\leq
4-\frac{n-2^{d}(d-1)}{2^{d-2}}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right).$

The term $\mathfrak{S}(f)$ is the motivic singular series and will emerge as
an infinite convergent sum, whose precise definition is given in (6.11). In
Sections 6.3.3 and 6.3.4 we will express $\mathfrak{S}(f)$ as a motivic Euler
product, using the construction of motivic Euler products found in [Bil23,
Section 3]. The relevant background facts about motivic Euler products will be
recalled in Section 2.5.

######  Remark 1.2.

In this paper the focus is on hypersurfaces of degree $d\geq 3$. However, on
combining Propositions 3.7 and 3.8 in (3.1), it is also possible to deduce a
statement for $d=2$, with the only difference being a weaker error term if
$e\leq n/2-7$.

The proof of Theorem 1.1 relies on the development of a motivic version of the
Hardy–Littlewood circle method. The arguments parallel some of the sheaf-
theoretic arguments in [BS20], but are much more attuned to the steps taken in
the classical circle method over $\mathbf{Q}$ in [Bir61], and its incarnation
over $\mathbf{F}_{q}(t)$ in [Lee13]. The motivic circle method is developed
over the course of several sections. In Section 3 we lay the foundations of
the method and in Section 5 we work out a Weyl differencing argument to bound
the weight of a general class of motivic exponential sums. In Sections 6 and 7
we analyse the contribution from the major and minor arcs, respectively. Our
treatment of the minor arcs relies on tools from the geometry of numbers over
the function field $\mathbf{C}(t)$, in the spirit of work by Mahler [Mah41].
We have collected together the necessary facts in Appendix A.

One notable feature of our work is that we work over $\mathbf{C}$ throughout
and never rely on the kind of counting arguments over finite fields that arise
from spreading out. In doing so, we believe that Theorem 1.1 has the potential
to unlock even more geometric information about the parameter space $M_{e}$,
and its cousins. We explain some applications in the remainder of this
introduction.

### The Hodge–Deligne polynomial

As explained in [CNS18, Prop. 3.2.14 of Chapter 2], the Hodge–Deligne
polynomial gives a unique motivic measure
$\mathrm{HD}:K_{0}(\mathrm{Var}_{\mathbf{C}})\to\mathbf{Z}[u,v]$, via

$\mathrm{HD}(X)=\sum_{p,q\geq 0}(-1)^{p+q}h^{p,q}(X)u^{p}v^{q},$

where $h^{p,q}(X)$ are the virtual Hodge numbers of $X$. One has
$\mathrm{HD}(\mathbf{L})=uv$. Moreover, $\mathrm{HD}$ induces a motivic
measure

$\widehat{\mathscr{M}_{\mathbf{C}}}\to\mathbf{Z}[u,v][[(uv)^{-1}]].$

Applying this motivic measure to both sides of the equality in Theorem 1.1, it
is possible to express a positive proportion of the coefficients of
$\mathrm{HD}(M_{e})$ in terms of the Hodge–Deligne polynomial of the motivic
Euler product $\mathfrak{S}(f)$. Following the categorification of
inclusion–exclusion, recently worked out by Das and Howe [DH22], it would be
interesting to discern whether this could give access to homological stability
results for the parameter space $M_{e}$.

While the use of spreading-out and counting procedures usually only gives
access to dimension and irreducibility results, which corresponds to the
dominant term of the Hodge–Deligne polynomial, our motivic approach computes a
positive proportion of the coefficients of this polynomial. We shall take this
point of view further in Corollary 1.6 in the special case $e=1$.

### The space of morphisms

Let $f\in\mathbf{C}[x_{1},\dots,x_{n}]$ be a non-singular homogeneous
polynomial of degree $d$, and let $\tilde{X}\subset\mathbf{P}^{n-1}$ be the
hypersurface it defines. We can also study the space of morphisms of degree
$e$ from $\mathbf{P}^{1}$ to $\tilde{X}$, given by

$\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})=\left\\{\mathbf{g}\in(\mathbf{C}[t]^{n}-\\{0\\})/\mathbf{C}^{\times}:\begin{array}[]{l}\max\deg
g_{i}=e,\leavevmode\nobreak\ \gcd(g_{1},\dots,g_{n})=1,\\\
f(\mathbf{g})=0\end{array}\right\\}.$

The restriction of $(\mathbf{C}[t]^{n}-\\{0\\})/\mathbf{C}^{\times}$ to
polynomials of degree $\leq e$ is isomorphic to $\mathbf{P}^{n(e+1)-1}$, and
so the expected dimension of $\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})$ is
$n(e+1)-1-(de+1)=\mu(e)-1,$ because it is cut out by $de+1$ equations.

Our next result is deduced from Theorem 1.1 using inclusion–exclusion and
supplies a similar result for the space of morphisms
$\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})$.

######  Theorem 1.3.

Let $f\in\mathbf{C}[x_{1},\dots,x_{n}]$ be a non-singular homogeneous
polynomial of degree $d\geq 3$, defining a hypersurface
$\tilde{X}\subset\mathbf{P}^{n-1}$. Assume $n>2^{d}(d-1)$ and let $e\geq 1$.
Then

$[\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})]=\frac{\mathbf{L}^{\mu(e)-1}}{1-\mathbf{L}^{-1}}\left(\prod_{v\in\mathbf{P}^{1}}c_{v}+S_{e}\right)$

in $\widehat{\mathscr{M}_{\mathbf{C}}}$, where

$c_{v}=(1-\mathbf{L}^{-1})\frac{[\tilde{X}]}{\mathbf{L}^{n-2}}$

and $S_{e}$ is an error term satisfying

$w(S_{e})\leq 4-\left(\frac{n-2^{d}(d-1)}{2^{d-1}(d-1)}\right)(e+1).$

### Irreducibility and dimension of the space of morphisms

We shall describe the properties of the weight function
$w:\mathscr{E}xp\mathscr{M}_{\mathbf{C}}\to\mathbf{Z}$ in Section 4. One of
its key properties is the identity $w([V])=2\dim V$, for any
$\mathbf{C}$-variety $V$, which is proved in Lemma 4.6. In fact, by Lemma 4.8,
we have

$w([V]-[W])\leq 2r-1,$

whenever $V,W$ are irreducible $\mathbf{C}$-varieties of equal dimension $r$.
Rather than apply the statement of Theorem 1.3, we shall instead invoke its
proof and the expression (8.2) in particular. Lemma 6.16 and Corollary 6.15
imply that $\Lambda_{N}(f,\infty)\subset\mathbf{A}^{Nn}$ is irreducible and of
dimension $N(n-1)$. Hence $\sigma_{\infty}(f)=1+E_{1}$ in (8.2), with
$w(E_{1})\leq-1$. Moreover, it follows from Remark 6.6 that
$\mathfrak{S}(f)=1+E_{2}$ in $\mathscr{E}xp\mathscr{M}_{\mathbf{C}}$, where
$w(E_{2})<0$, and

$\frac{(1-\mathbf{L}^{-(n-d)})(1-\mathbf{L}^{-(n-d)+1})}{1-\mathbf{L}^{-1}}=1+E_{3},$

where $w(E_{3})\leq-2$. Hence the following result is an easy consequence of
(8.2) in the proof of Theorem 1.3

######  Corollary 1.4.

Let $f\in\mathbf{C}[x_{1},\dots,x_{n}]$ be a non-singular homogeneous
polynomial of degree $d\geq 3$, defining a hypersurface
$\tilde{X}\subset\mathbf{P}^{n-1}$. Assume $n>2^{d}(d-1)$ and let $e\geq 1$.
Then $\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})$ is an irreducible variety of
dimension $\mu(e)-1$.

This refines [BS23, Thm. 1.1], which achieves the same conclusion under the
more stringent assumption $n>2^{d}\left(d-\frac{1}{2}\right)$.

### Weak approximation over $\mathbf{C}(t)$

Given a smooth hypersurface $\tilde{X}\subset\mathbf{P}^{n-1}$ and points
$p_{1},\dots,p_{m}\in\mathbf{P}^{1}$ and $x_{1},\dots,x_{m}\in\tilde{X}$, a
natural generalisation involves studying the space
$\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X};p_{1},\dots,p_{m};x_{1},\dots,x_{m})$
of degree $e$ morphisms $g:\mathbf{P}^{1}\to\tilde{X}$, such that
$g(p_{i})=x_{i}$ for $1\leq i\leq m$. Assuming that $n>2^{d}(d-1)$ and $e$ is
large enough in terms of $m$, the methods of this paper are robust enough to
study the class of this variety in $\widehat{\mathscr{M}_{\mathbf{C}}}$, which
would yield an analogue of Corollary 1.4 for
$\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X};p_{1},\dots,p_{m};x_{1},\dots,x_{m})$.
This would address a question raised by Will Sawin in his lecture at the Banff
workshop “Geometry via Arithmetic” in July 2021, who highlighted that methods
from algebraic geometry break down when $m$ is allowed to be arbitrary, even
for generic hypersurfaces.

The study of
$\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X};p_{1},\dots,p_{m};x_{1},\dots,x_{m})$
can also be interpreted as a weak approximation problem over $\mathbf{C}(t)$,
a topic that was studied by Hassett and Tschinkel [HT06] for rationally
connected varieties. Whereas Hassett and Tschinkel are only able to treat weak
approximation at the places of good reduction, our approach would allow one to
work with arbitrary places.

### Motivic Manin–Peyre

Our work allows us to address a question of Peyre [Pey20, Question 5.4] about
the convergence of

$[\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})]\mathbf{L}^{-e(n-d)},$

for a smooth hypersurface $\tilde{X}\subset\mathbf{P}^{n-1}$ of degree $d$
that is defined over $\mathbf{C}$. If $n>2^{d}(d-1)$, then Theorem 1.3 shows
that this sequence converges in the weight topology to

$\frac{\mathbf{L}^{n-2}}{1-\mathbf{L}^{-1}}\prod_{v\in\mathbf{P}^{1}}c_{v},$

where

$c_{v}=(1-\mathbf{L}^{-1})\frac{[\tilde{X}]}{\mathbf{L}^{n-2}}.$

This is consistent with a motivic analogue of the Manin–Peyre conjecture
described by Faisant [Fai22, p. 3], who suggests that the limit can be
interpreted as an adelic volume

$\frac{\mathbf{L}^{\dim(\tilde{X})}}{(1-\mathbf{L}^{-1})^{\mathrm{rk}\mathrm{Pic}(\tilde{X})}}\prod_{v\in\mathbf{P}^{1}}\mathfrak{c}_{v},$

where
$\mathfrak{c}_{v}=(1-\mathbf{L}^{-1})^{\mathrm{rk}\mathrm{Pic}(\tilde{X})}[\tilde{X}]\mathbf{L}^{-\dim\tilde{X}},$
at all but finitely many places. Indeed, we have $\dim(\tilde{X})=n-2$ and
$\mathrm{rk}\mathrm{Pic}(\tilde{X})=1$, by the Lefschetz hyperplane theorem.

### The variety of lines on hypersurfaces

Our method is robust enough to give non-trivial results already in the case
$e=1$ of lines. Let $\tilde{X}\subset\mathbf{P}^{n-1}$ as above be a smooth
hypersurface of degree $d$, and let $F_{1}(\tilde{X})$ be the Fano variety of
lines associated to $\tilde{X}$. We shall prove the following result in
Section 9.

######  Theorem 1.5.

Let $\tilde{X}\subset\mathbf{P}^{n-1}$ be a smooth hypersurface of degree
$d\geq 3$ with $n>2^{d}(d-1)$. Then

$\displaystyle[F_{1}(\tilde{X})]$
$\displaystyle=\frac{\mathbf{L}^{-d}[\tilde{X}]^{2}-\mathbf{L}^{n-d-2}[\tilde{X}]}{1+\mathbf{L}^{-1}}+\widehat{R}_{n,d}$

in $\widehat{\mathscr{M}_{\mathbf{C}}}$, where

$w(\widehat{R}_{n,d})\leq 4n-2d-6-\frac{n-2^{d}(d-1)}{2^{d-2}}.$

As a consequence, in Section 9 we may compute a positive proportion of the
coefficients of the Hodge–Deligne polynomial of $F_{1}(\tilde{X})$.

######  Corollary 1.6.

Let $\tilde{X}\subset\mathbf{P}^{n-1}$ be a smooth hypersurface of degree
$d\geq 3$, with $n>2^{d}(d-1)$. Let $p,q\in\mathbf{Z}$ such that $p,q\leq
2n-5-d$ and

$p+q>4n-2d-6-\frac{n-2^{d}(d-1)}{2^{d-2}}.$

Then

$h^{p,q}(F_{1}(\tilde{X}))=\begin{cases}\left\lfloor\frac{2n-d-5-p}{2}\right\rfloor+1&\text{
if}\ p=q,\\\ 0&\text{ otherwise.}\end{cases}$

Let us now discuss in a bit more detail the case $d=3$, that is, when
$\tilde{X}$ is a smooth cubic hypersurface. In this case, $F_{1}(\tilde{X})$
is known to be a smooth variety of dimension $2n-8$ as soon as $n\geq 4$, by
work of Altman and Kleiman [AK77]. Galkin and Schinder [GS14] have established
a relationship between the classes $[\tilde{X}]$ and $[F_{1}(\tilde{X})]$ in
$K_{0}(\mathrm{Var}_{\mathbf{C}})$: it follows from [GS14, Thm. 5.1] that

(1.4)
$\mathbf{L}^{2}[F_{1}(\tilde{X})]=[\mathrm{Sym}^{2}\tilde{X}]-(1+\mathbf{L}^{n-2})[\tilde{X}].$

In fact, in the cubic case, Corollary 1.6 can be deduced from [GS14, Thm.
6.1]. (See also Eq. (4.21) in the book by Huybrechts [Huy23]).

On taking $d=3$ in Theorem 1.5, we obtain

$\displaystyle\mathbf{L}^{2}[F_{1}(\tilde{X})]$
$\displaystyle=\frac{\mathbf{L}^{-1}[\tilde{X}]^{2}-\mathbf{L}^{n-3}[\tilde{X}]}{1+\mathbf{L}^{-1}}+\widehat{R}_{n},$

if $n\geq 17$, where $w(\widehat{R}_{n})\leq\frac{7}{2}n.$ Note that
$w(\mathbf{L}^{2}[F_{1}(\tilde{X})])=4+2(2n-8)=4n-12$, and so we do indeed get
non-trivial information. Comparing it with (1.4), and assuming $n\geq 17$,
this results in the expression

(1.5)
$[\mathrm{Sym}^{2}\tilde{X}]=\frac{\mathbf{L}^{-1}[\tilde{X}]^{2}-\mathbf{L}^{n-3}[\tilde{X}]}{1+\mathbf{L}^{-1}}+\mathbf{L}^{n-2}[\tilde{X}]+\widehat{R^{\prime}}_{n},$

where $w(\widehat{R^{\prime}}_{n})\leq\frac{7}{2}n$. In general there seems no
reason to expect a relation between the class of $\mathrm{Sym}^{2}\tilde{X}$
and the classes $[\tilde{X}]^{2}$ and $[\tilde{X}]$ in the Grothendieck group,
but (1.5) gives a good approximation of the class of
$\mathrm{Sym}^{2}\tilde{X}$ in the weight topology. In Remark 9.2 we use work
of Burillo [Bur90] to calculate part of the Hodge–Deligne polynomial of
$\mathrm{Sym}^{2}\tilde{X}$ and check that Corollary 1.6 is consistent with
(1.4) when $n\geq 17$.

It would be interesting to see whether our work could shed light on recent
questions of Popov [Pop18] around the class in
$K_{0}(\mathrm{Var}_{\mathbf{C}})$ of the moduli space that parameterises
twisted cubics in smooth cubic hypersurfaces
$\tilde{X}\subset\mathbf{P}^{n-1}$.

######  Acknowledgements.

The authors would like to thank Yohan Brunebarbe, Antoine Chambert-Loir, Loïs
Faisant and Will Sawin for useful discussions. M.B. received funding from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie Grant agreement No. 893012. T.B. was supported by FWF
grant P 36278 and by a grant from the Institute for Advanced Study School of
Mathematics.

##  2\. The Grothendieck ring of varieties with exponentials

In this section we define the Grothendieck ring of varieties with
exponentials, relative to an arbitrary noetherian scheme $S$. The Grothendieck
group of varieties with exponentials $K_{0}(\mathrm{ExpVar}_{S})$ is defined
by generators and relations. Generators are pairs $(X,f)$, where $X$ is a
variety over $S$ and $f:X\to\mathbf{A}^{1}$ is a morphism. We call such a pair
a variety with exponential. Relations are the following:

$(X,f)-(Y,f\circ u)$

whenever $X$, $Y$ are $S$-varieties, $f\colon X\to\mathbf{A}^{1}$ a morphism,
and $u\colon Y\to X$ is an $S$-isomorphism;

$(X,f)-(Y,f|_{Y})-(U,f|_{U})$

whenever $X$ is an $S$-variety, $f\colon X\to\mathbf{A}^{1}$ a morphism, $Y$ a
closed subscheme of $X$ and $U=X\setminus Y$ its open complement; and

$(X\times_{\mathbf{Z}}\mathbf{A}^{1},\mathrm{pr}_{2})$

where $X$ is a $S$-variety and $\mathrm{pr}_{2}$ is the second projection. We
will write $[X,f]$ (or $[X,f]_{S}$ if we want to keep track of the base scheme
$S$) for the class in $K_{0}(\mathrm{ExpVar}_{S})$ of a pair $(X,f)$. The
product
$[X,f][Y,g]=[X\times_{S}Y,f\circ\mathrm{pr}_{1}+g\circ\mathrm{pr}_{2}]$ endows
$K_{0}(\mathrm{ExpVar}_{S})$ with a ring structure. We denote by $\mathbf{L}$,
or $\mathbf{L}_{S}$, the class of $[\mathbf{A}^{1}_{S},0]$ in
$K_{0}(\mathrm{ExpVar}_{S})$. As for the usual Grothendieck ring
$K_{0}(\mathrm{Var}_{S})$, we may invert $\mathbf{L}$, which gives us a ring
denoted by $\mathscr{E}xp\mathscr{M}_{S}$. As proved in [CLL16, Lemma 1.1.3],
the natural morphism $K_{0}(\mathrm{Var}_{S})\to K_{0}(\mathrm{ExpVar}_{S})$
given by sending $[X]$ to $[X,0]$ is injective, and the same is true after
inverting $\mathbf{L}$.

Any morphism $u:T\to S$ of noetherian schemes naturally induces a group
homomorphism

$u_{!}:K_{0}(\mathrm{ExpVar}_{T})\to K_{0}(\mathrm{ExpVar}_{S})$

and a ring homomorphism

$u^{*}:K_{0}(\mathrm{ExpVar}_{S})\to K_{0}(\mathrm{ExpVar}_{T}).$

The latter endows $K_{0}(\mathrm{ExpVar}_{T})$ with a
$K_{0}(\mathrm{ExpVar}_{S})$-module structure: in particular, this allows us
to write expressions involving elements of both rings, which will implicitly
be viewed as living in $K_{0}(\mathrm{ExpVar}_{T})$.

###  2.1. Functional interpretation

An element $\varphi\in K_{0}(\mathrm{ExpVar}_{S})$ may be interpreted as a
motivic function with source $S$. More precisely, for every point $s\in S$
with residue field $k(s)$, denote by $\varphi(s)$ the element $s^{*}\varphi\in
K_{0}(\mathrm{ExpVar}_{k(s)})$. Then the following lemma says that a motivic
function is determined by its values:

######  Lemma 2.1.

Let $\varphi\in K_{0}(\mathrm{ExpVar}_{S})$. Assume that $\varphi(s)=0$ for
all $s\in S$. Then $\varphi=0$.

###### Proof.

See [CLL16, Lemma 1.1.8]. ∎

######  Remark 2.2.

Let $k$ be a field. Using the relations, we see that
$[\mathbf{A}^{1},\lambda\mathrm{id}]=0$ in $K_{0}(\mathrm{ExpVar}_{k})$ for
any non-zero $\lambda\in k$. More generally, for any variety $X$ over $k$ with
a morphism $u:X\to\mathbf{G}_{m}$, we have

$[X\times\mathbf{A}^{1},(x,t)\mapsto u(x)t]=0$

in $K_{0}(\mathrm{ExpVar}_{k})$. Indeed, using Lemma 2.1 we see that this
holds already in $K_{0}(\mathrm{ExpVar}_{X})$.

######  Lemma 2.3.

Let $V$ be a finite-dimensional $k$-vector space and $f:V\to k$ a linear form.
Then

$[V,f]=\begin{cases}\mathbf{L}^{\dim V}&\text{if}\ f=0,\\\
0&\text{otherwise.}\end{cases}$

###### Proof.

[CLL16, Lemma 1.1.11]. ∎

###  2.2. Interpretation as exponential sums

Let $\psi:\mathbf{F}_{q}\to\mathbf{C}^{*}$ be a non-trivial additive
character. Then there is a ring homomorphism

$K_{0}(\mathrm{ExpVar}_{\mathbf{F}_{q}})\to\mathbf{C},$

given by

$[X,f]\mapsto\sum_{x\in X(\mathbf{F}_{q})}\psi(f(x)).$

In general, even when the ground field $k$ is not finite, we will think of the
elements of $\mathrm{ExpVar}_{k}$ and $\mathscr{E}xp\mathscr{M}_{k}$ as
exponential sums.

###  2.3. A cohomological realisation

Let $k$ be a finite field, let $\ell\neq\mathrm{char}\,k$ be a prime, and let
$k^{s}$ be a separable closure of $k$. We denote
$G_{k}=\mathrm{Gal}(k^{s}/k)$. Consider the category
$\mathrm{Rep}_{G_{k}}\mathbf{Q}_{\ell}$ of continuous
$\mathbf{Q}_{\ell}$-representations of $G_{k}$ and the corresponding
Grothendieck ring $K_{0}(\mathrm{Rep}_{G_{k}}\mathbf{Q}_{\ell})$. We fix a
non-trivial additive character $\psi:k\to\mathbf{C}^{*}$, and denote by
$\mathcal{L}_{\psi}$ the corresponding Artin-Schreier sheaf on
$\mathbf{A}^{1}_{k}$. There is a ring homomorphism

$K_{0}(\mathrm{ExpVar}_{k})\to K_{0}(\mathrm{Rep}_{G_{k}}\mathbf{Q}_{\ell}),$

given by

$[X,f]\mapsto\sum_{i\geq
0}(-1)^{i}[H^{i}_{\text{ét},c}(X\times_{k}k^{s},f^{*}\mathcal{L}_{\psi})].$

This motivic measure provides a dictionary between some computations in this
paper and those carried out in [BS20].

###  2.4. Motivic functions and integrals

In this section, we follow the ideas of [CLL16, 1.2.1], except that we adapt
the notation and normalisations to make them more convenient for our setting.

The motivic functions we consider are motivic analogues of compactly supported
and locally constant functions on the locally compact field $k((t^{-1}))$,
where $k$ is a finite field. For such a function $\varphi$, there exist
integers $M\geq N$ such that $\varphi$ is zero outside $t^{M}k[[t^{-1}]]$, and
such that $\varphi$ is invariant modulo $t^{N}k[[t^{-1}]]$, so that $\varphi$
may be seen as a function on the quotient $t^{M}k[[t^{-1}]]/t^{N}k[[t^{-1}]]$.
We then say that $\varphi$ is of level $(M,N)$. The latter can be endowed with
the structure of an affine space of dimension $M-N$ over the field $k$,
through the identification

$\begin{array}[]{ccc}t^{M}k[[t^{-1}]]/t^{N}k[[t^{-1}]]&\to&\mathbf{A}_{k}^{M-N}(k)\\\
a_{M}t^{M}+a_{M-1}t^{M-1}+\dots+a_{N+1}t^{N+1}\bmod{t^{N}k[[t^{-1}]]}&\mapsto&(a_{M},\dots,a_{N+1})\end{array}.$

######  Notation 2.4.

For any field $k$ and for $M\geq N$ integers, we denote by
$\mathbf{A}^{(M,N)}_{k}$ the affine space $\mathbf{A}^{M-N}_{k}$, interpreted
as the domain of definition for functions which are zero outside
$t^{M}k[[t^{-1}]]$ and invariant modulo $t^{N}k[[t^{-1}]]$ as above. More
generally, for $n\geq 1$ an integer, we denote by $\mathbf{A}^{n(M,N)}_{k}$
the affine space $(\mathbf{A}^{(M,N)}_{k})^{n}$. If $S$ is a variety over $k$,
we denote by $\mathbf{A}^{n(M,N)}_{S}$ the extension
$\mathbf{A}^{n(M,N)}_{k}\times_{k}S$.

Writing $K_{\infty}=k((t^{-1}))$, we denote by
$\mathscr{F}_{S}(K_{\infty}^{n},M,N)$ the ring
$\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{n(M,N)}_{S}}$ and interpret it as the
ring of $S$-families of motivic functions of level $(M,N)$ in $n$ variables.
In the same way that is explained in [CLL16, 1.2.3], as $M$ and $N$ vary, the
rings $\mathscr{F}_{S}(K_{\infty}^{n},M,N)$ fit into a directed system with
direct limit denoted $\mathscr{F}_{S}(K_{\infty}^{n})$, the total ring of
$S$-families of motivic functions in $n$ variables.

If we have an element $\varphi\in\mathscr{F}_{S}(K_{\infty}^{n})$ we define
its integral in the following way: pick a pair $(M,N)$ such that
$\varphi\in\mathscr{F}_{S}(K_{\infty}^{n},M,N)$, and we put

$\int\varphi:=\mathbf{L}^{(N+1)n}[\varphi]_{S},$

where $[\varphi]_{S}$ is the image of $\varphi$ in
$\mathscr{E}xp\mathscr{M}_{S}$ via the forgetful morphism

$\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{n(M,N)}_{S}}\to\mathscr{E}xp\mathscr{M}_{S}.$

We may also write
$\int\varphi=\int_{\mathbf{x}}\varphi(\mathbf{x})\mathrm{d}\mathbf{x}$, or
simply $\int\varphi=\int_{\mathbf{x}}\varphi(\mathbf{x})$, if we want to
stress with respect to which variables the integral is taken. Analogously to
[CLL16, 1.2.3], this definition does not depend on the choice of $(M,N)$
(since the integral remains the same if we pick a larger $M$ or a smaller
$N$), and defines an $\mathscr{E}xp\mathscr{M}_{S}$-linear map

$\int:\mathscr{F}_{S}(K_{\infty})\to\mathscr{E}xp\mathscr{M}_{S}.$

######  Example 2.5.

Write $\mathbf{T}=t^{-1}k[[t^{-1}]]$, the analogue of the fundamental compact
interval $[0,1]$ in the function field circle method literature. Let
$\varphi=\mathbf{1}_{\mathbf{T}^{n}}$ be the characteristic function of
$\mathbf{T}^{n}$. Picking $M=N=-1$, we get $\int\varphi=1$. In other words,
with this normalisation, the motivic volume of $\mathbf{T}^{n}$ is 1.

######  Remark 2.6 (Change of variables).

Let $\varphi\in\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{n(M,N)}_{S}}$ and assume
that we want to perform a change of variables $\mathbf{x}\mapsto
t^{p}\mathbf{x}=:\mathbf{y}$ for some integer $p\in\mathbf{Z}$. This change of
variable induces an isomorphism

$\mathbf{A}^{n(M,N)}_{S}\to\mathbf{A}^{n(M+p,N+p)}_{S},$

which gives us an isomorphism between the corresponding Grothendieck rings,
sending $\varphi$ to the function
$\psi:\mathbf{y}\mapsto\varphi(t^{-p}\mathbf{y})$. Moreover, this induces an
isomorphism of the underlying $S$-varieties with exponentials, which implies
the equality $[\varphi]_{S}=[\psi]_{S}$. We may conclude that we have the
change of variable formula

$\displaystyle\int_{\mathbf{x}}\varphi(\mathbf{x})\mathrm{d}\mathbf{x}=\mathbf{L}^{n(N+1)}[\varphi]_{S}=\mathbf{L}^{n(N+1)}[\psi]_{S}$
$\displaystyle=\mathbf{L}^{-pn}\int_{\mathbf{y}}\psi(\mathbf{y})\mathrm{d}\mathbf{y}$
$\displaystyle=\mathbf{L}^{-pn}\int_{\mathbf{y}}\varphi(t^{-p}\mathbf{y})\mathrm{d}\mathbf{y}.$

###  2.5. Motivic Euler products

We will use the motivic Euler products of [Bil23, Section 3], more
specifically, the variant where the coefficients are varieties with
exponentials. We recall its definition here. Let $X$ be a variety over a field
$k$, and let

$(A_{i})_{i\geq 1}=(X_{i},f_{i}:X_{i}\to\mathbf{A}^{1})_{i\geq 1}$

be a family of quasi-projective varieties with exponentials over $X$. Let
$n\geq 1$ be an integer, and let $\omega=(n_{i})_{i\geq 1}$ be a partition of
$n$, with $n_{i}$ being the number of occurrences of $i$, so that
$n=\sum_{i\geq 1}in_{i}$. We define the variety with exponential
$\mathrm{Conf}^{\omega}(X_{i},f_{i})_{i\geq 1}$ in the following way: consider
the product

$\prod_{i\geq 1}X_{i}^{n_{i}}$

together with the morphism to $\prod_{i\geq 1}X^{n_{i}}$ induced by the
structural morphisms $X_{i}\to X$. We denote by $\left(\prod_{i\geq
1}X_{i}^{n_{i}}\right)_{*,X}$ the open subset lying above the complement of
the big diagonal in $\prod_{i\geq 1}X^{n_{i}}$ (that is, points with no two
coordinates being equal). Then the variety
$\mathrm{Conf}^{\omega}(X_{i})_{i\geq 1}$underlying
$\mathrm{Conf}^{\omega}(X_{i},f_{i})_{i\geq 1}$ is defined to be the quotient
of $\left(\prod_{i\geq 1}X_{i}^{n_{i}}\right)_{*,X}$ by the natural
permutation action of the product of symmetric groups $\prod_{i\geq
1}\mathfrak{S}_{i}$. The corresponding morphism
$f^{\omega}:\mathrm{Conf}^{\omega}(X_{i})_{i\geq 1}\to\mathbf{A}^{1}$ is
induced in the obvious way by the natural $\prod_{i\geq
1}\mathfrak{S}_{n_{i}}$-invariant morphism $\prod_{i\geq
1}f_{i}^{n_{i}}:\prod_{i\geq 1}X_{i}^{n_{i}}\to\mathbf{A}^{1}$, given by

$(x_{i,1},\dots,x_{i,n_{i}})_{i\geq 1}\mapsto\sum_{i\geq
1}\left(f_{i}(x_{i,1})+\dots+f(x_{i,n_{i}})\right).$

We define the motivic Euler product by

$\prod_{x\in X}\left(1+\sum_{i\geq 1}A_{i,x}T^{i}\right):=1+\sum_{n\geq
1}\left(\sum_{\omega}[\mathrm{Conf}^{\omega}(X_{i},f_{i})_{i\geq
1}]\right)T^{n}\in K_{0}(\mathrm{ExpVar}_{k})[[T]],$

where the inner sum is over partitions $\omega$ of $n$. Note that the left-
hand side is only a notation for the series on the right-hand side. In
[Bil23], it is shown that this notion of motivic Euler product satisfies many
good properties, in particular in terms of multiplicativity.

##  3\. The motivic circle method

We start by recalling the main ideas behind the function field version of the
circle method. Let $k=\mathbf{F}_{q}$ be a finite field. The field $k$ is
endowed with an additive character $\psi:k\to\mathbf{C}^{*}$ defined by

$x\mapsto\exp\left(\frac{2i\pi}{p}\mathrm{Tr}_{\mathbf{F}_{q}/\mathbf{F}_{p}}(x)\right).$

The role of the compact unit interval $[0,1]$ is played by the set

$\mathbf{T}=t^{-1}\mathbf{F}_{q}[[t^{-1}]]=\\{\alpha\in\mathbf{F}_{q}((t^{-1})),\
\operatorname{ord}(\alpha)\leq-1\\}.$

The field $k((t^{-1}))$ is locally compact, and we may normalise its Haar
measure so that $\int_{\mathbf{T}}\mathrm{d}\alpha=1$. The igniting spark of
the circle method over $\mathbf{F}_{q}(t)$ is the identity

$\int_{\mathbf{T}}\psi(\mathrm{res}(x\alpha))\mathrm{d}\alpha=\begin{cases}1&\text{
if}\ x=0,\\\ 0&\text{ otherwise},\end{cases}$

where, for a Laurent series $f\in k((t^{-1}))$, we denote by $\mathrm{res}(f)$
the coefficient of $t^{-1}$ in $f$.

If we want to count solutions of some polynomial equation over $k$, we may use
this identity to rewrite the quantity we are interested in as the integral

$\int_{\mathbf{T}}S(\alpha)\mathrm{d}\alpha$

of an appropriate exponential sum $S(\alpha)$. The method then proceeds, as in
the classical case, with cutting up $\mathbf{T}$ into major and minor arcs.
The contribution of the major arcs is rewritten as a product of local factors,
but one requires a suitably strong upper bound for the contribution from the
minor arcs.

The main obstruction in passing from $k=\mathbf{F}_{q}$ to $k=\mathbf{C}$ is
that we lose local compactness, and there is no theory of integration in the
classical sense that we could use to reproduce the above steps. This is why we
need to introduce ideas from motivic integration: exponential sums will be
replaced by elements in the Grothendieck rings of varieties with exponentials,
bearing in mind Section 2.2. Integrals will be replaced by the operation
defined in Section 2.4. We will use the cut-and-paste relations in the
Grothendieck ring of varieties to decompose our expression into major and
minor arcs. The decomposition into local factors of the major arc contribution
will be done via the notion of motivic Euler products, while the minor arcs
will be bounded in a topology defined using Hodge theory. Once this
translation into the motivic setting has been properly made, the motivic
circle method will bear many similarities with the function field circle
method, as implemented by Lee [Lee13].

######  Notation 3.1.

For any integer $m$, let $\mathrm{Poly}_{\leq m}$ be the space of polynomials
of degree $\leq m$ with coefficients in $\mathbf{C}$. We let
$\mathrm{MPoly}_{m}$ denote the space of monic polynomials of degree exactly
$m$ with coefficients in $\mathbf{C}$. Both spaces are simply affine spaces
over $\mathbf{C}$; the first one has dimension $m+1$ and the second one has
dimension $m$. We let $\mathbf{T}$ be the space of elements
$\alpha\in\mathbf{C}((t^{-1}))$ such that $\operatorname{ord}(\alpha)\leq-1$.
For an element $f\in\mathbf{C}((t^{-1}))$, we denote by $\\{f\\}\in\mathbf{T}$
its fractional part.

###  3.1. Approximation by rational functions

The classical circle method relies greatly on approximation of real numbers by
rationals. In the same way, its function field version depends on
approximation of power series in $\mathbf{T}$ by rational functions.

When we want to approximate $\alpha\in\mathbf{T}$ by a rational function
$\frac{h_{1}}{h_{2}}$, there are two parameters:

  * $\bullet$

the bound on the degree of $h_{2}$;

  * $\bullet$

the precision of the approximation, as quantified by a bound on the order of
$\alpha h_{2}-h_{1}$.

Thus, we define

$\mathbf{T}_{m,s}=\\{\alpha\in\mathbf{T},\ \exists h_{2}\in\mathrm{Poly}_{\leq
m},h_{1}\in\mathrm{Poly}_{<\deg h_{2}},\ \ \operatorname{ord}(\alpha
h_{2}-h_{1})<-s\\},$

for $m\geq 0$ and $s\geq 1$.

We can reformulate the definition of $\mathbf{T}_{m,s}$ purely in terms of
linear algebra, as follows. We have $\alpha\in\mathbf{T}_{m,s}$ if and only if
there exists a polynomial $c_{0}+c_{1}t+\dots+c_{m}t^{m}$ such that the
coefficients of $t^{-1},\dots,t^{-s}$ in the series

$\left(\sum_{i\geq 1}b_{i}t^{-i}\right)(c_{0}+c_{1}t+\dots+c_{m}t^{m})$

are zero. This gives us a linear system of $s$ equations in
$c_{0},\dots,c_{m}$:

$\displaystyle b_{1}c_{0}+b_{2}c_{1}+\dots+b_{m+1}c_{m}$ $\displaystyle=0,$
$\displaystyle b_{2}c_{0}+b_{3}c_{1}+\dots+b_{m+2}c_{m}$ $\displaystyle=0,$
$\displaystyle\vdots$ $\displaystyle
b_{s}c_{0}+b_{s+1}c_{1}+\dots+b_{m+s}c_{m}$ $\displaystyle=0,$

which translates into

$\left(\begin{array}[]{c}c_{0}\\\ \vdots\\\
c_{m}\end{array}\right)\in\mathrm{Ker}\left(\begin{array}[]{cccc}b_{1}&b_{2}&\dots&b_{m+1}\\\
\vdots&&&\vdots\\\ b_{s}&b_{s+1}&\dots&b_{m+s}\\\ \end{array}\right).$

In other words, denoting by $M$ the above $s\times(m+1)$ matrix, we have that
$\alpha\in\mathbf{T}_{m,s}$ if and only if $M$ has non-trivial kernel. Our
first observation, is that this is always satisfied if $m+1>s$, that is, if
$m\geq s$. In other words, whenever $m\geq s$ we have
$\mathbf{T}_{m,s}=\mathbf{T}$. This gives us the functional version of
Dirichlet approximation.

######  Lemma 3.2 (Dirichlet approximation).

Let $\alpha\in\mathbf{T}$. Then for any $m\geq 1$ there exist polynomials
$h_{1},h_{2}$ such that $\deg h_{1}<\deg h_{2}\leq m$ and
$\operatorname{ord}(\alpha h_{2}-h_{1})<-m.$

In general, the larger $s$ becomes, the more equations we impose on the
coefficients $c_{0},\dots,c_{m}$ of $h_{2}$, and we get a more and more
restrictive set.

In the motivic setting, there is an extra parameter, namely the number of
coefficients of $\alpha$ we actually take into account in our analysis. More
precisely, we will be looking at expressions of the form $\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))$ where $f$ is a polynomial of degree $d$ and
$g_{1},\dots,g_{n}$ have degrees at most $e$, and thus only the first $de+1$
coefficients of $\alpha$ will matter. If we want to make do with only
considering these coefficients, i.e. if we don’t want the matrix $M$ to
contain any additional coefficients of $\alpha$, we would need to impose the
condition $s+m\leq de+1.$ Motivated by the extra constraint, we define

$A^{de+1}_{m}=\\{(b_{1},\dots,b_{de+1})\in\mathbf{A}^{de+1}:\sum_{i=1}^{de+1}b_{i}t^{-i}\in\mathbf{T}_{m,de-m+1}\\}.$

######  Remark 3.3.

Note in particular that when $m\geq\frac{de+1}{2}$, then
$de+1-m\leq\frac{de+1}{2}\leq m$. Thus $A^{de+1}_{m}=\mathbf{A}^{de+1}$ in
this case.

######  Remark 3.4.

Note that $A^{de+1}_{m}$ is the space of
$(b_{1},\dots,b_{de+1})\in\mathbf{A}^{de+1}$ such that there exist coprime
polynomials $h_{1},h_{2}$ with $h_{2}$ monic, $\deg h_{1}<\deg h_{2}\leq m$
and $\operatorname{ord}\left(\alpha-\frac{h_{1}}{h_{2}}\right)\leq-de-2+m-\deg
h_{2}$.

######  Remark 3.5.

The sets $A^{de+1}_{m}$ can be stratified into sets
$A^{de+1}_{m,m^{\prime}}\subset A^{de+1}_{m}$ of those $\alpha$ such that
$h_{2}(t)$ is of degree exactly $m^{\prime}\leq m$. We may deduce

$\dim A^{de+1}_{m,m^{\prime}}\leq 2m^{\prime}+(m-m^{\prime})\leq
m+m^{\prime}\leq 2m,$

from which we in particular have that $\dim A^{de+1}_{m}\leq 2m.$

###  3.2. Activation of the circle method

######  Lemma 3.6.

We have

$\displaystyle[M_{e}]=\mathbf{L}^{-de-1}[(\mathrm{Poly}_{\leq
e})^{n}\times\mathbf{A}^{de+1},\mathrm{res}(\alpha f(g_{1},\dots,g_{n}))]$

in the Grothendieck ring $\mathscr{E}xp\mathscr{M}_{\mathrm{Poly}_{\leq
e}^{n}}.$

###### Proof.

Write

$f(g_{1},\dots,g_{n})=c_{0}(\mathbf{g})+c_{1}(\mathbf{g})t+\dots+c_{de}(\mathbf{g})t^{de},$

where $c_{0},\dots,c_{de}$ are regular functions in the coefficients of
$g_{1},\dots,g_{n}$, and

$\alpha=b_{1}t^{-1}+\dots+b_{de+1}t^{-de-1}.$

Then

$\mathrm{res}(\alpha f(g_{1},\dots,g_{n}))=b_{1}c_{0}+\dots+b_{de+1}c_{de}$

is a linear form in the coefficients of $\alpha$.

Consider now a point $\mathbf{g}\in(\mathrm{Poly}_{\leq e})^{n}$. Then on
evaluating the right-hand side at $\mathbf{g}$, we get

$\mathbf{L}^{-de-1}[\mathbf{A}^{de+1}_{k(\mathbf{g})},b_{1}c_{0}(\mathbf{g})+\dots+b_{de+1}c_{de}(\mathbf{g})].$

By Lemma 2.3, this is non-zero if and only if all of the coefficients
$c_{0}(\mathbf{g}),\dots,c_{de}(\mathbf{g})$ are zero. This occurs if and only
if $\mathbf{g}\in M_{e}$, and in this case the expression is equal to 1. We
conclude using Lemma 2.1 with

$\varphi=[M_{e}]-\mathbf{L}^{-de-1}[(\mathrm{Poly}_{\leq
e})^{n}\times\mathbf{A}^{de+1},\mathrm{res}(\alpha f(\mathbf{g}))]$

and $S=(\mathrm{Poly}_{\leq e})^{n}$. ∎

Adopting the notation in Section 2.4, we may also write

$[M_{e}]=\int_{\alpha}[(\mathrm{Poly}_{\leq
e})^{n}\times\mathbf{A}^{(-1,-de-2)},\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))]\mathrm{d}\alpha$

in $\mathscr{E}xp\mathscr{M}_{\mathrm{Poly}_{\leq e}^{n}}$, in order to
emphasise the similarity with the classical circle method. In fact, our plan
is to write

(3.1) $[M_{e}]=N_{\mathrm{major}}+N_{\mathrm{minor}},$

where $N_{\mathrm{major}}$ (resp. $N_{\mathrm{minor}}$) is the contribution of
the major (resp. minor) arcs. We want to compute $N_{\mathrm{major}}$ as
precisely as possible, and find a weight bound on $N_{\mathrm{minor}}$ of the
form $w(N_{\mathrm{minor}})<2\mu(e)$. Let

(3.2) $\tilde{\nu}:=\frac{n-2^{d}(d-1)}{2^{d-2}}>0.$

Although we delay defining the precise major and minor arcs that we work with
until the relevant sections, we proceed by recording the two main steps in the
motivic circle method, whose proofs will occupy the bulk of this paper.

######  Proposition 3.7.

Let $\tilde{\nu}>0$ be given by (3.2) and recall the varieties defined in
(1.2) and (1.3). Then

$N_{\mathrm{major}}=\mathbf{L}^{\mu(e)}\left(\mathfrak{S}(f)\cdot\lim_{N\to\infty}\mathbf{L}^{-(n-1)N}[\Lambda_{N}(f,\infty)]+R_{e}\right),$

where

$w(R_{e})\leq\begin{cases}4-\tilde{\nu}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right)&\text{
if $d\geq 3$,}\\\ \max\left(-\frac{(e+1)(n-2)}{2},\leavevmode\nobreak\
-\frac{n(e+2)}{2}+2e+8\right)&\text{ if $d=2$.}\end{cases}$

and

$\mathfrak{S}(f)=\prod_{x\in\mathbf{A}^{1}}\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)].$

######  Proposition 3.8.

Let $\tilde{\nu}>0$ be given by (3.2). Then

$w(N_{\mathrm{minor}})\leq
2\mu(e)+4-{\tilde{\nu}}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right).$

Note that when $e=1$, corresponding to the case of lines, we will have the
bound $w(N_{\mathrm{minor}})\leq 2\mu(1)-\frac{n-2^{d}d}{2^{d-2}},$ which is
interesting as soon as $n>2^{d}d$, if $d\geq 3$.

###### Proof of Theorem 1.1.

This follows on combining Propositions 3.7 and 3.8 in (3.1). ∎

##  4\. The weight function

Our weight function will be constructed with the aim of being able to prove an
analogue of the Weyl differencing argument [BS20, Proposition 5.5] in our
setting, which we do in Proposition 5.3. Unfortunately, the weight function
from [Bil23] is not in an appropriate form to carry this out. In order to
mimic the steps in the proof of [BS20, Proposition 5.5], which were done using
the properties of cohomology, we need a function that in some sense measures
the weights of the cohomology of the fibres, while the weight function from
[Bil23] used the global notion of weight of a mixed Hodge module. Thus, while
over $\mathbf{C}$ it coincides with the weight function from [Bil23], in the
relative setting we define it rather using the weights of the underlying
variations of Hodge structures (which introduces a shift by the dimension of
the support).

Note that while most of the intermediary estimates in the implementation of
the motivic circle method happen in relative Grothendieck rings and thus use
our new weight, the final results are stated in the Grothendieck ring over
$\mathbf{C}$, where our new weight coincides with the weight from [Bil23].

In Section 4.1, we recall some definitions and properties concerning mixed
Hodge modules and the corresponding Grothendieck rings. In Section 4.2 we
define the weight function at the level of Grothendieck rings of mixed Hodge
modules, and prove its fundamental properties. Then we pass to Grothendieck
rings of varieties in Section 4.3 using the same procedure as in [Bil23].
Finally, in Section 4.4 we explain what we mean by convergence of power series
in the topology defined by the weight.

###  4.1. Mixed Hodge modules

####  4.1.1. The category of mixed Hodge modules

If $S$ is a variety over $\mathbf{C}$, we denote by $\mathrm{MHM}_{S}$ the
abelian category of mixed Hodge modules on $S$ and by
$D^{b}(\mathrm{MHM}_{S})$ its bounded derived category. A morphism $f:S\to T$
between complex varieties induces functors $f_{!}:D^{b}(\mathrm{MHM}_{S})\to
D^{b}(\mathrm{MHM}_{T})$ and $f^{*}:D^{b}(\mathrm{MHM}_{T})\to
D^{b}(\mathrm{MHM}_{S})$.

In the case where $S$ is a point, the category $\mathrm{MHM}_{\mathrm{pt}}$ is
exactly the category of polarisable mixed Hodge structures. For any integer
$d\in\mathbf{Z}$, we denote by
$\mathbf{Q}_{\mathrm{pt}}^{\mathrm{Hdg}}(d)\in\mathrm{MHM}_{\mathrm{pt}}$ the
Hodge structure of type $(-d,-d)$ with underlying vector space $\mathbf{Q}$.
For $d=0$, it will be denoted simply by
$\mathbf{Q}^{\mathrm{Hdg}}_{\mathrm{pt}}$.

Every mixed Hodge module $M$ on a variety $S$ has a well-defined notion of
support, which will will be denoted by $\mathrm{supp}(M)$.

####  4.1.2. The complex $\mathbf{Q}_{X}^{\mathrm{Hdg}}$

For any complex variety $X$, we denote by $a_{X}:X\to\mathrm{Spec}\
\mathbf{C}$ its structural morphism, and by $\mathbf{Q}_{X}^{\mathrm{Hdg}}$
the complex of mixed Hodge modules given by
$a_{X}^{*}\mathbf{Q}_{\mathrm{pt}}^{\mathrm{Hdg}}$. In the case when $X$ is
smooth and connected, the complex of mixed Hodge modules
$\mathbf{Q}_{X}^{\mathrm{Hdg}}$ is concentrated in degree $\dim X$, and
$\mathcal{H}^{\dim X}\mathbf{Q}_{X}^{\mathrm{Hdg}}$ is pure of weight $\dim
X$, given by the pure Hodge module associated to the constant rank one
variation of Hodge structures of weight 0 on $X$.

####  4.1.3. Mixed Hodge modules with monodromy

We denote by $\mathrm{MHM}_{X}^{\mathrm{mon}}$ the category of mixed Hodge
modules $M$ on a complex variety $X$ endowed with commuting actions of a
finite order operator $T_{s}:M\to M$ and a locally nilpotent operator $N:M\to
M(-1)$. The category $\mathrm{MHM}_{X}$ can be identified with a full
subcategory of $\mathrm{MHM}_{X}^{\mathrm{mon}}$ via the functor

$\mathrm{MHM}_{X}\to\mathrm{MHM}_{X}^{\mathrm{mon}},$

sending a mixed Hodge module $M$ to itself with $T_{s}=\mathrm{id}$ and $N=0$.
We refer to [Bil23, Section 4.1.6] for a definition of the twisted external
tensor product ${\kern
1.00006pt\mathop{\boxtimes}\limits^{\scriptscriptstyle{T}}\kern 1.00006pt}$.

####  4.1.4. The weight filtration on Hodge modules

Each $M\in\mathrm{MHM}_{S}$ has a finite increasing weight filtration
$W_{\bullet}M$, the graded parts of which will be denoted
$\mathrm{Gr}^{W}_{\bullet}$. For a bounded complex of mixed Hodge modules
$M^{\bullet}$, we say $M^{\bullet}$ has weight $\leq n$ if
$\mathrm{Gr}^{W}_{i}\mathcal{H}^{j}(M^{\bullet})=0$ for all integers $i$ and
$j$ such that $i>j+n$.

For varieties $X$ and $Y$ over $\mathbf{C}$ we say that a functor
$F:D^{b}(\mathrm{MHM}_{X})\to D^{b}(\mathrm{MHM}_{Y})$ does not increase
weights if for every $n\in\mathbf{Z}$ and every $M^{\bullet}\in
D^{b}(\mathrm{MHM}_{X})$ with weight $\leq n$, the complex $F(M^{\bullet})$ is
also of weight $\leq n$. In particular, for any morphism of complex varieties
$f$, the functors $f_{!}$ and $f^{*}$ do not increase weights (see [Sai90,
(4.5.2)]).

####  4.1.5. Grothendieck ring of mixed Hodge modules

We refer to [Bil23, Section 4.1.7] for the definition of the Grothendieck
rings $K_{0}(\mathrm{MHM}_{X})$ and $K_{0}(\mathrm{MHM}_{X}^{\mathrm{mon}})$.
For every morphism $f:X\to Y$ of complex varieties, the functors $f_{!},f^{*}$
between the corresponding derived categories of mixed Hodge modules induce a
group morphism

$f_{!}:K_{0}(\mathrm{MHM}_{X}^{\mathrm{mon}})\to
K_{0}(\mathrm{MHM}_{Y}^{\mathrm{mon}})$

and a ring morphism

$f^{*}:K_{0}(\mathrm{MHM}_{Y}^{\mathrm{mon}})\to
K_{0}(\mathrm{MHM}_{X}^{\mathrm{mon}}).$

######  Remark 4.1.

Let $\mathrm{HM}_{X}$ be the subcategory of $\mathrm{MHM}_{X}$ of Hodge
modules of pure weight. By decomposing into graded pieces, one can see that
the natural inclusion $K_{0}(\mathrm{HM}_{X})\to K_{0}(\mathrm{MHM}_{X})$
induces an isomorphism.

####  4.1.6. Hodge modules and variations of Hodge structures

Saito’s structure theorem (see e.g. [Sai90, Theorem 3.21]) gives a
correspondence between pure polarisable Hodge modules on a variety $Z$ with
strict support in $Z$ and direct systems of polarisable variations of Hodge
structures (with quasi-unipotent local monodromies) over smooth open subsets
of $Z$, with a shift in weight (by the dimension of the support). Moreover,
this category of Hodge modules is semi-simple, with simple objects coming,
through this correspondence, from variations of Hodge structures for which the
monodromy representation is irreducible [PS08, Theorem 14.37]. We will use
these facts to give a particularly simple presentation of Grothendieck rings
of mixed Hodge modules.

We denote by $\mathrm{VHS}_{X}$ the disjoint union, over all irreducible
closed subvarieties $Z\subset X$, of the direct limit over smooth open dense
$U\subset Z$ of the sets of isomorphism classes of polarisable variations of
pure Hodge structures (satisfying the above condition of having quasi-
unipotent local monodromies with irreducible action) on $U$.

There is a well-defined morphism

$\mathbf{Z}^{(\mathrm{VHS}_{X})}\to K_{0}(\mathrm{MHM}_{X})$

sending a variation of Hodge structures over an open dense subset $U\subset Z$
of an irreducible closed subvariety $Z\subset X$ to the corresponding Hodge
module with strict support in $Z$.

######  Lemma 4.2.

The above morphism is an isomorphism, i.e. $K_{0}(\mathrm{MHM}_{X})$ is freely
generated by the elements of $\mathrm{VHS}_{X}$.

###### Proof.

By remark 4.1, we may replace $K_{0}(\mathrm{MHM}_{X})$ by
$K_{0}(\mathrm{HM}_{X})$. We construct the morphism in the other direction by
using decomposition into simple objects and the above correspondence. ∎

A similar property holds for $K_{0}(\mathrm{MHM}_{X}^{\mathrm{mon}})$,
considering the set $\mathrm{VHS}^{\mathrm{mon}}_{X}$ where the additional
datum of the operators $T_{s}$ and $N$ is added.

###  4.2. Definition of the weight function

We may define a filtration $W_{\leq\bullet}$ on the group
$K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{X})$ in the following way: $W_{\leq
n}K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{X})$ is the subgroup generated by

  * $\bullet$

classes of pure Hodge modules $(M,\mathrm{id},N)$ (i.e. with
$T_{s}=\mathrm{id}$), with irreducible support, of weight $m$ such that
$m-\mathrm{supp}(M)\leq n$, and

  * $\bullet$

classes of pure Hodge modules $(M,T_{s},N)$, with irreducible support, of
weight $m$ such that $m-\mathrm{supp}(M)\leq n-1$.

This dichotomy depending on the form of the semisimple monodromy is of the
same type as in the definition of the weight filtration in [Bil23, Section
4.5.1], which was motivated in particular by the definition of the twisted
exterior product.

Note that by 4.1.6, this corresponds to filtering
$K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{X})$ by the weights of the underlying
variations of Hodge structures, slightly twisted depending on whether the
semi-simple part $T_{s}$ of the monodromy is trivial or not. More precisely,
we may describe the associated gradeds of this filtration in the following
form:

######  Lemma 4.3.

We have

$\mathrm{Gr}^{W}_{n}K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{X})=W_{\leq
n}K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{X})/W_{\leq
n-1}K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{X})=\mathbf{Z}^{(VHS^{\mathrm{mon}}_{n,X})}$

where $\mathrm{VHS}^{\mathrm{mon}}_{n,X}$ is the subset of
$\mathrm{VHS}^{\mathrm{mon}}_{X}$ given by variations of Hodge structures
which have weight $n-1$ and correspond to Hodge modules with non-trivial
$T_{s}$, together with those of weight $n$ corresponding to Hodge modules with
trivial $T_{s}$.

###### Proof.

This follows from 4.1.6, using the fact that the shift in weight between Hodge
modules and the corresponding variations of Hodge structures is of exactly the
dimension of the support. ∎

We then define

$w_{X}(\mathfrak{a})=\min\\{n:\mathfrak{a}\in W_{\leq
n}K_{0}(\mathrm{MHM}_{X}^{\mathrm{mon}})\\}.$

This weight function satisfies the following properties.

######  Proposition 4.4.

Let $S$ be a complex variety.

  1. (1)

Let $(\mathfrak{a}_{i})_{i\in I}$ be a finite family of elements of
$K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{S})$. Then

$w_{S}\left(\sum_{i\in I}\mathfrak{a}_{i}\right)\leq\max_{i\in
I}w_{S}(\mathfrak{a}_{i}).$

  2. (2)

Let $p:T\to S$ be a dominant morphism with fibres of dimension $\leq d$. Let
$\mathfrak{a}\in K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{T})$. Then there exists an
open subset $S_{0}\subset S$ such that, denoting by $\mathfrak{b}$ the
pullback of $\mathfrak{a}$ to $p^{-1}(S_{0})$, we have

$w_{S_{0}}(p_{!}\mathfrak{b})\leq w_{p^{-1}(S_{0})}(\mathfrak{b})+2d.$

If $S$ is a point, we have

$w(p_{!}\mathfrak{a})\leq w_{T}(\mathfrak{a})+2d.$

  3. (3)

Let $T\subset S$ be a locally closed subset and $i:T\to S$ the corresponding
inclusion morphism. For every $\mathfrak{a}\in
K_{0}(\mathrm{MHM}^{\mathrm{mon}}_{T})$, we have

$w_{S}(i_{!}\mathfrak{a})\leq w_{T}(\mathfrak{a}).$

######  Remark 4.5.

Property (2) differs from the corresponding property of the weight in [Bil23,
Lemma 4.5.1.3(d)], where the upper bound holds with $d$ in place of $2d$.

###### Proof of Proposition 4.4.

  1. (1)

This follows immediately from the definition.

  2. (2)

For clarity, we first do the proof without the extra monodromy operators.

Let $V$ be a variation of Hodge structures of weight $n$ over an open subset
$U$ of a subvariety $Z$ of $T$, and let $M_{V}$ be the corresponding Hodge
module over $Z$.

We know that $p_{!}M_{V}$ is a complex of mixed Hodge modules over $S$, of
amplitude $\leq d$. Since the functor $p_{!}$ does not increase weights, this
complex moreover has weight $\leq n+\dim Z$, which means that

$\mathrm{Gr}_{i}^{W}\mathcal{H}^{j}(p_{!}M_{V})=0\ \ \ \ \text{if}\ \ \ \
i>j+n+\dim Z.$

Taking classes in $K_{0}(\mathrm{MHM}_{S}^{\mathrm{mon}})$,

$[p_{!}M_{V}]=\sum_{i\leq d}(-1)^{i}[\mathcal{H}^{i}p_{!}M_{V}]$

is a sum of Hodge modules of weight $\leq n+\dim Z+d$. When $S$ is a point, we
can conclude directly here, since these Hodge modules turn out to be simply
Hodge structures, of weights bounded by $n+\dim Z+d\leq n+2d.$

We now turn to the case where $S$ is not a point. Assume first that $p(U)$ is
contained in a closed subvariety of $S$. Then the result is trivially true by
taking $S_{0}$ to be the complement of that subvariety. Thus, we may now
assume that $p$ remains dominant when restricted to $U$, and we reduce to
proving the result for the induced map $p^{\prime}:U\to S$. We pick an open
subset $S_{0}$ of $S$ above which $p^{\prime}$ is a topological fibration.
Then above $S_{0}$, the Hodge modules $\mathcal{H}^{i}p^{\prime}_{!}M_{V}$ are
variations of Hodge structures, with supports of dimensions $\geq\dim Z-d$,
whence the result.

Now, assume there is also monodromy involved. We have shown that up to the
above restriction to open subsets, the underlying variations of Hodge
structures of $p_{!}M_{V}$ had weights $\leq n+2d$ if we start with $V$ of
weight $n$. If $M_{V}$ has non-trivial semisimple monodromy, then by
definition $w(M_{V})=n+1$ and in this situation the bound
$w_{S_{0}}(p^{\prime}_{!}M_{V})\leq n+1+2d$ holds. If now $M_{V}$ has trivial
monodromy, then so do all of the Hodge modules involved in $p_{!}M_{V}$. We
then have $w(M_{V})=n$ and the bound $w_{S_{0}}(p^{\prime}_{!}M_{V})\leq n+2d$
holds.

  3. (3)

We write $\mathfrak{a}=\sum a_{i}M_{i}$ where $a_{i}\in\mathbf{Z}$ and the
$M_{i}$ are pure Hodge modules (with monodromy) supported inside $S_{0}$, with
corresponding variations of Hodge structures $V_{i}$. Then
$w_{S}(i_{!}\mathfrak{a})$ is at most the weight of $\sum a_{i}i_{!}M_{i}$
which has the same variations of Hodge structures occurring, hence the result.

∎

###  4.3. The weight function on Grothendieck rings of varieties

We have defined a weight filtration on the Grothendieck ring of mixed Hodge
modules with monodromy $K_{0}(\mathrm{MHM}_{S}^{\mathrm{mon}})$. We now use
the same procedure as in [Bil23] to deduce from it a weight function on the
Grothendieck ring of varieties with exponentials
$\mathscr{E}xp\mathscr{M}_{S}$. We only give a brief overview of the
construction here, referring to [Bil23] and the references therein for the
details.

####  4.3.1. Motivic vanishing cycles

Let $k$ be a field of characteristic 0 and let $S$ be a variety over $k$. We
refer to [Bil23, Section 2.3.5] for the definition of the total vanishing
cycles measure

$\Phi_{S}:\mathscr{E}xp\mathscr{M}_{S}\to(\mathscr{M}_{S}^{\hat{\mu}},\ast),$

which is functorial in $S$. Here $\mathscr{M}_{S}^{\hat{\mu}}$ is the
Grothendieck ring of varieties over $S$ with $\hat{\mu}$-action (see [Bil23,
Section 2.1.5]), and $\ast$ denotes Looijenga’s convolution product (see
[Bil23, Section 2.2.1]). Recall moreover that the restriction of $\Phi_{S}$ to
$\mathscr{M}_{S}$ coincides with the natural inclusion
$\mathscr{M}_{S}\to\mathscr{M}_{S}^{\hat{\mu}}$.

####  4.3.2. The Hodge realisation

Let $S$ be a complex variety. There is a group morphism

$\begin{array}[]{rccc}\chi^{\mathrm{Hdg}}_{S}:&\mathscr{M}_{S}^{\hat{\mu}}&\to&K_{0}(\mathrm{MHM}_{S}^{\mathrm{mon}})\\\
&[X\xrightarrow{f}S,\sigma]&\mapsto&\sum_{i\in\mathbf{Z}}(-1)^{i}[\mathcal{H}^{i}(f_{!}\mathbf{Q}_{X}^{\mathrm{Hdg}}),T_{s}(\sigma),0]\end{array}$

called the Hodge realisation morphism (see [Bil23, Section 4.4.1] for a
discussion of its properties).

####  4.3.3. Weight function on Grothendieck rings of varieties

Let $S$ be a complex variety. The weight filtration on the ring
$\mathscr{E}xp\mathscr{M}_{S}$ is given by

$W_{\leq
n}\mathscr{E}xp\mathscr{M}_{S}:=(\chi_{S}^{\mathrm{Hdg}}\circ\Phi_{S})^{-1}(W_{\leq
n}K_{0}(\mathrm{MHM}_{S}^{\mathrm{mon}})),$

for every $n\in\mathbf{Z}$. The completion with respect to this filtration is
denoted by $\widehat{\mathscr{E}xp\mathscr{M}_{S}}$. The weight function on
$\mathscr{E}xp\mathscr{M}_{S}$, again denoted $w_{S}$, is given by the
composition

$\mathscr{E}xp\mathscr{M}_{S}\xrightarrow{\Phi_{S}}\mathscr{M}_{S}^{\hat{\mu}}\xrightarrow{\chi_{S}^{\mathrm{Hdg}}}K_{0}(\mathrm{MHM}_{S}^{\mathrm{mon}})\xrightarrow{w_{S}}\mathbf{Z}.$

####  4.3.4. Weight and dimension

######  Lemma 4.6.

Let $S$ be a complex variety and $X$ a variety over $S$. One has the equality

$w_{S}(X)=2\dim_{S}X.$

###### Proof.

Let $p:X\to S$ be the structural morphism, which we may assume to be dominant
by replacing $S$ with the closure of $p(X)$. Further, we pick a stratification
$(S_{i})_{i}$ of $S$ such that over every stratum $p$ is topologically a
fibration. For every $i$, put $X_{i}=p^{-1}(S_{i})$ and $p_{i}=p_{|X_{i}}$. Up
to refining our stratification, we may assume that over each $S_{i}$, the
Hodge modules involved in the complex
$(p_{i})_{!}\mathbf{Q}^{\mathrm{Hdg}}_{X_{i}}$ correspond to variations of
Hodge structures defined over $S_{i}$. Now pick a stratum $X_{i}$. By proper
base change, for every $s\in S_{i}$, the fibre
$((p_{i})_{!}\mathbf{Q}^{\mathrm{Hdg}}_{X_{i}})_{s}$ is given by a complex
with cohomology given by $H^{*}(X_{s},\mathbf{Q})$, which involves Hodge
structures of weights at most $\dim_{S_{i}}X_{i}$, with equality for the Hodge
structure in top weight. From this we may deduce, using Lemma 4.3, that the
weight of $p_{!}\mathbf{Q}_{X}^{\mathrm{Hdg}}$ is
$\max\dim_{S_{i}}X_{i}=\dim_{S}X.$ ∎

######  Lemma 4.7 (Triangular inequality for weights).

Let $S$ be a complex variety, $X$ a variety over $S$ and
$f:X\to\mathbf{A}^{1}$ a morphism. Then

$w_{S}([X,f])\leq w_{S}(X).$

###### Proof.

By [Bil23, Proposition 2.3.5.2], we have the triangular inequality for motivic
vanishing cycles

$\dim_{S}(\Phi_{S}([X,f]))\leq\dim_{S}X.$

On the other hand, following the proof of Lemma 4.6.3.2 in [Bil23], we see
that

$w_{S}(\Phi_{S}([X,f]))\leq 2\dim_{S}\Phi_{S}([X,f]).$

(Here we consider the weight function on $\mathscr{M}_{S}^{\hat{\mu}}$, as
defined in [Bil23].) We conclude by using Lemma 4.6. ∎

######  Lemma 4.8 (Cancellation of maximal weights).

Let $X$ and $Y$ be complex varieties, both irreducible and of dimension $d$.
Then

$w([X]-[Y])\leq 2d-1.$

###### Proof.

This follows from [Bil23, Lemma 4.6.3.4], given that over a point, our weight
and the weight in [Bil23] coincide. ∎

###  4.4. Convergence of power series and evaluation

Let $X$ be a variety over $\mathbf{C}$, and consider a power series

$F(T)=\sum_{i\geq 0}X_{i}T^{i}\in\mathscr{E}xp\mathscr{M}_{X}[[T]].$

The radius of convergence of $F$ is defined by

$\sigma_{F}=\limsup_{i\to\infty}\frac{w_{X}(X_{i})}{2i}.$

We say that $F(T)$ converges for $|T|<\mathbf{L}^{-r}$ if $r\geq\sigma_{F}$.
If $F(T)$ converges for $|T|<\mathbf{L}^{-r}$, then $F(\mathbf{L}^{-m})$
exists as an element of $\widehat{\mathscr{E}xp\mathscr{M}_{X}}$ for every
$m>r$.

If $F(T)=\prod_{x\in X}\left(1+\sum_{i\geq 1}X_{i,x}T^{i}\right)$ is a motivic
Euler product which converges for $|T|<\mathbf{L}^{-r}$, then for any integer
$M>r$ we write $F(\mathbf{L}^{-m})$ in the form

$\prod_{x\in X}\left(1+\sum_{i\geq
1}X_{i,x}T^{i}\right)_{|T=\mathbf{L}^{-m}}.$

######  Remark 4.9.

Special values of motivic Euler products should be handled with extreme care,
as motivic Euler products do not behave well with respect to non-monomial
substitutions. (See [BH21, Section 6.5] for a discussion of this fact.) In
particular, in principle, when writing a special value of a motivic Euler
product, one should always write out the formal product, and then specify at
what value of $T$ it has been evaluated.

With this caveat, to highlight the analogy with number theory, we will allow
ourselves the following abuse of notation (which is already employed in
[LVW22] and [Fai23]).

######  Notation 4.10.

We will denote by

$\prod_{x\in X}a_{x}$

the evaluation at $T=1$ of the motivic Euler product

$\prod_{x\in X}\left(1+(a_{x}-1)T\right).$

##  5\. Weyl differencing and a general exponential sum bound

Both the major and minor arc treatments in the circle method rely on a general
upper bound for certain exponential sums, which in the
$\mathbf{F}_{q}(t)$-setting take the form

$\sum_{(g_{1},\dots,g_{n})\in\mathrm{Poly}_{<E}^{n}}\psi(\mathrm{res}(\alpha
f(g_{1},\dots,g_{n})).$

The aim of this section is to provide such bounds in the motivic setting. We
start by proving a property of the weight in Section 5.1, which allows us to
adapt to our setting a geometric version of the classical Weyl differencing
argument provided in [BS20]. This is achieved in Proposition 5.3. The rest of
the section is dedicated to proving our general bound. After an application of
Proposition 5.3, we rewrite our bound in terms of the multilinear forms
associated to the polynomial $f$, and apply some inequalities coming from the
geometry of numbers over $\mathbf{C}(t)$. An interesting feature of our work
is that we never have to rely on any point counting estimates, unlike for the
procedure in [BS20]. Indeed, we circumvent the use of spreading out arguments
and rely purely on estimates for the dimensions of various spaces.

###  5.1. The fundamental property of the weight

######  Lemma 5.1.

For every variety $X$ over $S$ and every morphism $f:X\to\mathbf{A}^{1}$, we
have

$\Phi_{S}([X,f])=\Phi_{S}([X,-f]).$

###### Proof.

By Lemma 2.1 and [Bil23, Theorem 2.3.5.1] we may reduce to the case where $S$
is the spectrum of a field $k$ of characteristic zero. Now, by Bittner’s
presentation of the Grothendieck ring of varieties, we know that
$\mathscr{E}xp\mathscr{M}_{k}$ (which is a quotient of
$K_{0}(\mathrm{Var}_{\mathbf{A}^{1}_{k}})$) is generated by classes $[X,f]$
such that $X$ is smooth and $f$ is proper (as in [CLL16, Theorem 5.1.3]). For
such a class, using [Bil23, Theorem 2.3.5.1] we may write

$\Phi_{k}([X,f])=\epsilon_{!}(\varphi_{f}^{\mathrm{tot}}),$

where $\epsilon:\mathbf{A}^{1}\to k$ is the structural morphism and
$\varphi_{f}^{\mathrm{tot}}\in\mathscr{M}_{\mathbf{A}^{1}_{k}}^{\hat{\mu}}$
are the total motivic vanishing cycles defined in [Bil23, Section 2.3.1]. We
now remark that for a morphism $g:X\to\mathbf{A}^{1}$ defined on a smooth
variety $X$, the motivic vanishing cycles $\varphi_{g}$ only depend on a log-
resolution of the zero locus of $g$, and therefore we have
$\varphi_{g}=\varphi_{-g}$. Using this, we see that for every
$a\in\mathbf{A}^{1}_{k}$, we have

$(\varphi^{\mathrm{tot}}_{f})_{a}=\varphi_{f-a}=\varphi_{-f-a}=(\varphi^{\mathrm{tot}}_{-f})_{-a}$

in $\mathscr{M}_{k(a)}^{\hat{\mu}}$. We thus see that, denoting by
$i:\mathbf{A}^{1}\to\mathbf{A}^{1}$ the map $a\mapsto-a$, we have the identity
$\varphi^{\mathrm{tot}}_{f}=i_{!}\varphi^{\mathrm{tot}}_{-f}$. Pushing to
$\mathrm{Spec}\,k$ via $\epsilon$ we get the result. ∎

Throughout this section, let $S$ be a variety over $\mathbf{C}$ and let
$G\in\mathcal{O}_{S}[x_{1},\dots,x_{N}]$, for an integer $N\geq 1$. For every
point $s\in S$, we denote by $G_{s}$ the pullback of $G$ via $s$.

######  Lemma 5.2.

We have

$w_{S}([\mathbf{A}^{N}_{S},G][\mathbf{A}^{N}_{S},-G])=2w_{S}([\mathbf{A}^{N}_{S},G]).$

###### Proof.

We write

$\chi_{S}^{\mathrm{Hdg}}\circ\Phi_{S}([\mathbf{A}^{N}_{S},G])=\sum_{Z_{i}\subset
S}a_{i}H_{i},$

where each $H_{i}$ is a simple pure polarisable variation of Hodge structures
with monodromy over $Z_{i}$. Up to stratifying, we may assume that for every
$i,j$, either $Z_{i}=Z_{j}$ or $Z_{i}\cap Z_{j}=\varnothing$.

Since $H_{i}$ is polarisable, denoting by $n_{i}$ its weight we have the
selfduality

$H_{i}\simeq H_{i}^{\vee}(-n_{i}).$

From this, using Lemma 5.1 and the fact that $\chi_{S}^{\mathrm{Hdg}}$ and
$\Phi_{S}$ are ring morphisms, we get

$\displaystyle\chi_{S}^{\mathrm{Hdg}}\circ\Phi_{S}([\mathbf{A}^{N}_{S},G][\mathbf{A}^{N}_{S},-G])$
$\displaystyle=\chi_{S}^{\mathrm{Hdg}}\circ\Phi_{S}([\mathbf{A}^{N}_{S},G])\chi_{S}^{\mathrm{Hdg}}\circ\Phi_{S}([\mathbf{A}^{N}_{S},-G])$
$\displaystyle=\left(\sum_{Z_{i}\subset
S}a_{i}H_{i}\right)\left(\sum_{Z_{i}\subset
S}a_{i}H_{i}^{\vee}(-n_{i})\right)$
$\displaystyle=\sum_{i,j}a_{i}a_{j}\mathrm{Hom}(H_{i},H_{j})(-n_{j}).$

Because of the assumption on $Z_{i},Z_{j}$ for $i\neq j$, and because of the
simplicity of the $H_{i}$, we get that

$\chi_{S}^{\mathrm{Hdg}}\circ\Phi_{S}([\mathbf{A}^{N}_{S},G][\mathbf{A}^{N}_{S},-G])=\sum_{i}a_{i}^{2}\mathbf{Q}^{\mathrm{Hdg}}_{S}(-n_{i}),$

where $\mathbf{Q}_{S}^{\mathrm{Hdg}}$ denotes the constant rank one variation
of Hodge structures of weight 0 over $S$. We get a sum of variations of Hodge
structure with positive coefficients: by Lemma 4.3, its weight is therefore
$2\max_{i}{n_{i}}=2w_{S}([\mathbf{A}^{N}_{S},G]).$ ∎

###  5.2. The inductive argument

We define $V(G)$ to be the variety over $S$ given by

$(\mathbf{y}^{(1)},\dots,\mathbf{y}^{(d-1)},s)\in(\mathbf{A}^{N})^{d-1}_{S}$

such that

$\sum_{\epsilon_{1},\dots,\epsilon_{d-1}\in\\{0,1\\}}(-1)^{\epsilon_{1}+\dots+\epsilon_{d-1}}G_{s}(\mathbf{x}+\epsilon_{1}\mathbf{y}^{(1)}+\dots+\epsilon_{d-1}\mathbf{y}^{(d-1)})$

is a constant function of $\mathbf{x}$.

######  Proposition 5.3.

Let $G\in\mathcal{O}_{S}[x_{1},\dots,x_{N}]$ be a polynomial of degree $\leq
d$ in $N$ variables. The weight function
$w_{S}:\mathscr{E}xp\mathscr{M}_{S}\to\mathbf{Z}$ satisfies

$w_{S}([\mathbf{A}^{N}_{S},G])\leq\frac{\dim_{S}V(G)+N(2^{d-1}-(d-1))}{2^{d-2}}.$

###### Proof.

We essentially follow the proof of [BS20, Proposition 5.5], working by
induction on $d$. Assume first that $d=1$. Then $V(G)$ is a subvariety of $S$
given by the $s\in S$ such that $G_{s}(\mathbf{x})$ is constant. Assume first
that $V(G)$ is empty. Then for every $s\in S$, the polynomial
$G_{s}(\mathbf{x})$ is of degree exactly 1, and so
$[\mathbf{A}^{N}_{\kappa(s)},G_{s}]$ = 0 by Lemma 2.3. Lemma 2.1 therefore
yields $[\mathbf{A}_{S}^{N},G]=0$, and so both sides of the inequality in the
statement are equal to $-\infty$. On the other hand, if $V(G)$ is nonempty,
then $\dim_{S}V(G)=0$. By the triangular inequality of Lemma 4.7, we have

$w_{S}([\mathbf{A}^{N}_{S},G])\leq 2N,$

which is exactly the inequality that needed to be proved.

We now assume that the result is already known for polynomials of degree $\leq
d-1$. Let $G$ be a polynomial of degree $d\geq 2$. We begin with an
application of Lemma 5.2, which yields

$\displaystyle 2w_{S}([\mathbf{A}_{S}^{N},G])$ $\displaystyle\leq
w_{S}([\mathbf{A}_{S}^{N},G][\mathbf{A}_{S}^{N},-G])$
$\displaystyle=w_{S}([\mathbf{A}_{S}^{N}\times_{S}\mathbf{A}_{S}^{N},G(\mathbf{x})-G(\mathbf{y})])$
$\displaystyle=w_{S}([\mathbf{A}_{S}^{N}\times_{S}\mathbf{A}_{S}^{N},G(\mathbf{x})-G(\mathbf{x}+\mathbf{y})]),$

since $G(\mathbf{x})-G(\mathbf{y})$ is related to
$G(\mathbf{x})-G(\mathbf{x}+\mathbf{y})$ via an invertible change of
variables.

For any $\mathbf{y}_{0}=(y_{1},\dots,y_{N})$, the difference
$G(\mathbf{x})-G(\mathbf{x}+\mathbf{y}_{0})$ is a polynomial of degree $d-1$
in $x_{1},\dots,x_{N}$. The variety $V(G)$ admits a map
$p:V(G)\to\mathbf{A}_{S}^{N}$ along the coordinates $\mathbf{y}$, whose fibre
over a point $\mathbf{y}_{0}\in\mathbf{A}_{S}^{N}$ is the variety in the
definition of $V(G(\mathbf{x})-G(\mathbf{x}+\mathbf{y}_{0}))$. Choose a
stratification of $\mathbf{A}_{S}^{N}$, with strata $W_{j}$ being varieties
such that the fibre dimension of this map is constant on each stratum $W_{j}$.
Thus

(5.1)
$\dim_{W_{j}}V(G(\mathbf{x})-G(\mathbf{x}+\mathbf{y}_{0}))=\dim_{S}p^{-1}(W_{j})-\dim_{S}W_{j}\leq\dim_{S}V(G)-\dim_{S}W_{j},$

for any $\mathbf{y}_{0}\in W_{j}$. On appealing to part (1) of Proposition
4.4, it follows that

$2w_{S}([\mathbf{A}_{S}^{N},G])\leq\max_{j}w_{S}([\mathbf{A}_{S}^{N}\times_{S}W_{j},G(\mathbf{x})-G(\mathbf{x}+\mathbf{y})]).$

Fix a stratum $W_{j}$ and denote by $q_{j}:W_{j}\to S$ the restriction to
$W_{j}$ of the projection map $\mathbf{A}^{N}_{S}\to S$. Up to refining our
stratification, we may assume that $q_{j}:W_{j}\to S_{j}$ (where
$S_{j}=q_{j}(W_{j})$) has fibres of constant dimension
$\dim_{S}W_{j}=r\in[0,N]$, and that

$w_{S_{j}}([\mathbf{A}_{S_{j}}^{N}\times_{S_{j}}W_{j},G(\mathbf{x})-G(\mathbf{x}+\mathbf{y})])\leq
w_{W_{j}}([\mathbf{A}_{W_{j}}^{N},G(\mathbf{x})-G(\mathbf{x}+\mathbf{y}_{0})])+2r,$

by part (2) of Proposition 4.4. It follows from (5.1) and the inductive
hypothesis that

$w_{W_{j}}([\mathbf{A}_{W_{j}}^{N},G(\mathbf{x})-G(\mathbf{x}+\mathbf{y}_{0})])\leq\frac{\dim_{S}V(G)-r+N(2^{d-2}-(d-2))}{2^{d-3}},$

for any $\mathbf{y}_{0}\in W_{j}$. Hence by part (3) of Proposition 4.4,

$w_{S}([\mathbf{A}_{S}^{N}\times_{S}W_{j},G(\mathbf{x})-G(\mathbf{x}+\mathbf{y})])\leq\frac{\dim_{S}V(G)+(2^{d-2}-1)r+N(2^{d-2}-(d-2))}{2^{d-3}}.$

Taking the maximum over the different strata, we finally deduce that

$\displaystyle 2w_{S}([\mathbf{A}_{S}^{N},G])$ $\displaystyle\leq\max_{0\leq
r\leq N}\frac{\dim_{S}V(G)+(2^{d-2}-1)r+N(2^{d-2}-(d-2))}{2^{d-3}}$
$\displaystyle=\frac{\dim_{S}V(G)+N(2^{d-1}-(d-1))}{2^{d-3}},$

which suffices to complete the proof. ∎

###  5.3. A general bound for exponential sums

In this section, we want to prove a general upper bound for an exponential sum
of the form

$[\mathrm{Poly}_{<E}^{n}\times S,\mathrm{res}(\alpha f(g_{1},\dots,g_{n}))],$

when $S$ is a variety parameterising values of $\alpha$. We write our
polynomial $f$ in the form

$f(x_{1},\dots,x_{n})=\sum_{j_{1},\dots,j_{d}=1}^{n}c_{\mathbf{j}}x_{j_{1}}\dots
x_{j_{d}},$

for symmetric coefficients $c_{\mathbf{j}}$ (i.e.
$c_{\mathbf{j}}=c_{\sigma(\mathbf{j})}$ for any $\sigma\in S_{d}$). Associated
to $f$ are the multilinear forms

$\Psi_{j}(\mathbf{h}^{(1)},\dots,\mathbf{h}^{({d-1})})=d!\sum_{j_{1},\dots,j_{d-1}=1}^{n}c_{j_{1},\dots,j_{d-1},j}h_{j_{1}}^{(1)}\dots
h_{j_{d-1}}^{(d-1)},$

for $1\leq j\leq n$. Recall that for a Laurent series $h=\sum_{i\leq
M}h_{i}t^{i}\in k((t^{-1}))$ we denote by $\\{h\\}=\sum_{i\leq-1}h_{i}t^{i}$
its fractional part. The main goal of this section is to prove the following
general bound for the weight of the exponential sum.

######  Proposition 5.4.

We have

$w_{S}([\mathrm{Poly}_{<E}^{n}\times S,\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))])\leq\frac{\max_{\alpha\in S}\dim
N(\alpha)+En(2^{d-1}-(d-1))}{2^{d-2}},$

where

(5.2)
$N(\alpha)=\left\\{(\mathbf{u}^{(1)},\dots,\mathbf{u}^{(d-1)})\in\left(\mathrm{Poly}_{<E}^{n}\right)^{d-1}:\begin{array}[]{l}\operatorname{ord}\\{\alpha\Psi_{j}(\mathbf{u}^{(1)},\dots,\mathbf{u}^{(d-1)})\\}<-E\\\
\forall j\in\\{1,\dots,n\\}\end{array}\right\\}.$

####  5.3.1. Weyl differencing

Denote by $G_{\alpha}$ the polynomial

$G_{\alpha}=\mathrm{res}\left(\alpha f(g_{1},\dots,g_{n})\right),$

for $\alpha\in S$, considered as a function in the coefficients of
$g_{1},\dots,g_{n}$. Writing

$g_{j}(t)=a_{0,j}+a_{1,j}t+\dots+a_{E-1,j}t^{E-1},$

we see that $G_{\alpha}$ is a polynomial of degree $d$ in the $N=En$ variables
$(a_{i,j})$ for $0\leq i<E$ and $1\leq j\leq n$. In this notation we have

$[\mathrm{Poly}_{<E}^{n}\times S,\mathrm{res}(\alpha
f(g_{1},\dots,g_{n})]=[\mathbf{A}^{N}_{S},G_{\alpha}].$

Hence, on applying Proposition 5.3, we get

$w_{S}([\mathbf{A}^{N}_{S},G_{\alpha}])\leq\frac{\dim_{S}V(G_{\alpha})+N(2^{d-1}-(d-1))}{2^{d-2}}.$

Thus, to bound our exponential sum, it suffices to obtain a bound on $\dim
V(G_{\alpha})$ for every $\alpha\in S$.

####  5.3.2. Reformulation using multilinear forms

The polynomial $G_{\alpha}$ is the coefficient of $t^{-1}$ in

$(b_{1}t^{-1}+b_{2}t^{-2}+\dots)\sum_{j_{1},\dots,j_{d}=1}^{n}c_{\mathbf{j}}\prod_{i=1}^{d}(a_{0,j_{i}}+a_{1,j_{i}}t+\dots+a_{E-1,j_{i}}t^{E-1}),$
$=(b_{1}t^{-1}+b_{2}t^{-2}+\dots)\sum_{j_{1},\dots,j_{d}=1}^{n}c_{\mathbf{j}}\sum_{i_{1},\dots,i_{d}=0}^{E-1}a_{i_{1},j_{1}}\dots
a_{i_{d},j_{d}}t^{i_{1}+\dots+i_{d}},$

which is given by

$\sum_{j_{1},\dots,j_{d}=1}^{n}\sum_{i_{1},\dots,i_{d}=0}^{E-1}b_{i_{1}+\dots+i_{d}+1}c_{\mathbf{j}}a_{i_{1},j_{1}}\dots
a_{i_{d},j_{d}}.$

Using this expression, $V(G_{\alpha})$ is the set of
$(\mathbf{y}^{(1)},\dots,\mathbf{y}^{(d-1)})\in(\mathbf{A}^{N})^{d-1}$ such
that

$\sum_{\epsilon_{1},\dots,\epsilon_{d-1}\in\\{0,1\\}}(-1)^{\epsilon_{1}+\dots+\epsilon_{d-1}}\sum_{\mathbf{i,j}}b_{i_{1}+\dots+i_{d}+1}c_{\mathbf{j}}\prod_{k=1}^{d}(a_{i_{k},j_{k}}+\epsilon_{1}y^{(1)}_{i_{k},j_{k}}+\dots+\epsilon_{d-1}y_{i_{k},j_{k}}^{(d-1)})$

is a constant function in the $(a_{i,j})_{\begin{subarray}{c}0\leq i\leq
E-1\\\ 1\leq j\leq n\end{subarray}}$. Since we know that the latter is at most
linear in the $(a_{i,j})_{\begin{subarray}{c}0\leq i\leq E-1\\\ 1\leq j\leq
n\end{subarray}}$, it suffices to ensure that the coefficient of each
$a_{i,j}$ vanishes, which gives us equations

(5.3)
$\sum_{i_{1},\dots,i_{d-1}=0}^{E-1}\sum_{j_{1},\dots,j_{d-1}=1}^{n}b_{i_{1}+\dots+i_{d-1}+i+1}c_{j_{1},\dots,j_{d-1},j}y_{i_{1},j_{1}}^{(1)}\dots
y_{i_{d-1},j_{d-1}}^{(d-1)}=0,$

for every $i\in\\{0,\dots,E-1\\}$ and every $j\in\\{1,\dots,n\\}$. The
statement of Proposition 5.4 is now a direct consequence of the following
result.

######  Lemma 5.5.

There is a canonical identification $V(G_{\alpha})=N(\alpha),$ where
$N(\alpha)$ is given by (5.2).

###### Proof.

Writing $\mathbf{u}^{(i)}=\sum_{k=0}^{E-1}\mathbf{h}_{k}^{(i)}t^{k},$ with
$\mathbf{h}_{k}=(h_{k,1},\dots,h_{k,n})\in\mathbf{A}^{n}$, we see that

$\displaystyle\alpha$
$\displaystyle\Psi_{j}(\mathbf{u}^{(1)},\dots,\mathbf{u}^{(d-1)})$
$\displaystyle=\left(\sum_{r\geq
1}b_{r}t^{-r}\right)d!\sum_{j_{1},\dots,j_{d-1}=1}^{n}c_{j_{1},\dots,j_{d-1},j}u_{j_{1}}^{(1)}\dots
u_{j_{d-1}}^{(d-1)}$ $\displaystyle=d!\left(\sum_{r\geq
1}b_{r}t^{-r}\right)\sum_{j_{1},\dots,j_{d-1}=1}^{n}c_{j_{1},\dots,j_{d-1},j}\sum_{i_{1},\dots,i_{d-1}=0}^{E-1}h_{i_{1},j_{1}}^{(1)}\dots
h_{i_{d-1},j_{d-1}}^{(d-1)}t^{i_{1}+\dots+i_{d-1}}$
$\displaystyle=d!\sum_{r\geq
1}b_{r}\sum_{j_{1},\dots,j_{d-1}=1}^{n}c_{j_{1},\dots,j_{d-1},j}\sum_{i_{1},\dots,i_{d-1}=0}^{E-1}h_{i_{1},j_{1}}^{(1)}\dots
h_{i_{d-1},j_{d-1}}^{(d-1)}t^{i_{1}+\dots+i_{d-1}-r}.$

The space $N(\alpha)$ is defined by the vanishing of the coefficients of
$t^{-1},\dots,t^{-E}$. For $i\in\\{0,\dots,E-1\\}$, the coefficient of degree
$t^{-i-1}$ is given by

$d!\sum_{i_{1},\dots,i_{d-1}=0}^{E-1}\sum_{j_{1},\dots,j_{d-1}=1}^{n}b_{i_{1}+\dots+i_{d-1}+i+1}c_{j_{1},\dots,j_{d-1},j}h_{i_{1},j_{1}}^{(1)}\dots
h_{i_{d-1},j_{d-1}}^{(d-1)}.$

Comparing with equation (5.3), we get the required identification. ∎

Thus, it now remains to find a bound on $\dim N(\alpha)$. To be able to apply
it in both the major and in the minor arc setting, it will rely on parameters
associated to a rational approximation of $\alpha$.

####  5.3.3. A dimension bound on zero-sets defined by multilinear forms

For given $E\geq 1$, let $V_{E}\subset\mathbf{A}^{(d-1)En}$ denote the variety
of points

$(\mathbf{h}_{k}^{(1)},\dots,\mathbf{h}_{k}^{(d-1)})_{0\leq
k<E}\in\mathbf{A}^{(d-1)En},$

for which

$\Psi_{j}\left(\mathbf{h}_{0}^{(1)}+t\mathbf{h}_{1}^{(1)}+\dots+t^{E-1}\mathbf{h}_{E-1}^{(1)},\dots,\mathbf{h}_{0}^{(d-1)}+t\mathbf{h}_{1}^{(d-1)}+\dots+t^{E-1}\mathbf{h}_{E-1}^{(d-1)}\right)$

vanishes identically in $t$, for $1\leq j\leq n$. We prove the following
result.

######  Lemma 5.6.

We have $\dim V_{E}\leq(d-2)En$.

###### Proof.

The condition in the definition of $V_{E}$ is equivalent to the system of
polynomial equations

$\sum_{\begin{subarray}{c}0\leq i_{1},\dots,i_{d-1}<E\\\
i_{1}+\dots+i_{d-1}=\ell\end{subarray}}\Psi_{j}(\mathbf{h}_{i_{1}}^{(1)},\dots,\mathbf{h}_{i_{d-1}}^{({d-1})})=0,$

for $0\leq\ell\leq(d-1)(E-1)$ and $1\leq j\leq n$. Define the diagonal
$D\subset\mathbf{A}^{(d-1)En}$, via

$D=\\{\mathbf{h}_{k}^{(1)}=\mathbf{h}_{k}^{(2)}=\dots=\mathbf{h}_{k}^{(d-1)}:0\leq
k<E\\}.$

Then $D$ has affine dimension $En$. Moreover, $D\cap V_{E}$ consists precisely
of the points $(\mathbf{h}_{0},\dots,\mathbf{h}_{E-1})\in\mathbf{A}^{En}$ for
which

$\sum_{\begin{subarray}{c}0\leq i_{1},\dots,i_{d-1}<E\\\
i_{1}+\dots+i_{d-1}=\ell\end{subarray}}\Psi_{j}(\mathbf{h}_{i_{1}},\dots,\mathbf{h}_{i_{d-1}})=0,$

for $0\leq\ell\leq(d-1)(E-1)$ and $1\leq j\leq n$. When $\ell=0$ we deduce
that

$\Psi_{j}(\mathbf{h}_{0},\dots,\mathbf{h}_{0})=0,$

for $1\leq j\leq n$. Since $f$ is non-singular, the only solution to this
system of equations is $\mathbf{h}_{0}=\mathbf{0}$. When $\ell=d-1$ we obtain
the system of equations

$\Psi_{j}(\mathbf{h}_{1},\dots,\mathbf{h}_{1})=0,$

for $1\leq j\leq n$, since $\mathbf{h}_{0}=\mathbf{0}$ and the $\Psi_{j}$ are
multilinear. Thus also $\mathbf{h}_{1}=\mathbf{0}$. Proceeding in this way,
for $\ell$ running through multiples of $d-1$, one ultimately concludes that
$\mathbf{h}_{0}=\dots=\mathbf{h}_{E-1}=\mathbf{0}$, whence $\dim(D\cap
V_{E})=0$. It now follows from the affine dimension theorem that

$\displaystyle 0$ $\displaystyle=\dim(D\cap V_{E})$ $\displaystyle\geq\dim
V_{E}+\dim D-(d-1)En$ $\displaystyle\geq\dim V_{E}-(d-2)En,$

from which the lemma follows. ∎

####  5.3.4. Application of the shrinking lemma

Recall the definition (5.2) of $N(\alpha)$. We are now ready to prove the
following general result, which gives us a means of bounding the remaining
term in Proposition 5.4.

######  Lemma 5.7.

Let $\alpha=\frac{h_{1}}{h_{2}}+\theta$ such that
$\operatorname{ord}(\theta)<0$. Let $\rho=\deg(h_{2})$ and
$\psi=\operatorname{ord}(\theta)$. Assume $s\geq 0$ to be an integer chosen
such that

  1. (1)

We have $-E-1-(d-1)s<-\rho$ and $(d-1)(E-1-s)+\psi<-\rho$.

  2. (2)

We either have $(d-1)(E-1-s)<\rho$ or $-E-(d-1)s-\psi\leq\rho$.

Then

$\dim N(\alpha)\leq(d-2)En+sn.$

###### Proof.

We shall adopt the notation in (A.3) and Section A.2 in the proof of this
result. To begin with, the variety $N(\alpha)$ admits a projection
$N(\alpha)\to\mathbf{A}^{E(d-2)}$, along the coordinates
$(\mathbf{u}^{(1)},\dots,\mathbf{u}^{(d-2)})$. The fibre above such a point is
equal to the space whose dimension is taken in the definition of
$\nu(\mathsf{\Lambda}_{E,E}(\mathbf{U}),0)$, for an appropriate symmetric
$n\times n$ matrix $\mathbf{U}$ depending on
$(\mathbf{u}^{(1)},\dots,\mathbf{u}^{(d-2)})$. An application of Lemma A.3 now
yields

$\dim N(\alpha)\leq ns+\dim N_{s}^{\prime}(\alpha),$

for any integer $s\geq 0$, where

$N_{s}^{\prime}(\alpha)=\left\\{\underline{\mathbf{u}}\in\left(\mathrm{Poly}_{<E}^{n}\right)^{d-2}\times\mathrm{Poly}_{<E-s}^{n}:\operatorname{ord}\left\\{\alpha\Psi_{j}(\underline{\mathbf{u}})\right\\}<-E-s,\forall
j\in\\{1,\dots,n\\}\right\\}.$

Repeating this argument $d-2$ times, we finally conclude that

(5.4) $\dim N(\alpha)\leq(d-1)ns+\dim N_{s}(\alpha),$

for any integer $s\geq 0$, where

$N_{s}(\alpha)=\left\\{\underline{\mathbf{u}}\in\left(\mathrm{Poly}_{<E-s}^{n}\right)^{d-1}:\operatorname{ord}\left\\{\alpha\Psi_{j}(\underline{\mathbf{u}})\right\\}<-E-(d-1)s,\forall
j\in\\{1,\dots,n\\}\right\\}.$

Our goal is to choose $s$, depending on $\rho$ and $\psi$, such that any
$\underline{\mathbf{u}}$ appearing in the definition of $N_{s}(\alpha)$ must
in fact satisfy $\Psi_{j}(\underline{\mathbf{u}})=0$ for all
$j\in\\{1,\dots,n\\}$. Let $\underline{\mathbf{u}}\in N_{s}(\alpha)$ and let
$j\in\\{1,\dots,n\\}$. Put $m=\Psi_{j}(\underline{\mathbf{u}})\in k[t]$. The
conditions on the degrees of the components of $\underline{\mathbf{u}}$ give
us that $\deg(m)\leq(d-1)(E-s-1).$ Write $\\{\alpha
m\\}=\left\\{\frac{h_{1}}{h_{2}}m+\theta m\right\\}$, and note that we have
the upper bound

$\operatorname{ord}\\{\theta
m\\}\leq\deg(m)+\operatorname{ord}(\theta)\leq(d-1)(E-s-1)+\psi.$

We will proceed in two steps. First we will find conditions on $s$ such that
$h_{2}$ should divide $m$. For this, it suffices to bound
$\operatorname{ord}\left\\{\frac{h_{1}}{h_{2}}m\right\\}$ by something smaller
than $-\rho$. We write

$\operatorname{ord}\left\\{\frac{h_{1}}{h_{2}}m\right\\}=\operatorname{ord}\\{\alpha
m-\theta m\\}\leq\max\left(\operatorname{ord}\\{\alpha
m\\},\operatorname{ord}\\{\theta m\\}\right).$

Making use of the upper bounds we have on these quantities, we get

$\operatorname{ord}\left\\{\frac{h_{1}}{h_{2}}m\right\\}\leq\max\left(-E-1-(d-1)s,(d-1)(E-s-1)+\psi\right).$

Thus, we see that condition $(1)$ in the statement of the lemma forces
$\operatorname{ord}\\{\frac{h_{1}}{h_{2}}m\\}<-\rho,$ and therefore forces
$h_{2}$ to divide $m$.

The second step consists in finding conditions which would ensure that $m=0$.
Given that $h_{2}$ divides $m$, one possibility is just to force
$\deg(m)<\rho$. This is implied by the condition

$(d-1)(E-1-s)<\rho,$

which is the first inequality in part $(2)$ of the lemma. Alternatively, note
that since $h_{2}$ divides $m$, we have

$\operatorname{ord}\\{\theta m\\}=\operatorname{ord}\\{\alpha m\\}<-E-(d-1)s.$

On the other hand, $\operatorname{ord}(\theta m)\leq(d-1)(E-s-1)+\psi$, and
the latter is negative since condition $(1)$ is satisfied. This means that

$\operatorname{ord}(\theta m)=\operatorname{ord}\\{\theta m\\}<-E-(d-1)s$

and therefore $\deg(m)<-E-(d-1)s-\psi.$ Thus, if $-E-(d-1)s-\psi\leq\rho$,
which is the second inequality in part $(2)$ of the lemma, then $m=0$.

We may conclude that if $s$ is chosen so that conditions $(1)$ and $(2)$ in
the lemma are satisfied, then

$N_{s}(\alpha)=\left\\{\underline{\mathbf{u}}\in\left(\mathrm{Poly}_{<E-s}^{n}\right)^{d-1}:\Psi_{j}(\underline{\mathbf{u}})=0,\forall
j\in\\{1,\dots,n\\}\right\\}.$

Lemma 5.6 gives $\dim N_{s}(\alpha)\leq(d-2)(E-s)n$, and so the statement
follows from (5.4). ∎

######  Remark 5.8.

Assume $\frac{h_{1}}{h_{2}}=0$, so that $\rho=0$. Then we may take

$s=\max\left(0,E-1+\left\lceil\frac{1+\psi}{d-1}\right\rceil,\min\left(E,-\left\lfloor\frac{E+\psi}{d-1}\right\rfloor\right)\right).$

######  Remark 5.9.

Assume that $\alpha\in k(t)$, so that $\theta=0$ and $\psi=-\infty$. Then we
may take

$s=\max\left(0,\left\lceil\frac{\rho-E}{d-1}\right\rceil,E-1-\left\lfloor\frac{\rho-1}{d-1}\right\rfloor\right).$

##  6\. The motivic major arcs

###  6.1. Definition

Recall that the motivic analogue of the space $\mathbf{T}$ occurring in the
function field circle method is the space $\mathbf{A}^{(-1,-de-2)}$. It is an
affine space of dimension $de+1$, representing elements of $\mathbf{T}$ modulo
$t^{-de-2}k[[t^{-1}]]$. Fix a positive parameter $\gamma>0$. For every element
$\beta=(b_{1},\dots,b_{de+1})\in\mathbf{A}^{(-1,-de-2)}$ and every integer
$m\geq 0$, we define the matrices

$A_{\beta,m}=\left(\begin{array}[]{ccc}b_{1}&\dots&b_{m+1}\\\
\vdots&\ddots&\vdots\\\
b_{de+1-\gamma-m}&\dots&b_{de+1-\gamma}\end{array}\right)$

and

$A^{*}_{\beta,m}=\left(\begin{array}[]{ccc}b_{1}&\dots&b_{m}\\\
\vdots&\ddots&\vdots\\\
b_{de+1-\gamma-m}&\dots&b_{de-\gamma}\end{array}\right),$

the latter being obtained from the former by removing the last column. Note
that $\mathrm{Ker}A_{\beta,m}$ is generated by a vector $(c_{0},\dots,c_{m})$
satisfying $c_{m}\neq 0$ if and only if $A^{*}_{\beta,m}$ has full rank. We
define the subspaces $M_{m,\gamma}$ of $\mathbf{A}^{(-1,-de-2)}$ for $m\geq-1$
inductively in the following way. Let $M_{-1,\gamma}=\varnothing$, and let

$M_{m,\gamma}=M_{m-1,\gamma}\cup\left\\{(b_{1},\dots,b_{de+1}):\mathrm{rk}A_{\beta,m}\leq
m,\ \mathrm{rk}A^{*}_{\beta,m}=m\right\\}.$

Denote by $M^{*}_{m,\gamma}$ the second set in this union. By the discussion
in Section 3.1, we see that $(b_{1},\dots,b_{de+1})$ is an element of
$M^{*}_{m,\gamma}$ if and only if there exists $h_{2}$ of degree exactly $m$
and $h_{1}$ of degree $<m$ such that
$\alpha=b_{1}t^{-1}+\dots+b_{de+1}t^{-de-1}$ satisfies

$\operatorname{ord}(h_{2}\alpha-h_{1})\leq-de-2+\gamma+m.$

In other words, $\alpha=\frac{h_{1}}{h_{2}}+\theta$ with
$\operatorname{ord}\theta\leq-de-2+\gamma$. Moreover, the coefficients of
$h_{1},h_{2}$ and $\theta$ are algebraic functions of the coefficients of
$\alpha$.

######  Lemma 6.1.

For any integers $\gamma>0$ and $m\geq 0$, we have

$M_{m,\gamma}=\left\\{(b_{1},\dots,b_{de+1}):\begin{array}[]{l}b_{1}t^{-1}+\dots+b_{de+1}t^{-de-1}=\frac{h_{1}}{h_{2}}+\theta\\\
\gcd(h_{1},h_{2})=1\text{ and $h_{2}$ monic}\\\ \deg h_{1}<\deg h_{2}\leq m\\\
\operatorname{ord}\,\theta\leq-de-2+\gamma\end{array}\right\\}.$

###### Proof.

We proceed by induction on $m$. The statement is clearly true for $m=0$.
Assume it is true for $m-1$. Then by the above remark the inclusion of
$M_{m,\gamma}$ in the right hand side is obvious. Conversely, assume that
$b_{1}t^{-1}+\dots+b_{de+1}t^{-de-1}=\frac{h_{1}}{h_{2}}+\theta$ as in the
statement of the lemma. If $\deg h_{2}\leq m-1$, then we may conclude by
induction that $(b_{1},\dots,b_{de+1})\in M_{m-1,\gamma}\subset M_{m,\gamma}$.
If $\deg h_{2}=m$, then we have $(b_{1},\dots,b_{de+1})\in
M^{*}_{m,\gamma}\subset M_{m,\gamma}$, and so the inverse inclusion holds as
well. ∎

This characterisation of $M_{m,\gamma}$ is what we will use in practice. It is
similar to the set $A_{m}^{de+1}$ that was discussed in Remark 3.4, the chief
difference being that our bound on $\operatorname{ord}\theta$ doesn’t depend
on the degree of $h_{2}$.

The set $M_{m,\gamma}$ can be stratified into subsets
$M_{m,m^{\prime},\gamma}$, where $h_{2}$ has degree exactly $m^{\prime}$. (In
fact, there is an identification
$M_{m,m^{\prime},\gamma}=M^{*}_{m^{\prime},\gamma}$.) We are now ready to
define the contribution from our major arcs. Bearing in mind Lemma 3.6, we
shall take

(6.1) $N_{\mathrm{major}}=\sum_{0\leq m^{\prime}\leq
e+1-\gamma}N_{\mathrm{major}}(m^{\prime}),$

where

(6.2)
$N_{\mathrm{major}}(m^{\prime})=\mathbf{L}^{-de-1}\left[\mathrm{Poly}_{\leq
e}^{n}\times
M_{m,m^{\prime},\gamma},\mathrm{res}\left(\left(\frac{h_{1}}{h_{2}}+\theta\right)f(g_{1},\dots,g_{n})\right)\right],$

which is viewed as an element of
$\mathscr{E}xp\mathscr{M}_{\mathrm{Poly}_{\leq e}^{n}\times
M_{m,m^{\prime},\gamma}}.$ (We shall see that the condition $m^{\prime}\leq
e+1-\gamma$ arises naturally during the course of the argument.)

We shall ultimately take

(6.3) $\gamma=\left\lceil\frac{e+1}{2}\right\rceil,$

although we’ll often use $\gamma$ to ease notation.

For any $m\geq 0$, let

(6.4) $B_{m}=(\mathrm{Poly}_{<m}\times\mathrm{MPoly}_{m})_{*}$

denote the space of pairs $(h_{1},h_{2})$ of coprime polynomials such that
$\deg h_{1}<\deg h_{2}=m$, with $h_{2}$ monic. Then we may define the
“exponential sum”

(6.5) $S_{m}(f)=\left[\mathrm{Poly}_{<m}^{n}\times
B_{m},\mathrm{res}\left(\left(\frac{h_{1}}{h_{2}}\right)f(\overline{g}_{1},\dots,\overline{g}_{n})\right)\right].$

There is a piecewise isomorphism

$M_{m,m^{\prime},\gamma}\simeq
B_{m^{\prime}}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},$

where $B_{m^{\prime}}$ is given by (6.4), which is obtained by sending
$\alpha$ to $(h_{1},h_{2},\theta)$. This induces natural morphisms

(6.6)
$\mathscr{E}xp\mathscr{M}_{B_{m^{\prime}}}\to\mathscr{E}xp\mathscr{M}_{M_{m,m^{\prime},\gamma}}$

and

(6.7)
$\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{(-de-2+\gamma,-de-2)}}\to\mathscr{E}xp\mathscr{M}_{M_{m,m^{\prime},\gamma}}.$

With these, the sum $S_{m^{\prime}}(f)$ can be viewed as an element of
$\mathscr{E}xp\mathscr{M}_{M_{m,m^{\prime},\gamma}}$, for each $m^{\prime}\leq
m$.

###  6.2. Contribution of $M_{m,m^{\prime},\gamma}$

We will start by computing the contribution (6.2) for fixed $m^{\prime}\leq
m\leq e+1$. Note that $N_{\mathrm{major}}(m^{\prime})$ is viewed as an element
of $\mathscr{E}xp\mathscr{M}_{\mathrm{Poly}_{\leq e}^{n}\times
M_{m,m^{\prime},\gamma}}$, but we will allow ourselves the slight abuse of
notation of also denoting by $N_{\mathrm{major}}(m^{\prime})$ the image of
this element in some other Grothendieck rings. By Euclidean division, for each
$1\leq i\leq n$, we may write $g_{i}=h_{2}q_{i}+\overline{g}_{i}$, which gives
us an isomorphism

$\mathrm{Poly}_{\leq e}^{n}\simeq\mathrm{Poly}_{\leq
e-m^{\prime}}^{n}\times\mathrm{Poly}_{<m^{\prime}}^{n},$

sending $(g_{1},\dots,g_{n})$ to
$(q_{1},\dots,q_{n},\overline{g}_{1},\dots\overline{g}_{n})$. In terms of
classes in the Grothendieck ring, this implies

$\displaystyle N_{\mathrm{major}}(m^{\prime})$
$\displaystyle=\mathbf{L}^{-de-1}\left[\mathrm{Poly}_{<m^{\prime}}^{n}\times
M_{m,m^{\prime},\gamma},\mathrm{res}\left(\left(\frac{h_{1}}{h_{2}}\right)f(\overline{g}_{1},\dots,\overline{g}_{n})\right)\right]$
$\displaystyle\quad\cdot\left[\mathrm{Poly}_{\leq
e-m^{\prime}}^{n}\times\mathrm{Poly}_{<m^{\prime}}^{n}\times
M_{m,m^{\prime},\gamma},\mathrm{res}\left(\theta
f(\overline{g}_{1}+h_{2}q_{1},\dots,\overline{g}_{n}+h_{2}q_{n})\right)\right],$

where the product is taken in
$\mathscr{E}xp\mathscr{M}_{\mathrm{Poly}_{<m^{\prime}}^{n}\times
M_{m,m^{\prime},\gamma}}$. The following result shows that there is no
dependence on $\overline{g}_{1},\dots,\overline{g}_{n}$ in the second factor,
under our assumption on $m^{\prime}$.

######  Lemma 6.2.

Assume that $m^{\prime}\leq e+1-\gamma$. Then

$\operatorname{ord}(\theta(f(\overline{g}_{1}+h_{2}q_{1},\dots,\overline{g}_{n}+h_{2}q_{n})-f(h_{2}q_{1},\dots,h_{2}q_{n})))<-1.$

###### Proof.

We note that the difference
$f(\overline{g}_{1}+h_{2}q_{1},\dots,\overline{g}_{n}+h_{2}q_{n})-f(h_{2}q_{1},\dots,h_{2}q_{n})$
is a sum of monomials of degree $d$ in
$\overline{g}_{1},\dots,\overline{g}_{n},h_{2}q_{1},\dots,h_{2}q_{n}$.
Moreover, the degree in $\overline{g}_{1},\dots,\overline{g}_{n}$ is at least
1. We know that $\operatorname{ord}h_{2}q_{i}\leq e$ and
$\operatorname{ord}\overline{g}_{i}\leq m^{\prime}-1\leq m-1\leq e$. Thus,
using the bound $m^{\prime}-1$ for at least one of the
$\overline{g}_{i}$-factors of each monomial, and the bound $e$ for all the
other factors, we get

$\operatorname{ord}(f(\overline{g}_{1}+h_{2}q_{1},\dots,\overline{g}_{n}+h_{2}q_{n})-f(h_{2}q_{1},\dots,h_{2}q_{n}))\leq(d-1)e+m^{\prime}-1.$

Moreover, we know that $\operatorname{ord}(\theta)\leq-de-2+\gamma$. Combining
the two estimates, we get

$\displaystyle\operatorname{ord}\left(\theta(f(\overline{g}_{1}+h_{2}q_{1},\dots,\overline{g}_{n}+h_{2}q_{n})-f(h_{2}q_{1},\dots,h_{2}q_{n}))\right)$
$\displaystyle\leq(d-1)e+m^{\prime}-1-de-2+\gamma$
$\displaystyle\leq-e-3+m^{\prime}+\gamma.$

This is bounded by $-2$ if $m^{\prime}+\gamma\leq e+1$. ∎

We may now write the contribution of $M_{m.m^{\prime},\gamma}$ as

$N_{\mathrm{major}}(m^{\prime})=\mathbf{L}^{-de-1}S_{m^{\prime}}(f)\cdot[\mathrm{Poly}_{\leq
e-m^{\prime}}^{n}\times M_{m,m^{\prime},\gamma},\mathrm{res}(\theta
f(h_{2}q_{1},\dots,h_{2}q_{n})],$

now viewed in $\mathscr{E}xp\mathscr{M}_{M_{m,m^{\prime},\gamma}}$, where
$S_{m^{\prime}}(f)$ is given by (6.5). We now apply an averaging argument to
the rightmost factor, in order to remove the dependence in $m^{\prime}$
completely. We first of all re-introduce a factor
$\mathrm{Poly}_{<m^{\prime}}^{n}\simeq\mathbf{A}^{nm^{\prime}}$ to obtain

$\displaystyle[\mathrm{Poly}_{\leq e-m^{\prime}}^{n}\times$ $\displaystyle
M_{m,m^{\prime},\gamma},\mathrm{res}(\theta f(h_{2}q_{1},\dots,h_{2}q_{n})]$
$\displaystyle=\mathbf{L}^{-nm^{\prime}}[\mathrm{Poly}_{\leq
e-m^{\prime}}^{n}\times\mathrm{Poly}_{<m^{\prime}}^{n}\times
M_{m,m^{\prime},\gamma},\mathrm{res}(\theta f(h_{2}q_{1},\dots,h_{2}q_{n}))].$

Now we use Lemma 6.2, together with the isomorphism coming from Euclidean
division, in order to get that the right hand side is

$\displaystyle=\mathbf{L}^{-nm^{\prime}}[\mathrm{Poly}_{\leq
e-m^{\prime}}^{n}\times\mathrm{Poly}_{<m^{\prime}}^{n}\times
M_{m,m^{\prime},\gamma},\mathrm{res}(\theta
f(\overline{g}_{1}+h_{2}q_{1},\dots,\overline{g}_{2}+h_{2}q_{n}))]$
$\displaystyle=\mathbf{L}^{-nm^{\prime}}[\mathrm{Poly}_{\leq e}^{n}\times
M_{m,m^{\prime},\gamma},\mathrm{res}(\theta f(g_{1},\dots,g_{n}))]$
$\displaystyle=\mathbf{L}^{-nm^{\prime}}[\mathrm{Poly}_{\leq
e}^{n}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},\mathrm{res}(\theta
f(g_{1},\dots,g_{n}))],$

where the latter is an element of
$\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{(-de-2+\gamma,-de-2)}}$, which we view
as an element of $\mathscr{E}xp\mathscr{M}_{M_{m,m^{\prime},\gamma}}$ via the
morphism (6.7). We end up with

$\displaystyle N_{\mathrm{major}}(m^{\prime})$
$\displaystyle=\mathbf{L}^{-de-1}S_{m^{\prime}}(f)\mathbf{L}^{-nm^{\prime}}[\mathrm{Poly}_{\leq
e}^{n}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},\mathrm{res}(\theta
f(g_{1},\dots,g_{n}))]$
$\displaystyle=S_{m^{\prime}}(f)\mathbf{L}^{-nm^{\prime}}\int_{\theta}[\mathrm{Poly}_{\leq
e}^{n}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},\mathrm{res}(\theta
f(g_{1},\dots,g_{n}))],$

where the integral is the one from Section 2.4. Inserting this into (6.1), we
are therefore led to the following result.

######  Proposition 6.3.

We have

$N_{\mathrm{major}}=\left(\sum_{0\leq m^{\prime}\leq
e+1-\gamma}S_{m^{\prime}}(f)\mathbf{L}^{-nm^{\prime}}\right)\int_{\theta}U(\theta),$

in $\mathscr{E}xp\mathscr{M}_{\mathbf{C}}$, where

(6.8) $U(\theta):=[\mathrm{Poly}_{\leq
e}^{n}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},\mathrm{res}(\theta
f(g_{1},\dots,g_{n}))]\in\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{(-de-2+\gamma,-de-2)}}.$

The aim is now to analyse the two remaining factors to obtain motivic
analogues of the singular series and the singular integral.

###  6.3. Singular series

It is now time to treat the truncated singular series in Proposition 6.3.
Recalling the definition (6.3) of $\gamma$, we have

$\sum_{0\leq m^{\prime}\leq
e+1-\gamma}S_{m^{\prime}}(f)\mathbf{L}^{-nm^{\prime}}=\sum_{0\leq
m^{\prime}\leq\lfloor\frac{e+1}{2}\rfloor}S_{m^{\prime}}(f)\mathbf{L}^{-nm^{\prime}}.$

We begin with a useful upper bound, which will also prove valuable in the
treatment of the minor arcs.

######  Lemma 6.4.

Let $\Delta=\lfloor\frac{e+1}{2}\rfloor$. Then

$\max_{m\geq\Delta}\left(2^{d}m-\left\lfloor\frac{m}{d-1}\right\rfloor
n\right)\leq\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1).$

###### Proof.

To begin with, we note that

$\displaystyle 2^{d}m-\left\lfloor\frac{m}{d-1}\right\rfloor n$
$\displaystyle\leq
2^{d}\left(\left\lfloor\frac{m}{d-1}\right\rfloor+1\right)(d-1)-\left\lfloor\frac{m}{d-1}\right\rfloor
n$
$\displaystyle\leq\left\lfloor\frac{m}{d-1}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1).$

Since $n>2^{d}(d-1)$, the maximum is reached when
$m=\Delta=\lfloor\frac{e+1}{2}\rfloor$ is minimal.

If $2\mid e+1$, it now follows that

$\max_{m\geq\Delta}\left(2^{d}m-\left\lfloor\frac{m}{d-1}\right\rfloor
n\right)\leq\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1),$

as claimed in the lemma. Suppose next that $2\nmid e+1$, so that
$\Delta=\frac{e}{2}$. Then

$\displaystyle 2^{d}m-\left\lfloor\frac{m}{d-1}\right\rfloor n$
$\displaystyle\leq\left\lfloor\frac{e+2}{2d-2}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1)$
$\displaystyle\leq\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1),$

if $m\geq\Delta+1$, whereas

$2^{d}m-\left\lfloor\frac{m}{d-1}\right\rfloor
n=2^{d-1}e-\left\lfloor\frac{e}{2d-2}\right\rfloor n$

if $m=\Delta$. In order to complete the proof, it remains to prove that

(6.9) $2^{d-1}e-\left\lfloor\frac{e}{2d-2}\right\rfloor
n\leq\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1).$

Writing $e=(2d-2)q+r$ for $r<2d-2$, we must have $r\leq 2d-4$, since $e$ is
even. But then

$\left\lfloor\frac{e}{2d-2}\right\rfloor=\left\lfloor\frac{e+1}{2d-2}\right\rfloor=q,$

whence (6.9) is equivalent to $2^{d-1}r\leq 2^{d}(d-1)$, which is self-
evident. ∎

####  6.3.1. Convergence of the series

Recall from (6.5) that

$S_{m}(f)=\left[\mathrm{Poly}_{<m}^{n}\times
B_{m},\mathrm{res}\left(\left(\frac{h_{1}}{h_{2}}\right)f(\overline{g}_{1},\dots,\overline{g}_{n})\right)\right],$

where $B_{m}$ is given by (6.4). We can use Proposition 5.4 to bound this sum,
obtaining

$w_{B_{m}}(S_{m}(f))\leq\frac{\max_{(h_{1},h_{2})\in B_{m}}\dim
N(h_{1}/h_{2})+nm(2^{d-1}-(d-1))}{2^{d-2}}.$

We apply Lemma 5.7 with $\theta=0$ (so that $\psi=-\infty$) and $\rho=E=m$. It
follows from Remark 5.9 that we can take

$s=m-1-\left\lfloor\frac{m-1}{d-1}\right\rfloor.$

Using this, it therefore follows that

$\displaystyle\dim N(h_{1}/h_{2})$
$\displaystyle\leq(d-2)mn+\left(m-1-\left\lfloor\frac{m-1}{d-1}\right\rfloor\right)n$
$\displaystyle\leq(d-1)mn-\left(1+\left\lfloor\frac{m-1}{d-1}\right\rfloor\right)n,$

whence

$w_{B_{m}}(S_{m}(f))\leq\frac{2^{d-1}nm-(1+\lfloor\frac{m-1}{d-1}\rfloor)n}{2^{d-2}}=2nm-\frac{(1+\lfloor\frac{m-1}{d-1}\rfloor)n}{2^{d-2}}.$

From this, noticing that $\dim B_{m}=2m$, property (2) of Proposition 4.4
yields

(6.10) $w(S_{m}(f))-2nm\leq
4m-\frac{(1+\lfloor\frac{m-1}{d-1}\rfloor)n}{2^{d-2}}.$

We now compute the radius of convergence of the series $\sum_{m\geq
0}S_{m}(f)T^{m}$, in the sense of Section 4.4. We have
$1+\left\lfloor\frac{m-1}{d-1}\right\rfloor\geq\frac{m}{d-1},$ for any $m\geq
0$. Hence

$\frac{w(S_{m}(f))}{2m}\leq n+\left(\frac{2^{d}(d-1)-n}{2^{d-1}(d-1)}\right),$

and so as soon as the condition $n>2^{d}(d-1)$ is satisfied, we get

$\limsup_{m\to\infty}\frac{w(S_{m}(f))}{2m}\leq n+\text{negative constant}.$

We may deduce that the radius of convergence of the series $\sum_{m\geq
0}S_{m}(f)T^{m}$ is bounded by $n-\epsilon$ for some $\epsilon>0$, and the
series converges for $|T|<\mathbf{L}^{-n+\epsilon}$. In particular, the
motivic singular series

(6.11) $\mathfrak{S}(f):=\sum_{m\geq 0}S_{m}(f)\mathbf{L}^{-nm}$

is well-defined as an element of
$\widehat{\mathscr{E}xp\mathscr{M}_{\mathbf{C}}}$

####  6.3.2. Replacing by full series

These bounds also allow us to evaluate the error made by replacing the finite
sum

$\sum_{0\leq m\leq\lfloor\frac{e+1}{2}\rfloor}S_{m}(f)\mathbf{L}^{-nm}$

by the full sum $\mathfrak{S}(f)$ defined in (6.11). For this, it suffices to
bound the weight of a term $S_{m}(f)\mathbf{L}^{-mn}$ for
$m\geq\lfloor\frac{e+1}{2}\rfloor+1$. It follows from (6.10) that

$w(S_{m}(f)\mathbf{L}^{-mn})\leq\frac{2^{d}(m-1)-\lfloor\frac{m-1}{d-1}\rfloor
n}{2^{d-2}}+4-\frac{n}{2^{d-2}}.$

Hence Lemma 6.4 implies that

$\max_{m\geq\lfloor\frac{e+1}{2}\rfloor+1}w(S_{m}(f)\mathbf{L}^{-mn})\leq\frac{\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n)+2^{d}(d-1)}{2^{d-2}}+4-\frac{n}{2^{d-2}}.$

Property (1) of Proposition 4.4 allows us to conclude as follows.

######  Lemma 6.5.

Recall the definition (3.2) of $\tilde{\nu}$. Then

$w\left(\mathfrak{S}(f)-\sum_{0\leq
m\leq\lfloor\frac{e+1}{2}\rfloor}S_{m}(f)\mathbf{L}^{-nm}\right)\leq
4-\tilde{\nu}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right).$

######  Remark 6.6.

We can also conclude from our argument that $\mathfrak{S}(f)$ has weight zero,
equal to $1$ plus an element of
$\widehat{\mathscr{E}xp\mathscr{M}_{\mathbf{C}}}$ of negative weight. To see
this we use $1+\left\lfloor\frac{m-1}{d-1}\right\rfloor\geq\frac{m}{d-1},$ for
any $m\geq 1$, so that (6.10) yields

(6.12) $w(S_{m}(f)\mathbf{L}^{-nm})\leq
4m-\frac{nm}{2^{d-2}(d-1)}=\left(\frac{2^{d}(d-1)-n}{2^{d-2}(d-1)}\right)m.$

The condition $n>2^{d}(d-1)$ therefore implies that

$w(\mathfrak{S}(f)-1)=w\left(\sum_{m\geq
1}S_{m}(f)\mathbf{L}^{-mn}\right)\leq-\nu,$

for a constant $\nu>0$.

####  6.3.3. Motivic Euler product decomposition

######  Proposition 6.7 (Factorisation property).

Let $m_{1},m_{2}$ be positive integers, and let $V_{m_{1},m_{2}}$ be the space
of pairs of coprime monic polynomials $l_{1},l_{2}$ such that
$\deg(l_{i})=m_{i}$ for $i=1,2$. Let
$\pi_{i}:V_{m_{1},m_{2}}\to\mathrm{MPoly}_{m_{i}}$ be given by
$(l_{1},l_{2})\mapsto l_{i}$, and let
$\pi_{12}:V_{m_{1},m_{2}}\to\mathrm{MPoly}_{m_{1}+m_{2}}$ be given by
$(l_{1},l_{2})\mapsto l_{1}l_{2}$. Then, there is an isomorphism

$\pi_{12}^{*}S_{m_{1}+m_{2}}(f)\simeq\pi_{1}^{*}S_{m_{1}}(f)\times_{V_{m_{1},m_{2}}}\pi_{2}^{*}S_{m_{2}}(f)$

of varieties with exponentials over $V_{m_{1},m_{2}}$ . Moreover, when
$m_{1}=m_{2}$, this isomorphism commutes with switching $l_{1}$ and $l_{2}$.

###### Proof.

We follow the argument in [BS20, Lemma 3.5]. Recall from (6.5) that

$S_{m}(f)=\left[\mathrm{Poly}_{<m}^{n}\times
B_{m},\mathrm{res}\left(\left(\frac{h_{1}}{h_{2}}\right)f({g}_{1},\dots,{g}_{n})\right)\right],$

for any positive integer $m$, where
$B_{m}=(\mathrm{Poly}_{<m}\times\mathrm{MPoly}_{m})_{*}$.

To begin with, there is an isomorphism

(6.13)
$(\mathrm{Poly}_{<m_{1}+m_{2}})^{n}\to(\mathrm{Poly}_{<m_{1}})^{n}\times(\mathrm{Poly}_{<m_{2}})^{n},$

which is given by sending a tuple of polynomials $(g_{1},\dots,g_{n})$ of
degree $<m_{1}+m_{2}$ to the pair of tuples

$((g_{1,1},\dots,g_{n,1}),(g_{1,2},\dots,g_{n,2})),$

where each $g_{i,1}$ has degree $<m_{1}$ (respectively, each $g_{i,2}$ has
degree $<m_{2}$), and such that $g_{i}\equiv g_{i,1}\bmod l_{1}$ and
$g_{i}\equiv g_{i,2}\bmod l_{2}$, for $1\leq i\leq n$. The inverse map is
given by

$((g_{1,1},\dots,g_{n,1}),(g_{1,2},\dots,g_{n,2}))\mapsto\left(l_{2}(g_{i,1}l_{2}^{-1}\bmod
l_{1})+l_{1}(g_{i,2}l_{1}^{-1}\bmod l_{2})\right)_{1\leq i\leq n},$

where the inverses are understood to be modulo $l_{1}$ and $l_{2}$,
respectively. Since $l_{1}$ and $l_{2}$ are coprime, their inverses modulo
each other are polynomial. Moreover, since they are monic it follows from
Euclid’s algorithm that the modulo operation is also polynomial.

We also obtain an isomorphism of $V_{m_{1},m_{2}}$-varieties

(6.14) $B_{m_{1}+m_{2}}\times_{\pi_{12}}V_{m_{1},m_{2}}\to
B_{m_{1}}\times_{\pi_{1}}V_{m_{1},m_{2}}\times_{\pi_{2}}B_{m_{2}},$

given by

$\left((h_{1},l_{1}l_{2}),(l_{1},l_{2})\right)\to\left((h_{1}\bmod{l_{1}},l_{1}),(l_{1},l_{2}),(h_{2}\bmod{l_{2}},l_{2})\right),$

and with inverse

$\left((h_{1,1},l_{1}),(l_{1},l_{2}),(h_{1,2},l_{2})\right)\to\left((h_{1,1}l_{2}+h_{1,2}l_{1},l_{1}l_{2}),(l_{1},l_{2})\right).$

To check that the isomorphisms (6.13) and (6.14) induce isomorphisms of
varieties with exponentials, we compute, using the same notation:

$\displaystyle\mathrm{res}$
$\displaystyle\left(\frac{h_{1,1}}{l_{1}}f\left(g_{1,1},\dots,g_{n,1}\right)+\frac{h_{1,2}}{l_{2}}f\left(g_{1,2},\dots,g_{n,2}\right)\right)$
$\displaystyle=\mathrm{res}\left(\frac{h_{1,1}}{l_{1}}f\left(g_{1},\dots,g_{n}\right)+\frac{h_{1,2}}{l_{2}}f\left(g_{1},\dots,g_{n}\right)\right)$
$\displaystyle=\mathrm{res}\left(\frac{h_{1,1}l_{2}+h_{1,2}l_{1}}{l_{1}l_{2}}f\left(g_{1},\dots,g_{n}\right)\right)$
$\displaystyle=\mathrm{res}\left(\frac{h}{l_{1}l_{2}}f\left(g_{1},\dots,g_{n}\right)\right).$

The lemma follows on putting these together. ∎

We can now write the series $\sum_{m\geq 0}S_{m}(f)T^{m}$ as a motivic Euler
product. For this, for every $m\geq 0$, we need to write $S_{m}(f)$ as the
$m$-th coefficient of a motivic Euler product; that is, as the sum of some
configuration spaces. For $i\geq 1$, define

(6.15) $U_{i}(f)=\left[\mathrm{Poly}_{<i}^{n}\times
C_{i},\mathrm{res}\left(\frac{h}{(t-x)^{i}}f(g_{1},\dots,g_{n})\right)\right],$

where $C_{i}=\\{(h,x)\in\mathrm{Poly}_{<i}\times\mathbf{A}^{1}:h(x)\neq 0\\}.$
We view this as an element of $\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{1}}$ via
the second projection $C_{i}\to\mathbf{A}^{1}$. Note that if $S_{i}(f)$ is
viewed as an element of $\mathscr{E}xp\mathscr{M}_{\mathrm{MPoly}_{i}}$ via
the projection morphism $B_{i}\to\mathrm{MPoly}_{i}$, given by
$(h_{1},h_{2})\mapsto h_{2}$, then $U_{i}(f)$ is the restriction of $S_{i}(f)$
to the locus $\Delta_{i}\simeq\mathbf{A}^{1}$ of polynomials of the form
$(t-x)^{i}$.

Recalling the definition of motivic Euler products from Section 2.5, we have
the following result.

######  Proposition 6.8.

We have the motivic Euler product decomposition

(6.16) $\mathfrak{S}(f)=\prod_{x\in\mathbf{A}^{1}}\left(1+\sum_{i\geq
1}U_{i}(f)_{x}T^{i}\right)\Big{|}_{T=\mathbf{L}^{-n}}.$

###### Proof.

Recalling the definition (6.11) of $\mathfrak{S}(f)$, it suffices to prove
that

$1+\sum_{m\geq 1}S_{m}(f)T^{m}=\prod_{x\in\mathbf{A}^{1}}\left(1+\sum_{i\geq
1}U_{i}(f)_{x}T^{i}\right).$

The element $S_{m}(f)$ comes from a variety with exponential over
$\mathrm{MPoly}_{m}$. On the other hand, there is an identification
$\mathrm{MPoly}_{m}\simeq\mathrm{Sym}^{m}\mathbf{A}^{1}$, giving rise to a
stratification of $\mathrm{MPoly}_{m}$ into pieces
$\mathrm{Conf}^{\omega}(\mathbf{A}^{1})$ indexed by partitions
$\omega=(m_{i})_{i}$ of $m$. We are going to exhibit an isomorphism

$S_{m}(f)_{|\mathrm{Conf}^{\omega}(\mathbf{A}^{1})}\to\mathrm{Conf}^{\omega}(U_{i}(f))_{i\geq
1}.$

We define the variety

$V_{\omega}=\\{(l_{i,1},\dots,l_{i,m_{i}})_{i\geq 1}:l_{i,j}\ \text{relatively
prime, monic},\ \deg l_{i,j}=i\\}=\left(\prod_{i\geq
1}(\mathrm{MPoly}_{i})^{m_{i}}\right)_{*}$

where we denote by $()_{*}$ the fact that we take only tuples of relatively
prime polynomials. We also define $W_{\omega}\subset V_{\omega}$ by

$W_{\omega}=\\{(l_{i,j})_{i,j}\in V_{\omega}:l_{i,j}\ \text{is of the form}\
(T-x_{i,j})^{i}\\}=\left(\prod_{i\geq 1}\Delta_{i}^{m_{i}}\right)_{*}.$

Finally, we consider the morphisms

$\pi_{i,j}:V_{\omega}\to\mathrm{MPoly}_{i},\ \ \ (l_{p,q})_{p,q}\mapsto
l_{i,j},$

which restrict to $\pi_{i,j}:W_{\omega}\to\Delta_{i}$, as well as

$\pi_{\omega}:V_{\omega}\to\mathrm{MPoly}_{m},\ \ \
(l_{p,q})_{p,q}\mapsto\prod_{i,j}l_{i,j}.$

By iterating the factorisation property in Proposition 6.7, we obtain an
isomorphism

$\pi_{\omega}^{*}S_{m}(f)\simeq\prod_{i\geq 1}\prod_{1\leq j\leq
m_{i}}\pi_{i,j}^{*}S_{i}(f),$

which is compatible with any permutation of the $l_{i,j}$ with fixed $i$.
Restriction to $W_{\omega}$ now gives

$\pi_{\omega}^{*}S_{m}(f)_{|W_{\omega}}\simeq\prod_{i\geq 1}\prod_{1\leq j\leq
m_{i}}\pi_{i,j}^{*}U_{i}(f),$

There is a natural permutation action of
$\mathfrak{S}_{\omega}=\prod_{i}\mathfrak{S}_{m_{i}}$ on $V_{\omega}$, which
restricts to $W_{\omega}$, and such that via the isomorphism
$\Delta_{i}\simeq\mathbf{A}^{1}$, the quotient $W_{\omega}$ is naturally
identified with $\mathrm{Conf}^{\omega}(\mathbf{A}^{1})$. Taking the quotient
by this permutation action, we get the result. ∎

####  6.3.4. Interpretation as local densities

######  Lemma 6.9.

Let $x\in\mathbf{A}^{1}.$ Then, for every $N\geq 1$, we have

$1+\sum_{i=1}^{N}U_{i}(f)_{x}\mathbf{L}^{-in}=\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)],$

where $\Lambda_{N}(f,x)$ is given by (1.2).

###### Proof.

Recalling (6.15), we begin by observing the relation

$U_{i}(f)_{x}=\mathbf{L}^{i-1}(\mathbf{L}-1)[\Lambda_{i}(f,x)]-\mathbf{L}^{i-1}[\Lambda^{\prime}_{i}(f,x)],$

in $\mathscr{E}xp\mathscr{M}_{\mathrm{Poly}_{<i}^{n}}$, where

$\Lambda^{\prime}_{i}(f,x)=\\{\mathbf{g}\in\mathrm{Poly}_{<i}^{n}:f(\mathbf{g})\equiv
0\bmod{(t-x)^{i-1}}\ \text{but}\ f(\mathbf{g})\not\equiv
0\bmod{(t-x)^{i}}\\}.$

Next, on observing that
$[\Lambda^{\prime}_{i}(f,x)]=\mathbf{L}^{n}[\Lambda_{i-1}(f,x)]-[\Lambda_{i}(f,x)]$,
the left-hand side in the lemma leads to a telescopic sum, and the only term
remaining is the one on the right-hand side. ∎

######  Remark 6.10.

Recall the motivic Euler product introduced in Proposition 6.8. Consider the
companion motivic Euler product

(6.17) $\prod_{x\in\mathbf{A}^{1}}\left(1+\sum_{i\geq
1}U_{i}(f)_{x}\mathbf{L}^{-in}ST^{i}\right),$

with an extra variable $S$. Then, by definition, the motivic Euler product
$\mathfrak{S}(f)$ is equal to the product (6.17) evaluated at $S=1$ and then
at $T=1$. On the other hand, the exponential sum estimates from Section 5.3
apply in the same way as in the proof of (6.12), and allow us to deduce the
bound

$w(U_{i}(f)_{x}\mathbf{L}^{-in})\leq\frac{(2^{d-1}(d-1)-n)i}{2^{d-2}(d-1)}.$

On assuming that $n>2^{d-1}(d-1)$, this shows that we may evaluate the product
(6.17) at $T=1$ and then Lemma 6.9 shows that we obtain the motivic Euler
product

$\prod_{x\in\mathbf{A}^{1}}\left(1+\left(\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)]-1\right)S\right).$

Hence, on evaluating at $S=1$, we get that

$\mathfrak{S}(f)=\prod_{x\in\mathbf{A}^{1}}\left(1+\left(\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)]-1\right)S\right)_{|S=1}.$

Bearing in mind Notation 4.10, the following result is a consequence of Remark
6.10.

######  Corollary 6.11.

We have

$\mathfrak{S}(f)=\prod_{x\in\mathbf{A}^{1}}\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)],$

where $\Lambda_{N}(f,x)$ is given by (1.2).

###  6.4. Singular integral

Returning to Proposition 6.3, we proceed by analysing the term

$\int_{\theta}U(\theta),$

where $U(\theta)$ is given by (6.8). We want to rewrite it in a way that draws
an obvious comparison with the singular integral in the classical circle
method.

####  6.4.1. Rewriting the sum (over polynomials) as an integral (over power
series in $t^{-1}$)

For this, we show that the function
$(g_{1},\dots,g_{n})\mapsto\mathrm{res}(\theta f(g_{1},\dots,g_{n}))$ is
invariant modulo $\mathbf{T}^{n}$; i.e., for every $\mathbf{x}\in
t^{e}k[[t^{-1}]]$ and every $\sigma\in\mathbf{T}^{n}$, we claim that

$\operatorname{ord}(\theta(f(\mathbf{x}+\sigma)-f(\mathbf{x})))<-1.$

The polynomial $f(\mathbf{x}+\sigma)-f(\mathbf{x})$ is a sum of monomials of
global degree $d$, with the degree in $\mathbf{x}$ being at most $d-1$. Thus,
bounding the order of the components of $\mathbf{x}$ by $e$ and the order of
the components of $\sigma$ by $-1$, we get

$\operatorname{ord}((f(\mathbf{x}+\sigma)-f(\mathbf{x})))\leq(d-1)e-1,$

whence

$\operatorname{ord}(\theta(f(\mathbf{x}+\sigma)-f(\mathbf{x})))\leq-
de-2+\gamma+(d-1)e-1=-e-3+\gamma<-1,$

since $\gamma\leq e.$ Thus, it makes sense to consider the element

$[\mathbf{A}^{n(e,-1)}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},(\mathbf{x},\theta)\mapsto\mathrm{res}(\theta
f(\mathbf{x}))]\in\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{n(e,-1)}\times\mathbf{A}^{(-de-2+\gamma,-de-2)}},$

and moreover, since there is an obvious isomorphism
$\mathbf{A}^{n(e,-1)}\simeq\mathrm{Poly}_{\leq e}^{n}$, we get the equality

$U(\theta)=\int_{\mathbf{x}}[\mathbf{A}^{n(e,-1)}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},\mathrm{res}(\theta
f(\mathbf{x}))]$

in $\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{(-de-2+\gamma,-de-2)}}.$

####  6.4.2. Change of variables to get an integral over $\mathbf{T}^{n}$

Using Remark 2.6, the change of variables $\mathbf{x}=t^{e+1}\mathbf{y}$,
together with the homogeneity of the polynomial $f$, gives the equality

$U(\theta)=\mathbf{L}^{n(e+1)}\int_{\mathbf{y}}[\mathbf{A}^{n(-1,-e-2)}\times\mathbf{A}^{(-de-2+\gamma,-de-2)},\mathrm{res}(\theta
t^{(e+1)d}f(\mathbf{y}))],$

in $\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{(-de-2+\gamma,-de-2)}}.$

####  6.4.3. Integrating over the space of $\theta$, and a change of variable

Now, using the change of variable $\beta=t^{(e+1)d}\theta$ and Remark 2.6, we
get

(6.18)
$\int_{\theta}U(\theta)=\mathbf{L}^{n(e+1)}\mathbf{L}^{-d(e+1)}\int_{\mathbf{y}}\mathcal{J}(\mathbf{y}),$

where

$\mathcal{J}(\mathbf{y})=\int_{\beta}[\mathbf{A}^{n(-1,-e-2)}\times\mathbf{A}^{(d-2+\gamma,d-2)},\mathrm{res}(\beta
f(\mathbf{y}))].$

We may now establish the following result.

######  Proposition 6.12.

We have

$\int_{\theta}U(\theta)=\mathbf{L}^{-de-1+\gamma}[V_{d+\gamma-1}],$

where

$V_{d+\gamma-1}=\\{\mathbf{y}\in\mathbf{A}^{n(-1,-e-2)}:\operatorname{ord}f(\mathbf{y})<-d-\gamma+1\\}.$

###### Proof.

By definition, we have

$\mathcal{J}(\mathbf{y})=\mathbf{L}^{d-1}[\mathbf{A}^{n(-1,-e-2)}\times\mathbf{A}^{(d-2+\gamma,d-2)},\mathrm{res}(\beta
f(\mathbf{y}))]$

in $\mathscr{E}xp\mathscr{M}_{\mathbf{A}^{n(-1,-e-2)}}.$ We write

$\beta=b_{d-2+\gamma}t^{d-2+\gamma}+\dots+b_{d-1}t^{d-1}+t^{d-2}k[[t^{-1}]]$

and

$f(\mathbf{y})=c_{d}(\mathbf{y})t^{-d}+c_{d+1}(\mathbf{y})t^{-d-1}+\cdots,$

where we have used the fact that $f$ is homogeneous of degree $d$ to deduce
that $f(\mathbf{y})$ only has terms of degrees $\leq-d$. Then

$\mathrm{res}(\beta
f(\mathbf{y}))=b_{d-1}c_{d}(\mathbf{y})+b_{d}c_{d+1}(\mathbf{y})+\dots+b_{d+\gamma-2}c_{d+\gamma-1}(\mathbf{y}).$

Thus, by Lemma 2.3, for every $\mathbf{y}\in\mathbf{A}^{n(-1,-e-2)}$, the
value of $\mathcal{J}(\mathbf{y})$ at $\mathbf{y}$ is
$\mathbf{L}^{d-1+\gamma}$ if and only if

$c_{d}(\mathbf{y})=c_{d+1}(\mathbf{y})=\dots=c_{d+\gamma-1}(\mathbf{y})=0,$

that is, if and only if $\operatorname{ord}f(\mathbf{y})<-d-\gamma+1$, and 0
otherwise. Using Lemma 2.1, it now follows from (6.18) that

$\displaystyle\int_{\theta}U(\theta)$
$\displaystyle=\mathbf{L}^{n(e+1)}\mathbf{L}^{-de-1}\mathbf{L}^{-d+1}\int_{\mathbf{y}}[V_{d+\gamma-1}]\mathbf{L}^{d-1+\gamma}$
$\displaystyle=\mathbf{L}^{n(e+1)}\mathbf{L}^{-de-1}\mathbf{L}^{-d+1}\mathbf{L}^{-n(e+1)}[V_{d+\gamma-1}]\mathbf{L}^{d-1+\gamma},$

with $V_{d+\gamma-1}$ as in the statement. The proposition easily follows. ∎

###  6.5. Connection to jet spaces

For any integer $N$, recall that the $N$th jet space of $X$ is given by

(6.19)
$\mathcal{L}_{N}(X)=\\{\mathbf{x}=\mathbf{x}_{0}+\mathbf{x}_{1}t+\dots+\mathbf{x}_{N}t^{N}:f(\mathbf{x})\equiv
0\bmod t^{N+1}\\},$

where $\mathbf{x}_{0},\dots,\mathbf{x}_{N}$ run over $\mathbf{A}^{n}$. We take
$\mathcal{L}_{N}(X)=\varnothing$ when $N<0$, and note that
$\mathcal{L}_{0}(X)=X$.

Jet spaces of smooth varieties are very well understood. In our setting, the
variety $X$ has one singular point at 0, so we decided it would be useful to
give a few details on the structure of jet spaces for hypersurfaces defined by
non-singular homogeneous polynomials $f\in\mathbf{C}[x_{1},\dots,x_{n}]$ of
degree $d$.

######  Lemma 6.13.

For any $N\geq 1$, we have

$\frac{[\mathcal{L}_{N}(X)]}{\mathbf{L}^{(N+1)(n-1)}}=\frac{[\mathcal{L}_{N-1}(X)]}{\mathbf{L}^{N(n-1)}}+\mathbf{L}^{N-n-n\lfloor
N/d\rfloor}\cdot\begin{cases}\mathbf{L}-1&\text{ if $d\nmid N$,}\\\
\mathbf{L}\left([X]-\mathbf{L}^{n-1}\right)&\text{ if $d\mid N$.}\end{cases}$

in $\mathscr{E}xp\mathscr{M}_{\mathbf{C}}$.

###### Proof.

We shall write $Z_{N}$ for the locus of points in $\mathcal{L}_{N}(X)$ with
$\mathbf{x}_{0}=\mathbf{0}$ and we let $U_{N}=\mathcal{L}_{N}(X)\setminus
Z_{N}$. We claim that

$Z_{N}\simeq\begin{cases}\mathbf{A}^{Nn}&\text{ if $0\leq N\leq d-1$,}\\\
\mathcal{L}_{N-d}(X)\times\mathbf{A}^{(d-1)n}&\text{ if $N\geq
d$.}\end{cases}$

To see this, we note that $Z_{N}$ is the set of
$\mathbf{x}_{1}t+\dots+\mathbf{x}_{N}t^{N}$ such that
$f(\mathbf{x}_{1}t+\dots+\mathbf{x}_{N}t^{N})\equiv 0\bmod{t^{N+1}}$. When
$N\leq d-1$ it is clear that $Z_{N}\simeq\mathbf{A}^{Nn}$. Suppose that $N\geq
d$. On relabelling the variables, $Z_{N}$ is equal to space of
$\mathbf{x}=\mathbf{x}_{0}+\mathbf{x}_{1}t+\dots+\mathbf{x}_{N-1}t^{N-1}$ such
that $f(\mathbf{x})\equiv 0\bmod{t^{N+1-d}}$, since $f$ is homogeneous of
degree $d$. The claim is now obvious.

Now let $N\geq 1$ and consider the truncation morphism $\pi:U_{N}\to U_{N-1}.$
For each
$\mathbf{x}=\mathbf{x}_{0}+\mathbf{x}_{1}t+\dots+\mathbf{x}_{N-1}t^{N-1}\in
U_{N-1}$, the fibre $\pi^{-1}(\mathbf{x})$ is the set of
$\mathbf{x}_{N}\in\mathbf{A}^{n}$ such that
$f(\mathbf{x}+\mathbf{x}_{N}t^{N})\equiv 0\bmod t^{N+1}$. Note that
$f(\mathbf{x})\equiv 0\bmod t^{N}$ and

$f(\mathbf{x}+\mathbf{x}_{N}t^{N})=f(\mathbf{x})+t^{N}\mathbf{x}_{N}\cdot\nabla
f(\mathbf{x}_{0})+O(t^{N+1}),$

by Taylor’s theorem. Since $f$ is non-singular, $\nabla f(\mathbf{x}_{0})$
doesn’t vanish for $\mathbf{x}_{0}\neq\mathbf{0}$. Hence
$\pi^{-1}(\mathbf{x})\simeq\mathbf{A}^{n-1}$. Hence it follows that

(6.20) $[U_{N}]=[U_{N-1}]\mathbf{L}^{n-1},$

for any $N\geq 1$, since $\pi$ is a Zariski locally trivial fibration with
fibre $\mathbf{A}^{n-1}$.

To prove the lemma we write

$\Psi_{N}=\frac{[\mathcal{L}_{N}(X)]}{\mathbf{L}^{(N+1)(n-1)}}-\frac{[\mathcal{L}_{N-1}(X)]}{\mathbf{L}^{N(n-1)}},$

for any $N\geq 1$. To begin with, we assume that $N>d$. Then

$\displaystyle[\mathcal{L}_{N}(X)]$
$\displaystyle=[U_{N}]+[Z_{N}]=[U_{N}]+[\mathcal{L}_{N-d}(X)]\mathbf{L}^{(d-1)n},$

and the same expression holds with $N-1$ in place of $N$. Thus

$\displaystyle\Psi_{N}$
$\displaystyle=\mathbf{L}^{-(N+1)(n-1)}\left([U_{N}]-[U_{N-1}]\mathbf{L}^{n-1}\right)+\mathbf{L}^{-(n-d)}\Psi_{N-d}=\mathbf{L}^{-(n-d)}\Psi_{N-d},$

by (6.20). We next calculate $\Psi_{N}$ when $N\leq d$, for which we observe
that

$[Z_{N}]=\begin{cases}\mathbf{L}^{Nn}&\text{ if $0\leq N<d$,}\\\
[X]\mathbf{L}^{(d-1)n}&\text{ if $N=d$},\end{cases}$

since $\mathcal{L}_{0}(X)=X$. It easily follows from (6.20) that

$\displaystyle\Psi_{d}$
$\displaystyle=\mathbf{L}^{-(d+1)(n-1)}\left([Z_{d}]-[Z_{d-1}]\mathbf{L}^{n-1}\right)$
$\displaystyle=\mathbf{L}^{-(d+1)(n-1)}\left([X]\mathbf{L}^{(d-1)n}-\mathbf{L}^{dn-1}\right)$
$\displaystyle=\mathbf{L}^{-(2n-d-1)}\left([X]-\mathbf{L}^{n-1}\right).$

Similarly,

$\displaystyle\Psi_{N}$
$\displaystyle=\mathbf{L}^{-(N+1)(n-1)}\left([Z_{N}]-[Z_{N-1}]\mathbf{L}^{n-1}\right)$
$\displaystyle=\mathbf{L}^{-(N+1)(n-1)}\left(\mathbf{L}^{Nn}-\mathbf{L}^{Nn-1}\right)$
$\displaystyle=\mathbf{L}^{-(n-N)}\left(\mathbf{L}-1\right),$

if $1\leq N\leq d-1$.

We are now ready to complete the proof of the lemma. Any $N\geq 1$ can be
written $N=qd+r$, where $q=\lfloor N/d\rfloor$ and $0\leq r<d$. If $r>0$ then
we may put the above calculations together to deduce that

$\Psi_{N}=\mathbf{L}^{-(n-d)q}\Psi_{r}=\mathbf{L}^{N-n(q+1)}(\mathbf{L}-1)=\mathbf{L}^{N-n-n\lfloor
N/d\rfloor}(\mathbf{L}-1)$

On the other hand, if $r=0$ then $q\geq 1$ and

$\Psi_{N}=\mathbf{L}^{-(n-d)(q-1)}\Psi_{d}=\mathbf{L}^{N-n+1-nN/d}\left([X]-\mathbf{L}^{n-1}\right).$

This completes the proof of the lemma. ∎

######  Remark 6.14.

Let $N\geq 1$. If $d\nmid N$ then

$\dim\left(\mathbf{L}^{N-n-n\lfloor
N/d\rfloor}\cdot(\mathbf{L}-1)\right)\leq(N+1)(1-n/d),$

since $\lfloor N/d\rfloor\geq N/d-1+1/d$. Moreover, if $d\mid N$ then

$\dim\left(\mathbf{L}^{N-n-n\lfloor
N/d\rfloor}\cdot\mathbf{L}\left([X]-\mathbf{L}^{n-1}\right)\right)\leq
N(1-n/d).$

######  Corollary 6.15.

Assume that $n>d$. Then the $N$-th jet space $\mathcal{L}_{N}(X)$ is
irreducible and has dimension $(n-1)(N+1)$.

###### Proof.

Since we are working over a field of characteristic $0$, and since $X$ is
irreducible, the irreducibility of $\mathcal{L}_{N}(X)$ follows from a theorem
of Kolchin [Kol73, Proposition 10 in Chapter IV]. Turning to the dimension,
the lower bound $\dim\mathcal{L}_{N}(X)\geq(n-1)(N+1)$ is trivial. It remains
to show that $\dim\mathcal{L}_{N}(X)\leq(n-1)(N+1)$, which we shall do by
induction on $N$. When $N=0$ we have $\mathcal{L}_{0}(X)=X$ and the claim is
obvious. If $N\geq 1$ it follows from Lemma 6.13, the induction hypothesis and
Remark 6.14 that

$\dim\left(\frac{\mathcal{L}_{N}(X)}{\mathbf{L}^{(N+1)(n-1)}}\right)\leq(N+1)(1-n/d)<0,$

since $n>d$. This confirms that $\dim\mathcal{L}_{N}(X)\leq(n-1)(N+1)$. ∎

We now proceed by relating the jet spaces to the spaces that we met in
Corollary 6.11 and Proposition 6.12.

######  Lemma 6.16.

Let $x\in\mathbf{A}^{1}$ and let $N\geq 1$. Then we have the relation

$[\Lambda_{N}(f,x)]=[\Lambda_{N}(f,\infty)]=[\mathcal{L}_{N-1}(X)]$

in the Grothendieck ring of varieties, where $\Lambda_{N}(f,x)$ is given by
(1.2) and $\Lambda_{N}(f,\infty)$ is given by (1.3).

###### Proof.

The first equality is clear and so it suffices to prove that
$\Lambda_{N}(f,x)\simeq\mathcal{L}_{N-1}(X)$, for any $x\in\mathbf{A}^{1}$.
According to (1.2), $\Lambda_{N}(f,x)$ is the space of
$\mathbf{g}=\mathbf{g}_{0}+\mathbf{g}_{1}t+\dots+\mathbf{g}_{N-1}t^{N-1}$ such
that $f(\mathbf{g})\equiv 0\bmod{(t-x)^{N}}$, for
$\mathbf{g}_{0},\dots,\mathbf{g}_{N-1}\in\mathbf{A}^{n}$. This is isomorphic
to the space of
$\mathbf{g}^{\prime}=\mathbf{g}_{0}^{\prime}+\mathbf{g}_{1}^{\prime}(t-x)+\dots+\mathbf{g}_{N-1}^{\prime}(t-x)^{N-1}$
such that $f(\mathbf{g}^{\prime})\equiv 0\bmod{(t-x)^{N}}$, via the map
$\mathbf{g}\mapsto\mathbf{g}^{\prime}$ given by

$\displaystyle\mathbf{g}_{0}^{\prime}$
$\displaystyle=\mathbf{g}_{0}+x\mathbf{g}_{1}+\dots+x^{N-1}\mathbf{g}_{N-1},$
$\displaystyle\mathbf{g}_{1}^{\prime}$
$\displaystyle=\mathbf{g}_{1}+2x\mathbf{g}_{2}+3x^{2}\mathbf{g}_{3}+\dots+(N-1)x^{N-1}\mathbf{g}_{N-1},$
$\displaystyle\mathbf{g}_{2}^{\prime}$
$\displaystyle=\mathbf{g}_{2}+\tbinom{3}{2}x\mathbf{g}_{3}+\tbinom{4}{2}x^{2}\mathbf{g}_{4}+\dots+\tbinom{N-1}{2}x^{N-1}\mathbf{g}_{N-1},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \vdots$
$\displaystyle\mathbf{g}_{N-1}^{\prime}$ $\displaystyle=\mathbf{g}_{N-1}.$

But this is just the jet space $\mathcal{L}_{N-1}(X)$, which thereby completes
the proof. ∎

Turning to the space $V_{d+\gamma-1}$ that appears in Proposition 6.12, we
shall prove the following result, relating it to the variety
$\Lambda_{N}(f,\infty)$ in (1.3) for a suitable choice of $N$.

######  Lemma 6.17.

We have the relation

$[V_{d+\gamma-1}]=[\Lambda_{\gamma}(f,\infty)]\mathbf{L}^{n(e+1-\gamma)}$

in the Grothendieck ring of varieties.

###### Proof.

For any
$\mathbf{y}=\mathbf{y}_{1}t^{-1}+\dots+\mathbf{y}_{e+1}\mathbf{t}^{-e-1}\in
V_{d+\gamma-1}$, the condition $\operatorname{ord}f(\mathbf{y})<-d-\gamma+1$
says that the coefficients of the terms in $f(\mathbf{y})$ of degrees
$-d,\dots,-d-\gamma+1$ should all be zero. This does not affect the
coordinates $\mathbf{y}_{\gamma+1},\dots,\mathbf{y}_{e+1}$, because $f$ is
homogeneous of degree $d$. Thus, there is an isomorphism

$V_{d+\gamma-1}\simeq\\{\mathbf{y}\in\mathbf{A}^{n(-1,-\gamma)}:\operatorname{ord}f(\mathbf{y})<-d-\gamma+1\\}\times\mathbf{A}^{n(e+1-\gamma)},$

sending $\mathbf{y}$ to
$(\mathbf{y}_{1}t^{-1}+\dots+\mathbf{y}_{\gamma}t^{-\gamma},(\mathbf{y}_{\gamma+1},\dots,\mathbf{y}_{e+1})).$
Writing
$\mathbf{y}_{1}t^{-1}+\dots+\mathbf{y}_{\gamma}t^{-\gamma}=t^{-1}\mathbf{g}$,
where
$\mathbf{g}=\mathbf{y}_{1}+\mathbf{y}_{2}t^{-1}+\dots+\mathbf{y}_{\gamma}t^{-(\gamma-1)}$,
we see that $\operatorname{ord}f(\mathbf{y})<-d-\gamma+1$ if and only if
$f(\mathbf{g})\in t^{-\gamma}\mathbf{C}[t^{-1}]$. Hence
$V_{d+\gamma-1}\simeq\Lambda_{\gamma}(f,\infty)\times\mathbf{A}^{n(e+1-\gamma)}$,
in the notation of (1.3). ∎

###  6.6. Major arcs: conclusion

We are now ready to conclude the proof of Proposition 3.7. We return to
Proposition 6.3, where $\gamma\geq 1$ is given by (6.3). It follows from
Proposition 6.12 and Lemma 6.17 that

$\displaystyle N_{\mathrm{major}}$
$\displaystyle=\mathbf{L}^{\mu(e)}\left(\sum_{0\leq
m\leq\lfloor\frac{e+1}{2}\rfloor}S_{m}(f)\mathbf{L}^{-nm}\right)\mathbf{L}^{-(n-1)\gamma}[\Lambda_{\gamma}(f,\infty)],$

where $\mu(e)$ is given by (1.1).

Next, we observe that
$w(\mathbf{L}^{-(n-1)\gamma}[\Lambda_{\gamma}(f,\infty)])\leq 0$, as follows
from Lemma 6.16 and Corollary 6.15. Appealing to Lemma 6.5, we conclude that

$N_{\mathrm{major}}=\mathbf{L}^{\mu(e)}\left(\mathfrak{S}(f)\cdot\mathbf{L}^{-(n-1)\gamma}[\Lambda_{\gamma}(f,\infty)]+R_{e}^{\prime}\right),$

where $\mathfrak{S}(f)$ is the motivic Euler product described in Corollary
6.11 and

$w\left(R_{e}^{\prime}\right)\leq
4-\tilde{\nu}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right),$

with $\tilde{\nu}$ given by (3.2). Inserting Lemma 6.16 into Lemma 6.13, it
therefore follows from Remark 6.14 that

$\mathbf{L}^{-(n-1)\gamma}[\Lambda_{\gamma}(f,\infty)]=\lim_{N\to\infty}\mathbf{L}^{-(n-1)N}[\Lambda_{N}(f,\infty)]+E,$

in $\widehat{\mathscr{E}xp\mathscr{M}_{\mathbf{C}}}$, where

$w(E)\leq\begin{cases}-2\gamma(n/d-1)&\text{ if $d\mid\gamma$,}\\\
-2(\gamma+1)(n/d-1)&\text{ if $d\nmid\gamma$.}\end{cases}$

We now return to our expression for the major arcs and we recall from Remark
6.6 that $w(\mathfrak{S}(f))=0$. This allows us to write

$N_{\mathrm{major}}=\mathbf{L}^{\mu(e)}\left(\mathfrak{S}(f)\cdot\lim_{N\to\infty}\mathbf{L}^{-(n-1)N}[\Lambda_{N}(f,\infty)]+R_{e}^{\prime\prime}\right),$

where

(6.21)
$w\left(R_{e}^{\prime\prime}\right)\leq\max\left(-\frac{(e+1+\kappa_{d,e})(n-d)}{d},\leavevmode\nobreak\
4-\tilde{\nu}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right)\right)$

and

$\kappa_{d,e}=\begin{cases}0&\text{ if $d\mid\gamma$,}\\\ 2&\text{ if
$d\nmid\gamma$.}\end{cases}$

Suppose first that $d\geq 3$. Then we claim that the second term always
exceeds the first term in the maximum (6.21), as claimed in Proposition 3.7.
To check the claim, we note that it is equivalent to

$\tilde{\nu}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right)\leq\frac{(e+1+\kappa_{d,e})(n-d)}{d}+4.$

On inserting the definition (3.2) of $\tilde{\nu}$, and collecting together
the coefficients of $n$, we see that this holds if $A_{d,e}\leq nB_{d,e},$
where $A_{d,e}=\kappa_{d,e}-(e+1)-4$ and

$B_{d,e}=\frac{e+1+\kappa_{d,e}}{d}-\frac{1}{2^{d-2}}\left(1+\left\lfloor\frac{e+1}{2d-2}\right\rfloor\right).$

Since $A_{d,e}<0$, the claim will follow if we can prove that $B_{d,e}\geq 0$
for $e\geq 1$ and $d\geq 3$. If $d\nmid\gamma$ then $\kappa_{d,e}=2$ and

$\displaystyle B_{d,e}$
$\displaystyle\geq(e+1)\left(\frac{1}{d}-\frac{1}{2^{d-1}(d-1)}\right)+\frac{2}{d}-\frac{1}{2^{d-2}}\geq\frac{4}{d}-\frac{d}{2^{d-2}(d-1)},$

on taking $e\geq 1$. Thus we deduce that $B_{d,e}\geq 0$ for all $d\geq 2$. If
$d\mid\gamma$ then $\kappa_{d,e}=0$ and we must have $e+1\geq 2d-1\geq 5$, if
$d\geq 3$. But then it follows that

$\displaystyle B_{d,e}$
$\displaystyle\geq(e+1)\left(\frac{1}{d}-\frac{1}{2^{d-1}(d-1)}\right)-\frac{1}{2^{d-2}}\geq\frac{5}{d}-\frac{2d+3}{2^{d-1}(d-1)}.$

This is non-negative for all $d\geq 3$.

Finally, we suppose that $d=2$. We may also suppose that $2\mid\gamma$, since
the previous paragraph suffices to handle the case $2\nmid\gamma$, even when
$d=2$. Thus (6.21) becomes

$\displaystyle w\left(R_{e}^{\prime\prime}\right)$
$\displaystyle\leq\max\left(-\frac{(e+1)(n-2)}{2},\leavevmode\nobreak\