4-\tilde{\nu}\left(1+\left\lfloor\frac{e+1}{2}\right\rfloor\right)\right)$
$\displaystyle\leq\max\left(-\frac{(e+1)(n-2)}{2},\leavevmode\nobreak\
-\frac{n(e+2)}{2}+2e+8\right)$

since $\kappa_{2,e}=0$, $\tilde{\nu}=n-4$ and
$1+\lfloor\frac{e+1}{2}\rfloor\geq e/2+1$. This completes the proof of
Proposition 3.7.

##  7\. The motivic minor arcs

Recall the expression for $M_{m,\gamma}$ in Lemma 6.1 and put
$\Delta=\left\lfloor\frac{e+1}{2}\right\rfloor.$ In the previous section we
took the full set of major arcs to be $\mathfrak{M}:=M_{\Delta,\Delta}$. Our
aim in this section is therefore to get a bound for

$N_{\mathrm{minor}}=\mathbf{L}^{-de-1}[\mathrm{Poly}_{\leq e}^{n}\times
S,\mathrm{res}(\alpha f(g_{1},\dots,g_{n}))].$

where $S=\mathbf{A}^{de+1}-\mathfrak{M}$. Now it follows from Proposition 5.4
that

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

where $N=(e+1)n$ and

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

###  7.1. The minor arc bound

We recall the definition of $A_{m}^{de+1}$ from Remark 3.4. In particular
$A_{m_{0}}^{de+1}=\mathbf{A}^{de+1}$, where
$m_{0}=\lfloor\frac{de+1}{2}\rfloor$. Bearing this in mind, we begin by
proving the following result.

######  Lemma 7.1.

Let $\alpha\in A^{de+1}_{m+1}-A^{de+1}_{m}$, with $m\leq m_{0}-1$. Then

$\dim
N(\alpha)\leq\left(de+d-e-2-\left\lfloor\frac{m}{d-1}\right\rfloor\right)n.$

###### Proof.

We seek to apply Lemma 5.7. Since $\alpha\in A^{de+1}_{m+1}$, there exist
coprime $h_{1},h_{2}$ with $\deg h_{1}<\deg h_{2}\leq m+1$ and $\theta$ such
that $\operatorname{ord}\theta\leq-de-2+m+1-\deg h_{2}$. Moreover, since
$\alpha\not\in A^{de+1}_{m}$, we have the following dichotomy: either $\deg
h_{2}=m+1$, or if $\deg h_{2}\leq m$, then we can’t have
$\operatorname{ord}\theta\leq-de-2+m-\deg h_{2}$. Thus

$\operatorname{ord}\theta=-de-2+m+1-\deg h_{2}$

if $\deg h_{2}\leq m$. Using this, we now prove a bound on $N(\alpha)$ using
Lemma 5.7 with $E=e+1$.

We place ourselves in the first case, where, using the notation of the lemma,
$\rho=m+1$ and $\psi\leq-de-2.$ We start by checking condition $(1)$ in the
lemma. The first inequality will be satisfied provided that $(d-1)s\geq m-e.$
Using $\psi\leq-de-2$, we see that the second inequality is also implied by
$(d-1)s\geq m-e$. We now check condition $(2)$ in the lemma. In fact, as we
have no longer have a useful lower bound on $\psi$ in this setting, we will
find a condition for the first inequality to hold, which is equivalent to

$(d-1)e-m\leq(d-1)s.$

We now treat the second case, where $\rho\leq m$ and
$\psi=-de-2+m+1-\rho=-de-1+m-\rho$. We start by checking condition $(1)$ in
the lemma. The first inequality is implied by $(d-1)s\geq m-e$. The second
inequality is equivalent to $(d-1)s\geq m-e$. We now check condition $(2)$ in
the lemma, noting that we have no useful lower bound on $\rho$ in this
setting. But we see that the second inequality holds if and only if

$(d-1)e-m\leq(d-1)s.$

In conclusion, we are able to apply Lemma 5.7 provided that $s$ is an integer
chosen to satisfy

$(d-1)s\geq\max\left\\{0,m-e,(d-1)e-m\right\\}.$

Since $d\geq 2$, one checks that the maximum is $(d-1)e-m$ if $m\leq m_{0}-1$.
Hence, taking $s=e-\left\lfloor\frac{m}{d-1}\right\rfloor$, we arrive at the
lower bound

$\dim
N(\alpha)\leq(d-2)(e+1)n+\left(e-\left\lfloor\frac{m}{d-1}\right\rfloor\right)n=\left(de+d-e-2-\left\lfloor\frac{m}{d-1}\right\rfloor\right)n,$

as claimed. ∎

The bound in Lemma 7.1 becomes stronger as $m$ increases. Thus, to optimise
our minor arc estimate, we will stratify the minor arcs. We note from Remark
3.4 and Lemma 6.1 that $A^{de+1}_{\Delta}\subset
M_{\Delta,\Delta}\subset\mathfrak{M}$ and
$A^{de+1}_{m_{0}}=\mathbf{A}^{de+1}$, where
$m_{0}=\lfloor\frac{de+1}{2}\rfloor$. Since the spaces $A^{de+1}_{m}$ form an
increasing sequence, we may therefore cut up the complement of the major arcs
as

$\mathbf{A}^{de+1}-\mathfrak{M}=\left(\bigsqcup_{\Delta\leq m\leq
m_{0}-1}\left(A^{de+1,\star}_{m+1}-A^{de+1,\star}_{m}\right)\right),$

where $A^{de+1,\star}_{m}:=A^{de+1}_{m}-\mathfrak{M}.$

We are now ready to produce our final bound for the minor arc contribution

$N_{\mathrm{minor}}=\mathbf{L}^{-de-1}\left[\mathrm{Poly}_{\leq
e}^{n}\times\left(\mathbf{A}^{de+1}-\mathfrak{M}\right),\mathrm{res}(\alpha
f(g_{1},\dots,g_{n})\right].$

Thus, using property (1) in Proposition 4.4, we get

$\displaystyle w(N_{\mathrm{minor}})$
$\displaystyle\leq-2de-2+w\left(\sum_{m=\Delta}^{m_{0}-1}[\mathrm{Poly}_{\leq
e}^{n}\times(A^{de+1}_{m+1}-A^{de+1}_{m}),\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))]\right)$ $\displaystyle\leq-2de-2+\max_{\Delta\leq m\leq
m_{0}-1}w([\mathrm{Poly}_{\leq
e}^{n}\times(A^{de+1}_{m+1}-A^{de+1}_{m}),\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))]).$

According to Remark 3.5 we have $\dim(A^{de+1}_{m+1}-A^{de+1}_{m})\leq
2(m+1)$. We now use successively property (2) of Proposition 4.4 and the
inequality (7.1), to write

$\displaystyle w([$ $\displaystyle\mathrm{Poly}_{\leq
e}^{n}\times(A^{de+1}_{m+1}-A^{de+1}_{m}),\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))])$ $\displaystyle\leq
w_{A^{de+1}_{m+1}-A^{de+1}_{m}}\left([\mathrm{Poly}_{\leq
e}^{n}\times(A^{de+1}_{m+1}-A^{de+1}_{m}),\mathrm{res}(\alpha
f(g_{1},\dots,g_{n}))]\right)+4(m+1)$
$\displaystyle\leq\frac{\dim_{A^{de+1}_{m+1}-A^{de+1}_{m}}N(\alpha)+(e+1)n(2^{d-1}-(d-1))}{2^{d-2}}+4(m+1),$

for any $m$. It now follows from Lemma 7.1 that $w(N_{\mathrm{minor}})$ is

$\displaystyle\leq 2-2de+\max_{\Delta\leq m\leq
m_{0}-1}\left(\frac{\left(de+d-e-2-\left\lfloor\frac{m}{d-1}\right\rfloor\right)n+(e+1)n(2^{d-1}-(d-1)))}{2^{d-2}}+4m\right)$
$\displaystyle\leq
2-2de-\frac{n}{2^{d-2}}+2n(e+1)+\frac{1}{2^{d-2}}\max_{\Delta\leq m\leq
m_{0}-1}\left(2^{d}m-\left\lfloor\frac{m}{d-1}\right\rfloor n\right).$

We now appeal to Lemma 6.4, to conclude that

$\displaystyle w(N_{\mathrm{minor}})$ $\displaystyle\leq
2-2de-\frac{n}{2^{d-2}}+2n(e+1)+4(d-1)+\frac{1}{2^{d-2}}\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n).$
$\displaystyle=\underbrace{2n(e+1)-2(de+1)}_{2\times\text{expected
dimension}}+4d-\frac{n}{2^{d-2}}+\frac{1}{2^{d-2}}\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n).$
$\displaystyle=2\mu(e)+\frac{1}{2^{d-2}}\left((2^{d}d-n)+\left\lfloor\frac{e+1}{2d-2}\right\rfloor(2^{d}(d-1)-n)\right)$

The statement of Proposition 3.8 now follows.

##  8\. The space of morphisms

In this section we prove Theorem 1.3, which deals with the space

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

where $(\mathrm{Poly}^{n}_{\leq e}-\\{0\\})/\mathbf{C}^{\times}$ is the space
of non-zero $n$-tuples of polynomials of degree $\leq e$, viewed modulo the
multiplication by a non-zero scalar. We begin with the following result, which
allows us to pass from the classes of the naive moduli spaces $M_{e}$ to the
class of the moduli space $\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})$.

######  Lemma 8.1.

Let $e\geq 1$. Then

$[\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})]=\frac{[M_{e}]-(\mathbf{L}+1)[M_{e-1}]+\mathbf{L}[M_{e-2}]}{\mathbf{L}-1},$

with the convention that $M_{-1}$ is a point.

###### Proof.

For every $e\geq 1$, note that $M_{e-1}$ is a subset of $M_{e}$, and that a
point $g=(g_{1},\dots,g_{n})\in M_{e}-M_{e-1}$ may be written in the form

$g=(hg^{\prime}_{1},\dots,hg^{\prime}_{n})$

where $h=\gcd(g_{1},\dots,g_{n})$ and $(g^{\prime}_{1},\dots,g^{\prime}_{n})$,
defines, up to multiplication by a non-zero scalar multiple, an element of
$\mathrm{Mor}_{e-\deg h}(\mathbf{P}^{1},\tilde{X})$. Thus, we have the
decomposition

$\left(M_{e}-M_{e-1}\right)/\mathbf{C}^{\times}=\bigsqcup_{i=0}^{e}\mathrm{MPoly}_{i}\times\mathrm{Mor}_{e-i}(\mathbf{P}^{1},\tilde{X}),$

where we recall that $\mathrm{MPoly}_{i}$ is the space of degree $i$ monic
polynomials. Thus, in the Grothendieck ring, we have the relation

$[M_{e}]-[M_{e-1}]=(\mathbf{L}-1)\sum_{i=0}^{e}\left([\mathrm{MPoly}_{i}]\times[\mathrm{Mor}_{e-i}(\mathbf{P}^{1},\tilde{X})]\right),$

for every $e\geq 1$.

In terms of generating series, we get the relation

$\sum_{e\geq 1}([M_{e}]-[M_{e-1}])T^{e}=(\mathbf{L}-1)\sum_{e\geq
1}\left(\sum_{i=0}^{e}[\mathrm{MPoly}_{i}]\times[\mathrm{Mor}_{e-i}(\mathbf{P}^{1},\tilde{X})]\right)T^{e}.$

The left-hand side may be rewritten as

$\sum_{e\geq 1}[M_{e}]T^{e}-\sum_{e\geq 0}[M_{e}]T^{e+1}=(1-T)\sum_{e\geq
0}[M_{e}]T^{e}-[X],$

using the fact that $M_{0}=X$. As for the right-hand side, we rewrite it as

$(\mathbf{L}-1)\left(\left(\sum_{i\geq
0}[\mathrm{MPoly}_{i}]T^{i}\right)\left(\sum_{i\geq
0}[\mathrm{Mor}_{i}(\mathbf{P}^{1},\tilde{X})]T^{i}\right)-[\tilde{X}]\right),$

since $\mathrm{Mor}_{0}(\mathbf{P}^{1},\tilde{X})=\tilde{X}$. Noting that
$[\mathrm{MPoly}_{i}]=\mathbf{L}^{i}$, we get that

$\sum_{i\geq 0}[\mathrm{MPoly}_{i}]T^{i}=\sum_{i\geq
0}\mathbf{L}^{i}T^{i}=\frac{1}{1-\mathbf{L}T}.$

We have $[\tilde{X}]=(\mathbf{L}-1)^{-1}([X]-1)$, from which it follows that

$(1-T)(1-\mathbf{L}T)\left(\sum_{e\geq
0}[M_{e}]T^{e}\right)=(\mathbf{L}-1)\sum_{e\geq
0}[\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})]T^{e}+1-\mathbf{L}T.$

We expand and deduce that the left-hand side is

$\displaystyle=(1-(1+\mathbf{L})T+\mathbf{L}T^{2})\left(\sum_{e\geq
0}[M_{e}]T^{e}\right)$ $\displaystyle=\sum_{e\geq
0}[M_{e}]T^{e}-(\mathbf{L}+1)\sum_{e\geq
0}[M_{e}]T^{e+1}+\mathbf{L}\sum_{e\geq 0}[M_{e}]T^{e+2}$
$\displaystyle=\sum_{e\geq 2}[M_{e}]T^{e}-(\mathbf{L}+1)\sum_{e\geq
2}[M_{e-1}]T^{e}+\mathbf{L}\sum_{e\geq
2}[M_{e-2}]T^{e}+[M_{0}]+[M_{1}]T-(\mathbf{L}+1)[M_{0}]T$
$\displaystyle=[M_{0}]+([M_{1}]-(\mathbf{L}+1)[M_{0}])T+\sum_{e\geq
2}([M_{e}]-(\mathbf{L}+1)[M_{e-1}]+\mathbf{L}[M_{e-2}])T^{e}.$

The statement of the lemma easily follows. ∎

Our next result expresses the product of local densities in a more convenient
form.

######  Lemma 8.2.

Assume that $n>d$ and let $x\in\mathbf{A}^{1}$. Then

$\displaystyle\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)]$
$\displaystyle=\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,\infty)]$
$\displaystyle=\mathbf{L}^{-(n-2)}[\tilde{X}](1-\mathbf{L}^{-1})(1-\mathbf{L}^{-(n-d)})^{-1}.$

###### Proof.

We prove the part involving $\Lambda_{N}(f,x)$, the proof for
$\Lambda_{N}(f,\infty)$ being identical. Given $x\in\mathbf{A}^{1}$ and
integer $N\geq 0$ we define

$\Lambda_{N}^{*}(f,x)=\left\\{\mathbf{g}\in\mathbf{C}[t]^{n}:\begin{array}[]{l}\deg(g_{1}),\dots,\deg(g_{n})<N,\leavevmode\nobreak\
f(\mathbf{g})\equiv 0\bmod{(t-x)^{N}}\\\
(g_{1},\dots,g_{n})\not\equiv(0,\dots,0)\bmod(t-x)\end{array}\right\\},$

as an analogue of (1.2). Next, let us put

$\Lambda_{N,i}(f,x)=\left\\{\mathbf{g}\in\mathbf{C}[t]^{n}:\begin{array}[]{l}\deg(g_{1}),\dots,\deg(g_{n})<N,\leavevmode\nobreak\
f(\mathbf{g})\equiv 0\bmod{(t-x)^{N}}\\\
\mathbf{g}\equiv\mathbf{0}\bmod(t-x)^{i},\leavevmode\nobreak\
\mathbf{g}\not\equiv\mathbf{0}\bmod(t-x)^{i+1}\end{array}\right\\},$

for any $0\leq i<N.$ If $N>id+i$ then $\Lambda_{N,i}(f,x)$ is isomorphic to
the set of polynomials $\mathbf{g}\in\mathbf{C}[t]^{n}$ of degree $<N-i$ such
that $f(\mathbf{g})\equiv 0\bmod{(t-x)^{N-di}}$ and
$\mathbf{g}\not\equiv\mathbf{0}\bmod(t-x)$. Hence
$\Lambda_{N,i}(f,x)\simeq\mathbf{A}^{ni(d-1)}\times\Lambda_{N-i}^{*}(f,x)$ if
$N>id+i$. We conclude that

$[\Lambda_{N}(f,x)]=\mathbf{L}^{n(N-\lceil\frac{N-1}{d}\rceil)}+\sum_{0\leq
i\leq(N-1)/d}\mathbf{L}^{n(d-1)i}[\Lambda_{N-di}^{*}(f,x)],$

from which it follows that

(8.1)
$\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)]=\mathbf{L}^{N-n\lceil\frac{N-1}{d}\rceil}+\sum_{0\leq
i\leq(N-1)/d}\mathbf{L}^{-(n-d)i}\mathbf{L}^{-(N-di)(n-1)}[\Lambda_{N-di}^{*}(f,x)].$

The proof of Lemma 6.16 goes through and yields
$[\Lambda_{N}^{*}(f,x)]=[U_{N-1}]$, where we recall that

$U_{N-1}=\\{\mathbf{x}=\mathbf{x}_{0}+\mathbf{x}_{1}t+\dots+\mathbf{x}_{N-1}t^{N-1}:f(\mathbf{x})\equiv
0\bmod t^{N},\leavevmode\nobreak\ \mathbf{x}_{0}\neq\mathbf{0}\\}.$

It now follows from (6.20) that

$\mathbf{L}^{-(N+1)(n-1)}[\Lambda_{N+1}^{*}(f,x)]=\mathbf{L}^{-N(n-1)}[\Lambda_{N}^{*}(f,x)],$

for any $N\geq 1$, whence

$\mathbf{L}^{-(N-di)(n-1)}[\Lambda_{N-di}^{*}(f,x)]=\mathbf{L}^{-(n-1)}[\Lambda_{1}^{*}(f,x)]=\mathbf{L}^{-(n-1)}(\mathbf{L}-1)[\tilde{X}]$

in (8.1). Observing that

$N-n\left\lceil\frac{N-1}{d}\right\rceil\leq
N-\frac{n(N-1)}{d}=-N(n/d-1)+n/d,$

and taking the limit $N\to\infty$, it follows that

$\displaystyle\lim_{N\to\infty}\mathbf{L}^{-N(n-1)}[\Lambda_{N}(f,x)]$
$\displaystyle=\mathbf{L}^{-(n-1)}(\mathbf{L}-1)[\tilde{X}]\sum_{i\geq
0}\mathbf{L}^{-(n-d)i}$
$\displaystyle=\mathbf{L}^{-(n-2)}[\tilde{X}](1-\mathbf{L}^{-1})(1-\mathbf{L}^{-(n-d)})^{-1}.$

The statement of the lemma follows. ∎

###### Proof of Theorem 1.3.

Assume that $d\geq 3$, $e\geq 1$ and $n>2^{d}(d-1)$. It will ease notation if
we put

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

where $\Lambda_{N}(f,\infty)$ is given by (1.3). Then it follows from Theorem
1.1 that

$\displaystyle[M_{e}]-$
$\displaystyle(\mathbf{L}+1)[M_{e-1}]+\mathbf{L}[M_{e-2}]$
$\displaystyle=\mathbf{L}^{\mu(e)}(\mathfrak{S}(f)\sigma_{\infty}(f)+R_{e})-(\mathbf{L}+1)\mathbf{L}^{\mu(e-1)}(\mathfrak{S}(f)\sigma_{\infty}(f)+R_{e-1})$
$\displaystyle\quad+\mathbf{L}^{\mu(e-2)+1}(\mathfrak{S}(f)\sigma_{\infty}(f)+R_{e-2})$
$\displaystyle=\mathbf{L}^{n-1+e(n-d)}\mathfrak{S}(f)\sigma_{\infty}(f)\
(1-(\mathbf{L}+1)\mathbf{L}^{-(n-d)}+\mathbf{L}^{-2(n-d)+1})$
$\displaystyle\quad+\mathbf{L}^{n-1+e(n-d)}(R_{e}-(\mathbf{L}+1)\mathbf{L}^{-(n-d)}R_{e-1}+\mathbf{L}^{-2(n-d)+1}R_{e-2}).$

Dividing by $\mathbf{L}-1$, it follows from Lemma 8.1 that

(8.2)
$[\mathrm{Mor}_{e}(\mathbf{P}^{1},\tilde{X})]=\mathbf{L}^{\mu(e)-1}\left(\mathfrak{S}(f)\sigma_{\infty}(f)\frac{(1-\mathbf{L}^{-(n-d)})(1-\mathbf{L}^{-(n-d)+1})}{1-\mathbf{L}^{-1}}+S_{e}\right),$

where

$S_{e}=\frac{R_{e}-(\mathbf{L}+1)\mathbf{L}^{-(n-d)}R_{e-1}+\mathbf{L}^{-2(n-d)+1}R_{e-2}}{1-\mathbf{L}^{-1}}.$

We analyse the main term using Lemma 8.2. Denoting by $F_{v}(T)$ the local
factors of the motivic Euler product (6.16) of $\mathfrak{S}(f)$, by
multiplicativity of motivic Euler products and the fact that Kapranov’s zeta
function for $\mathbf{A}^{1}$ is

$Z_{\mathbf{A}^{1}}(T)=\prod_{v\in\mathbf{A}^{1}}(1-T)^{-1}=\frac{1}{(1-\mathbf{L}T)},$

we therefore obtain

$\displaystyle\mathfrak{S}(f)(1-\mathbf{L}^{-(n-d)+1})$
$\displaystyle=\prod_{v\in\mathbf{A}^{1}}F_{v}(\mathbf{L}^{-n})\prod_{v\in\mathbf{A}^{1}}(1-\mathbf{L}^{-{n-d}})^{-1}$
$\displaystyle=\left(\prod_{x\in\mathbf{A}^{1}}F_{v}(\mathbf{L}^{-n}T)(1-\mathbf{L}^{-(n-d)}T)\right)_{|T=1}.$

Also used here is the compatibility of Euler products with transformations of
the form $T\mapsto\mathbf{L}^{r}T.$ By Remark 6.10 and Lemma 8.2, each local
factor of the latter motivic Euler product is equal to
$\mathbf{L}^{-(n-2)}[\tilde{X}](1-\mathbf{L}^{-1})$. Combining this with an
application of Lemma 8.2 to the contribution of $\sigma_{\infty}$ and
compatibility of motivic Euler products with finite products, we may write

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

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

Turning to the error term, we deduce from Theorem 1.1 that

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

for any $e\geq 1$. By convention $R_{e}=\varnothing$ if $e\leq 0$. Hence

$w(S_{e})\leq\max\left(4-\nu(e+1),6-2(n-d)-\nu
e,6-4(n-d)-\nu(e-1)\right)=4-\nu(e+1),$

where

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

The statement of Theorem 1.3 is now clear. ∎

##  9\. The variety of lines

Our task in this section is to prove Theorem 1.5. It follows from Lemma 8.1
that

$[\mathrm{Mor}_{1}(\mathbf{P}^{1},\tilde{X})]=\frac{[M_{1}]-(\mathbf{L}+1)[M_{0}]+\mathbf{L}}{\mathbf{L}-1},$

where $[M_{0}]=[X]=(\mathbf{L}-1)[\tilde{X}]+1$. The variety of lines
$F_{1}(\tilde{X})$ is obtained as a quotient of
$\mathrm{Mor}_{1}(\mathbf{P}^{1},\tilde{X})$ by the action of the automorphism
group of $\mathbf{P}^{1}$. The quotient map

$\mathrm{Mor}_{1}(\mathbf{P}^{1},\tilde{X})\to F_{1}(\tilde{X})$

is a Zariski-locally trivial fibration with fibre $\mathrm{PGL}_{2}$. Since
$[\mathrm{PGL}_{2}]=\mathbf{L}^{3}-\mathbf{L}$, it follows that

(9.1)
$\begin{split}[F_{1}(\tilde{X})]=\frac{[\mathrm{Mor}_{1}(\mathbf{P}^{1},\tilde{X})]}{\mathbf{L}^{3}-\mathbf{L}}&=\frac{[M_{1}]-(\mathbf{L}^{2}-1)[\tilde{X}]-1}{(\mathbf{L}^{3}-\mathbf{L})(\mathbf{L}-1)}\\\
&=\frac{[M_{1}]-1}{(\mathbf{L}^{3}-\mathbf{L})(\mathbf{L}-1)}-\frac{[\tilde{X}]}{\mathbf{L}(\mathbf{L}-1)}.\end{split}$

We proceed by summarising what our work says about the class of $M_{1}$. Let
$d\geq 3$ and assume that $n>2^{d}(d-1)$. Rather than Theorem 1.1, we shall
invoke a version in which the truncated singular series appears. This amounts
to combining Proposition 3.8 with the contents of Section 6.6 in (3.1). Thus

$[M_{1}]=\mathbf{L}^{\mu(1)}\left((1+S_{1}(f)\mathbf{L}^{-n})\mathbf{L}^{-(n-1)}[X]+R_{1}\right),$

where $R_{1}$ is an error term satisfying

$w(R_{1})\leq 4-\frac{n-2^{d}(d-1)}{2^{d-2}}.$

Recalling that $\mu(1)=2n-d-1$, by (1.1), we deduce that

(9.2)
$[M_{1}]=\mathbf{L}^{n-d}(\mathbf{L}-1)[\tilde{X}]\left(1+S_{1}(f)\mathbf{L}^{-n}\right)+\widetilde{R_{1}},$

where

(9.3)
$\begin{split}w(\widetilde{R_{1}})&\leq\max\left(2n-2d,4n-2d+2-\frac{n-2^{d}(d-1)}{2^{d-2}}\right)\\\
&=4n-2d+2-\frac{n-2^{d}(d-1)}{2^{d-2}}.\end{split}$

We proceed by computing $S_{1}(f)$. Note that in this case
$B_{1}\simeq\mathbf{G}_{m}\times\mathbf{A}^{1}$ is just the space of pairs
$(h,T-x)$ where $h$ is a nonzero constant, and the polynomials
$g_{1},\dots,g_{n}\in\mathrm{Poly}_{<1}$ are just constants. Thus

$\mathrm{res}\left(\frac{h}{T-x}f(g_{1},\dots,g_{n})\right)=hf(g_{1},\dots,g_{n}).$

We therefore get

$\displaystyle S_{1}(f)$ $\displaystyle=\left[\mathrm{Poly}_{<1}^{n}\times
B_{1}:\mathrm{res}\left(\frac{h}{T-x}f(g_{1},\dots,g_{n})\right)\right]$
$\displaystyle=[\mathrm{Poly}_{<1}^{n}\times\mathbf{G}_{m}\times\mathbf{A}^{1},hf(g_{1},\dots,g_{n})].$

As an element of $\mathscr{E}xp\mathscr{M}_{B_{1}}$, this is the pullback of

$[\mathrm{Poly}_{<1}^{n}\times\mathbf{G}_{m},hf(g_{1},\dots,g_{n})]\in\mathscr{E}xp\mathscr{M}_{\mathbf{G}_{m}}$

via $B_{1}\to\mathbf{G}_{m}$ given by the projection $(h,T-x)\mapsto h$.

Applying the orthogonality relation, we see that

$\displaystyle[\mathrm{Poly}_{<1}^{n}\times\mathbf{G}_{m},hf(g_{1},\dots,g_{n})]$
$\displaystyle=[\mathrm{Poly}_{<1}^{n}\times\mathbf{A}^{1},hf(g_{1},\dots,g_{n})]-[\mathrm{Poly}_{<1}^{n}\times\\{0\\},0]$
$\displaystyle=[X]\mathbf{L}-\mathbf{L}^{n},$

whence

(9.4) $S_{1}(f)=[X]\mathbf{L}^{2}-\mathbf{L}^{n+1}.$

######  Remark 9.1.

Applying (6.10) with $m=1$, it follows that our work provides the bound

$w([X]\mathbf{L}^{2}-\mathbf{L}^{n+1})\leq 2n+4-\frac{n}{2^{d-2}}.$

That is, for any hypersurface $X\subset\mathbf{A}^{n}$ defined by a non-
singular form of degree $d$, we have the weight bound

$w([X]-\mathbf{L}^{n-1})\leq 2(n-1)-\frac{n-2^{d-1}}{2^{d-2}}.$

When working over a finite field $k=\mathbf{F}_{q}$, this should be compared
with the bound

$\\#X(\mathbf{F}_{q})-q^{n-1}=O_{d,n}(q^{n/2}),$

that follows from Deligne’s resolution of the Weil conjectures.

###### Proof of Theorem 1.5.

It follows from (9.4) that
$S_{1}(f)=\mathbf{L}^{2}(\mathbf{L}-1)[\tilde{X}]+\mathbf{L}^{2}-\mathbf{L}^{n+1}.$
Combining this with (9.2), we deduce that

$\displaystyle[M_{1}]$
$\displaystyle=\mathbf{L}^{n-d}(\mathbf{L}-1)[\tilde{X}]\left(1+\mathbf{L}^{-n+2}(\mathbf{L}-1)[\tilde{X}]+\mathbf{L}^{-n+2}-\mathbf{L}\right)+\widetilde{R_{1}}$
$\displaystyle=(\mathbf{L}-1)^{2}\left(\mathbf{L}^{-d+2}[\tilde{X}]^{2}-\mathbf{L}^{n-d}[\tilde{X}]\right)+\widetilde{\widetilde{R_{1}}},$

where $\widetilde{\widetilde{R_{1}}}$ has the same weight as
$\widetilde{R}_{1}$ in (9.3). Inserting this into (9.1), it follows that

$\displaystyle\mathbf{L}^{2}[F_{1}(\tilde{X})]$
$\displaystyle=\frac{\mathbf{L}([M_{1}]-1)}{(\mathbf{L}^{2}-1)(\mathbf{L}-1)}-\frac{\mathbf{L}[\tilde{X}]}{\mathbf{L}-1}$
$\displaystyle=\frac{\mathbf{L}^{-d+2}[\tilde{X}]^{2}-\mathbf{L}^{n-d}[\tilde{X}]}{1+\mathbf{L}^{-1}}-\frac{[\tilde{X}]}{1-\mathbf{L}^{-1}}-\frac{1}{(\mathbf{L}^{2}-1)(1-\mathbf{L}^{-1})}+\frac{\mathbf{L}\widetilde{\widetilde{R_{1}}}}{(\mathbf{L}^{2}-1)(\mathbf{L}-1)}.$

The term $(1-\mathbf{L}^{-1})^{-1}[\tilde{X}]$ has weight $2n-4$ and the third
term has weight $-4$. Both of these are dominated by the upper bound we have
for the weight of the last term on the right hand side. Dividing both sides by
$\mathbf{L}^{2}$, we thereby arrive at the statement of Theorem 1.5. ∎

###### Proof of Corollary 1.6.

As described in [Ara12, Corollary 17.2.2], for example, it is well-known that
the Hodge–Deligne polynomial of the hypersurface $\tilde{X}$ takes the shape

$\mathrm{HD}(\tilde{X})=(uv)^{n-2}(1+(uv)^{-1}+(uv)^{-2}+\cdots)+g_{1}(u,v),$

where $g_{1}\in\mathbf{Z}[u,v][[(uv)^{-1}]]$ has only terms of total degree at
most $n-2$ in $u$ and $v$. Noting that

$(uv)^{n-2}(1+(uv)^{-1}+(uv)^{-2}+\cdots)=\frac{(uv)^{n-2}}{1-(uv)^{-1}},$

we thereby deduce that

$\mathrm{HD}([\tilde{X}]^{2})=\frac{(uv)^{2n-4}}{(1-(uv)^{-1})^{2}}+g_{2}(u,v),$

where $g_{2}\in\mathbf{Z}[u,v][[(uv)^{-1}]]$ only has terms of total degree at
most $2(n-2)+n-2=3n-6$.

Thus, there exists $g_{3}\in\mathbf{Z}[u,v][[(uv)^{-1}]]$ with terms of total
degree at most $3n-2d-6$ such that

$\displaystyle\mathrm{HD}\left(\frac{\mathbf{L}^{-d}[\tilde{X}]^{2}-\mathbf{L}^{n-d-2}[\tilde{X}]}{1+\mathbf{L}^{-1}}\right)$
$\displaystyle=\frac{(uv)^{2n-d-5}}{(1-(uv)^{-1})^{2}(1+(uv)^{-1})}+g_{3}(u,v)$
$\displaystyle=\frac{(uv)^{2n-d-5}}{(1-(uv)^{-1})(1-(uv)^{-2})}+g_{3}(u,v)$
$\displaystyle=(uv)^{2n-d-5}\sum_{k,l\geq 0}(uv)^{-k-2l}+g_{3}(u,v)$
$\displaystyle=\sum_{m\geq
0}\left(\left\lfloor\frac{m}{2}\right\rfloor+1\right)(uv)^{2n-d-5-m}+g_{3}(u,v).$

Using Theorem 1.5, we see that this should agree with the Hodge–Deligne
polynomial of $F_{1}(\tilde{X})$ up to terms $u^{p}v^{q}$, with $p+q\leq
4n-2d-6-\frac{n-2^{d}(d-1)}{2^{d-2}}$, which thereby concludes the proof. ∎

######  Remark 9.2.

We now assume $d=3$, so that we are in the case of a smooth cubic hypersurface
$\tilde{X}\subset\mathbf{P}^{n-1}$. Recall Galkin and Shinder’s relation
(1.4). We can use work of Burillo to calculate
$\mathrm{HD}(\mathrm{Sym}^{2}(\tilde{X}))$ and so show that Corollary 1.6 is
consistent with it. Indeed, Burillo’s formula [Bur90, (2.4)] states that
$\mathrm{HD}(\mathrm{Sym}^{2}(\tilde{X}))$ is given by the coefficient of
$t^{2}$ in

$\prod_{p,q\geq 0}(1-(-1)^{p+q}u^{p}v^{q}t)^{(-1)^{p+q+1}h^{p,q}(\tilde{X})}.$

Since $h^{p,q}(\tilde{X})=\delta_{p,q}$, for $p+q>n-2$ and $p+q\leq 2(n-2)$,
we see that modulo terms of total degree at most $n-2$ in $u,v$, the
polynomial $\mathrm{HD}(\mathrm{Sym}^{2}(\tilde{X}))$ is the coefficient of
$t^{2}$ in the expansion

$\displaystyle\prod_{0\leq p\leq n-2}(1-(uv)^{p}t)^{-1}=\leavevmode\nobreak\ $
$\displaystyle(1+(uv)^{n-2}t+(uv)^{2(n-2)}t^{2}+\cdots)$
$\displaystyle\times(1+(uv)^{n-3}t+(uv)^{2(n-3)}t^{2}+\cdots)\cdots.$

Thus there exists a polynomial $g_{1}\in\mathbf{Z}[u,v]$ of degree $\leq n-2$
such that

$\mathrm{HD}(\mathrm{Sym}^{2}(\tilde{X}))=\sum_{n-2\geq p\geq p^{\prime}\geq
0}(uv)^{p+p^{\prime}}+g_{1}(u,v).$

This may be rewritten

$\displaystyle\mathrm{HD}(\mathrm{Sym}^{2}(\tilde{X}))$
$\displaystyle=\sum_{0\leq q\leq
q^{\prime}}(uv)^{2(n-2)-q-q^{\prime}}+g_{2}(u,v)$ $\displaystyle=\sum_{m\geq
0}\left(\left\lfloor\frac{m}{2}\right\rfloor+1\right)(uv)^{2n-4-m}+g_{2}(u,v),$

where $g_{2}(u,v)\in\mathbf{Z}[u,v][[(uv)^{-1}]]$ has terms of total degree at
most $n-2$ in $u$ and $v$.

Taking $d=3$ and $n\geq 17$ in Corollary 1.6, we compute the Hodge–Deligne
polynomial of
$\mathbf{L}^{2}[F_{1}(\tilde{X})]+(1+\mathbf{L}^{n-2})[\tilde{X}],$ modulo
degree $\leq\frac{7}{2}n$ in $u,v$. This shows that there exists
$h\in\mathbf{Z}[u,v][[(uv)^{-1}]]$ with terms of degrees $\leq\frac{7}{2}n$
such that

$\displaystyle\mathrm{HD}(\mathbf{L}^{2}[F_{1}(\tilde{X})]+(1+\mathbf{L}^{n-2})[\tilde{X}])=\leavevmode\nobreak\
$ $\displaystyle\sum_{m\geq
0}\left(\left\lfloor\frac{m}{2}\right\rfloor+1\right)(uv)^{2n-6-m}$
$\displaystyle+(uv)^{2(n-2)}(1+(uv)^{-1}+(uv)^{-2}+\cdots)+h(u,v).$

This is easily seen to agree with our expression for
$\mathrm{HD}(\mathrm{Sym}^{2}(\tilde{X}))$ above.

##  Appendix A Geometry of numbers over function fields

###  A.1. Basic facts

Let $k$ be a field. In this note we discuss basic facts from the geometry of
numbers in the setting of the function field $k(t)$. A non-archimedean
absolute value $|\cdot|:k(t)\to\mathbf{R}_{\geq 0}$ is given by taking $|0|=0$
and $|x|=2^{\deg(p)-\deg(q)},$ if $x=p/q$ for polynomials $p,q\in k[t]$, with
$q\neq 0$. The completion of $k(t)$ with respect to this absolute value is the
field of Laurent series $K_{\infty}=k((t^{-1}))$, whose elements take the
shape

$x=\sum_{-\infty<i\leq M}x_{i}t^{i},$

for $x_{i}\in k$ and $M\in\mathbf{Z}$. The absolute value is extended to
$K_{\infty}$ by taking

$|x|=\begin{cases}0&\text{ if $x=0$,}\\\ 2^{M}&\text{ if
$x=\sum_{-\infty<i\leq M}x_{i}t^{i}\in K_{\infty}$ with $x_{M}\neq 0$.
}\end{cases}$

We shall extend this to vectors by setting

$|\mathbf{x}|=\max(|x_{1}|,\dots,|x_{n}|),$

for any $\mathbf{x}\in K_{\infty}^{n}$. This provides a distance function on
$K_{\infty}^{n}$.

Mahler [Mah41] initiated an extensive investigation of lattices
$\mathsf{\Lambda}\subset K_{\infty}^{n}$, proving analogues of many results
from the classical setting of lattices in $\mathbf{R}^{n}$. In this section we
summarise some of the basic facts that will be needed in our application. (In
doing so, we shall recover work of Lee [Lee13], which is concerned with the
special case $k=\mathbf{F}_{q}$.)

A (full rank) lattice $\mathsf{\Lambda}\subset K_{\infty}^{n}$ is defined to
be a set of the form

$\mathsf{\Lambda}=\\{u_{1}\mathbf{x}_{1}+\dots+u_{n}\mathbf{x}_{n}:u_{1},\dots,u_{n}\in
k[t]\\},$

where $\mathbf{x}_{1},\dots,\mathbf{x}_{n}\in K_{\infty}^{n}$ are
$k(t)$-linearly independent vectors. This set of vectors is called the basis
of the lattice. Let $\mathbf{M}=(\mathbf{x}_{1},\dots,\mathbf{x}_{n})$ be the
associated $n\times n$ matrix of basis vectors. The determinant of the lattice
is defined to be

$\det(\mathsf{\Lambda})=|\det\mathbf{M}|.$

This does not depend on the choice of basis for $\mathsf{\Lambda}$, as proved
in [Mah41, Section 8]. Next, the successive minima
$\sigma_{1},\dots,\sigma_{n}$ associated to $\mathsf{\Lambda}$ are defined as
follows. We take $2^{\sigma_{1}}$ to be minimum of $|\mathbf{x}_{1}|$, for
non-zero $\mathbf{x}_{1}\in\mathsf{\Lambda}$. Next, $2^{\sigma_{2}}$ is the
minimum of $|\mathbf{x}_{2}|$, for
$\mathbf{x}_{2}\in\mathsf{\Lambda}\setminus\operatorname{Span}_{k(t)}(\mathbf{x}_{1})$.
One continues in this way, ultimately defining $2^{\sigma_{n}}$ to be the
minimum of $|\mathbf{x}_{n}|$, for
$\mathbf{x}_{n}\in\mathsf{\Lambda}\setminus\operatorname{Span}_{k(t)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n-1})$.
It is clear from the construction that $\sigma_{1},\dots,\sigma_{n}$ are
integers satisfying

(A.1) $-\infty<\sigma_{1}\leq\dots\leq\sigma_{n}.$

The analogue of Minkowski’s theorem is

(A.2) $\det(\mathsf{\Lambda})=2^{\sigma_{1}+\dots+\sigma_{n}},$

which is established in [Mah41, Section 9].

For lattices in $\mathbf{R}^{n}$ there is a wealth of literature around the
problem of counting the number of lattice points that are constrained to lie
in Euclidean balls of growing radius in $\mathbf{R}^{n}$. In the function
field setting, we shall be interested in the “size” of the set

$\left\\{\mathbf{x}\in\mathsf{\Lambda}:|\mathbf{x}|<2^{R}\right\\}.$

This set forms a finitely generated $k$-vector space and we may consider its
dimension as a $k$-vector space. The key object of interest is then the
quantity

(A.3)
$\nu(\mathsf{\Lambda},R)=\dim\left\\{\mathbf{x}\in\mathsf{\Lambda}:|\mathbf{x}|<2^{R}\right\\},$

as $R\to\infty$. For example, when $\mathsf{\Lambda}=k[t]^{n}$ we see that the
monomials $1,t,t^{2},\dots,t^{R-1}$ are linearly independent over $k$, whence
$\nu(k[t]^{n},R)=nR.$

Our starting point for the analysis of $\nu(\mathsf{\Lambda},R)$ is the
identification of a suitable basis for the lattice $\mathsf{\Lambda}$. The
following result is adapted from an argument of Davenport [Dav02, Lemma 12.3]
(as already adapted to the setting $k=\mathbf{F}_{q}$ by Lee [Lee13]).

######  Lemma A.1.

Let $\mathsf{\Lambda}\subset K_{\infty}^{n}$ be a lattice and let
$\sigma_{1},\dots,\sigma_{n}$ be the successive minima of $\mathsf{\Lambda}$.
Then there exists a basis $\mathbf{x}_{1},\dots,\mathbf{x}_{n}$ of
$\mathsf{\Lambda}$ with

$\mathbf{x}_{i}=(x_{i,1},\dots,x_{i,i},0,\dots,0),\quad\text{for $1\leq i\leq
n$},$

such that $|\mathbf{x}_{i}|=|x_{i,i}|=2^{\sigma_{i}},$ for $1\leq i\leq n$.

###### Proof.

Choose vectors $\mathbf{y}_{1},\dots,\mathbf{y}_{n}\in\mathsf{\Lambda}$ such
that $|\mathbf{y}_{i}|=2^{\sigma_{i}}$, for $1\leq i\leq n$. On applying a
suitable unimodular linear transformation we can assume that these vectors
take the shape

$\mathbf{y}_{i}=(y_{i,1},\dots,y_{i,i},0,\dots,0),$

for $1\leq i\leq n$. We take $\mathbf{x}_{1}=\mathbf{y}_{1}$. Next, we choose
$\mathbf{x}_{2}\in\mathsf{\Lambda}\cap\operatorname{Span}_{k(t)}(\mathbf{y}_{1},\mathbf{y}_{2})$
such that

$\operatorname{Span}_{k[t]}(\mathbf{x}_{1},\mathbf{x}_{2})=\mathsf{\Lambda}\cap\operatorname{Span}_{k(t)}(\mathbf{y}_{1},\mathbf{y}_{2}).$

It is clear that there exists $q,u_{1},u_{2}\in k[t]$, with $q\neq 0$, such
that

(A.4) $q\mathbf{x}_{2}=u_{1}\mathbf{y}_{1}+u_{2}\mathbf{y}_{2}.$

This implies that $qx_{2,2}=u_{2}y_{2,2}$, from which it follows that
$u_{2}\neq 0$, since $q,x_{2,2},y_{2,2}$ are all non-zero.

We claim that the choice of $\mathbf{x}_{2}$ can be made in such a way that
$|u_{1}|,|u_{2}|\leq|q|$, which will then imply that

$|\mathbf{x}_{2}|\leq|q|^{-1}\max\left\\{|u_{1}||\mathbf{y}_{1}|,|u_{2}||\mathbf{y}_{2}|\right\\}\leq|\mathbf{y}_{2}|=2^{\sigma_{2}},$

by the ultrametric inequality. To prove the claim, we note that

$\mathbf{y}_{2}\in\operatorname{Span}_{k[t]}(\mathbf{x}_{1},\mathbf{x}_{2})=\operatorname{Span}_{k[t]}(\mathbf{y}_{1},\mathbf{x}_{2}).$

Hence, there exist $v_{1},v_{2}\in k[t]$ such that
$\mathbf{y}_{2}=v_{1}\mathbf{y}_{1}+v_{2}\mathbf{x}_{2}.$ But then (A.4)
implies that

$(q-u_{2}v_{2})\mathbf{x}_{2}=(u_{1}+u_{2}v_{1})\mathbf{y}_{1},$

whence $(q-u_{2}v_{2})x_{2,1}=(u_{1}+u_{2}v_{1})y_{1,1}$ and
$(q-u_{2}v_{2})x_{2,2}=0$, since $y_{1,2}=0$. Thus it follows that
$q=u_{2}v_{2}$ and $u_{1}=-u_{2}v_{1}$. Once inserted into (A.4) and recalling
that $u_{2}\neq 0$, we obtain

$v_{2}\mathbf{x}_{2}=-v_{1}\mathbf{y}_{1}+\mathbf{y}_{2}.$

Moreover, $v_{2}\neq 0$. Writing $-v_{1}=w_{1}v_{2}+r_{1}$, for
$w_{1},r_{1}\in k[t]$ such that $|r_{1}|<|v_{2}|$, we therefore obtain

$v_{2}(\mathbf{x}_{2}-w_{1}\mathbf{y}_{1})=r_{1}\mathbf{y}_{1}+\mathbf{y}_{2}.$

The claim follows on redefining
$\mathbf{x}_{2}-w_{1}\mathbf{y}_{1}=\mathbf{x}_{2}-w_{1}\mathbf{x}_{1}$ to be
$\mathbf{x}_{2}$.

In a similar way, for $3\leq i\leq n$,
$\mathbf{x}_{i}\in\mathsf{\Lambda}\cap\operatorname{Span}_{k(t)}(\mathbf{y}_{1},\dots,\mathbf{y}_{i})$
can be chosen so that

$\operatorname{Span}_{k[t]}(\mathbf{x}_{1},\dots,\mathbf{x}_{i})=\mathsf{\Lambda}\cap\operatorname{Span}_{k(t)}(\mathbf{y}_{1},\dots,\mathbf{y}_{i}),$

with $|\mathbf{x}_{i}|\leq 2^{\sigma_{i}}.$ Proceeding in this way, we have

$|x_{i,i}|\leq|\mathbf{x}_{i}|\leq 2^{\sigma_{i}},$

for $1\leq i\leq n$.

It remains to prove the lower bound

$|x_{i,i}|\geq 2^{\sigma_{i}},$

for $1\leq i\leq n$. The vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{n}$ form a
basis for $\mathsf{\Lambda}$. Thus

$\det(\mathsf{\Lambda})=|\det(\mathbf{x}_{1},\dots,\mathbf{x}_{n})|=|x_{1,1}\dots
x_{n,n}|.$

It follows that

$\det(\mathsf{\Lambda})\leq
2^{\sigma_{1}+\dots+\sigma_{i-1}}|x_{i,i}|2^{\sigma_{i+1}+\dots+\sigma_{n}},$

for any $i\in\\{1,\dots,n\\}$. Appealing to (A.2), we deduce that
$|x_{i,i}|\geq 2^{\sigma_{i}}$, as required. ∎

We are now ready to prove our key lattice point counting result.

######  Lemma A.2.

Let $\mathsf{\Lambda}\subset K_{\infty}^{n}$ be a lattice and let
$\sigma_{1},\dots,\sigma_{n}$ be the successive minima of $\mathsf{\Lambda}$.
Then for any $R\in\mathbf{Z}_{>0}$, we have

$\nu(\mathsf{\Lambda},R)=\sum_{i=1}^{n}\max\\{0,R-\sigma_{i}\\}.$

###### Proof.

Let $\mathsf{\Lambda}\subset K_{\infty}^{n}$ be a lattice and let
$\sigma_{1},\dots,\sigma_{n}$ be the successive minima of $\mathsf{\Lambda}$.
Choose a basis $\mathbf{x}_{1},\dots,\mathbf{x}_{n}$ for $\mathsf{\Lambda}$,
as in Lemma A.1. Thus

$\mathbf{x}_{i}=(x_{i,1},\dots,x_{i,i},0,\dots,0),\quad\text{for $1\leq i\leq
n$},$

with $|\mathbf{x}_{i}|=|x_{i,i}|=2^{\sigma_{i}},$ for $1\leq i\leq n$. Any
$\mathbf{x}\in\mathsf{\Lambda}$ has the form

$\mathbf{x}=u_{1}\mathbf{x}_{1}+\dots+u_{n}\mathbf{x}_{n},$

for $u_{1},\dots,u_{n}\in k[t]$. The condition $|\mathbf{x}|<2^{R}$ is then
equivalent to the system of inequalities

$\displaystyle|u_{1}+u_{2}x_{2,1}/x_{1,1}+\dots+u_{n}x_{n,1}/x_{1,1}|$
$\displaystyle<2^{R-\sigma_{1}},$ $\displaystyle\hskip 5.69046pt\vdots$
$\displaystyle|u_{n-1}+u_{n}x_{n,n-1}/x_{n,n}|$
$\displaystyle<2^{R-\sigma_{n-1}},$ $\displaystyle|u_{n}|$
$\displaystyle<2^{R-\sigma_{n}}.$

The set of polynomials $u_{1},\dots,u_{n}\in k[t]$ satisfying these
constraints clearly defines a $k$-vector space of dimension
$\sum_{i=1}^{n}\max\\{0,R-\sigma_{i}\\}$, as claimed in the statement of the
lemma. ∎

Let $\mathsf{\Lambda}\subset K_{\infty}^{n}$ be a lattice with basis matrix
$\mathbf{M}$. The dual lattice $\mathsf{\Lambda}^{*}$ is the lattice with
basis matrix $\mathbf{M}^{\text{adj}}$, obtained by taking the adjoint matrix
of $\mathbf{M}$. We clearly have
$\det(\mathsf{\Lambda}^{*})=1/\det(\mathsf{\Lambda})$. In fact the successive
minima of $\mathsf{\Lambda}$ share a close correspondence with the successive
minima $\sigma_{1}^{*},\dots,\sigma_{n}^{*}$ of the dual lattice. The relation

(A.5) $\sigma_{i}=-\sigma_{n-i+1}^{*},\quad\text{for $1\leq i\leq n$},$

is established in [Mah41, Section 10].

###  A.2. Davenport’s shrinking lemma

Given any $x=\sum_{-\infty<i\leq M}x_{i}t^{i}\in K_{\infty}$ we define the
distance to the nearest integer function to be

$\|x\|=\left|\sum_{-\infty<i\leq-1}x_{i}t^{i}\right|.$

Associated to any $a,b\in\mathbf{Z}$ such that $b>0$, and any symmetric
$n\times n$ matrix $\mathbf{U}$ with entries in $K_{\infty}$, we set
$\mathsf{\Lambda}_{a,b}(\mathbf{U})\subset K_{\infty}^{2n}$ for the lattice
with underlying basis matrix

$\mathbf{M}_{a,b}(\mathbf{U})=\begin{pmatrix}t^{-a}\mathbf{I}_{n}&\mathbf{0}\\\
t^{b}\mathbf{U}&t^{b}\mathbf{I}_{n}\end{pmatrix},$

where $\mathbf{I}_{n}$ is the $n\times n$ identity matrix. Recalling (A.3), we
then see that $\nu(\mathsf{\Lambda}_{a,b}(\mathbf{U}),0)$ is the dimension of
the $k$-vector space of $\mathbf{x}\in k[t]^{n}$ for which $|{\bf x}|<2^{a}$
and $\|\mathbf{U}{\bf x}\|<2^{-b}$. The following version of Davenport’s
shrinking lemma generalises [BS23, Lemma 5.3] to arbitrary function fields.

######  Lemma A.3.

Let $\mathbf{U}$ be a symmetric $n\times n$ matrix with entries in
$K_{\infty}$. Let $a,b,s\in\mathbb{Z}$ such that $b>0$ and $s\geq 0$. Then

$\nu(\mathsf{\Lambda}_{a,b}(\mathbf{U}),0)\leq\nu(\mathsf{\Lambda}_{a-s,b+s}(\mathbf{U}),-s)+ns+n\max\left\\{\left\lfloor\frac{a-b}{2}\right\rfloor,0\right\\}.$

###### Proof.

The inequality is trivial when $a\leq 0$ since then the left hand side is $0$.
Hence we may assume that $a>0$. Let
$-\infty<\sigma_{1}\leq\dots\leq\sigma_{2n}$ be the successive minima of
$\mathsf{\Lambda}_{a,b}(\mathbf{U})$ and let
$-\infty<\sigma_{1}^{*}\leq\dots\leq\sigma_{2n}^{*}$ be the successive minima
of the dual lattice $\mathsf{\Lambda}_{a,b}(\mathbf{U})^{*}\subset
K_{\infty}^{2n}$, with underlying matrix

$\mathbf{M}_{a,b}(\mathbf{U})^{\text{adj}}=\begin{pmatrix}t^{a}\mathbf{I}_{n}&-t^{a}\mathbf{U}\\\
\mathbf{0}&t^{-b}\mathbf{I}_{n}\end{pmatrix}.$

We note that

$\displaystyle t^{b-a}\mathbf{M}_{a,b}(\mathbf{U})^{\text{adj}}$
$\displaystyle=\begin{pmatrix}t^{b}\mathbf{I}_{n}&-t^{b}\mathbf{U}\\\
\mathbf{0}&t^{-a}\mathbf{I}_{n}\end{pmatrix}=\begin{pmatrix}\mathbf{0}&\mathbf{I}_{n}\\\
-\mathbf{I}_{n}&\mathbf{0}\end{pmatrix}\mathbf{M}_{a,b}(\mathbf{U})\begin{pmatrix}\mathbf{0}&\mathbf{I}_{n}\\\
-\mathbf{I}_{n}&\mathbf{0}\end{pmatrix}^{-1}.$

It follows that the lattice with underlying basis matrix
$t^{b-a}\mathbf{M}_{a,b}(\mathbf{U})^{\text{adj}}$ is equal to the one with
basis matrix $\mathbf{M}_{a,b}(\mathbf{U})$, up to left and right
multiplication by a matrix in $\mathrm{SL}_{2n}(k)$. Hence the associated
lattices share the same successive minima, whence
$2^{\sigma_{i}}=2^{b-a+\sigma_{i}^{*}},$ for $1\leq i\leq 2n$. Appealing to
(A.5), it follows that $\sigma_{i}+\sigma_{2n-i+1}=b-a,$ for $1\leq i\leq 2n$.
Taking $i=n+1$, we deduce that

(A.6) $\sigma_{n+1}\geq\left\lceil\frac{b-a}{2}\right\rceil.$

We now apply Lemma A.2 to deduce that

$\nu(\mathsf{\Lambda}_{a,b}(\mathbf{U}),0)=\sum_{i=1}^{2n}\max\\{0,-\sigma_{i}\\}$

and

$\nu(\mathsf{\Lambda}_{a-s,b+s}(\mathbf{U}),-s)=\sum_{i=1}^{2n}\max\\{0,-s-\sigma_{i}\\}.$

For $1\leq i\leq n$, it is clear that

$\max\\{0,-\sigma_{i}\\}-\max\\{0,-s-\sigma_{i}\\}\leq s.$

Moreover, for $n+1\leq i\leq 2n$ we have

$\max\\{0,-\sigma_{i}\\}-\max\\{0,-s-\sigma_{i}\\}\leq\max\left\\{\left\lfloor\frac{a-b}{2}\right\rfloor,0\right\\},$

by (A.6). The statement of the lemma is now clear. ∎

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Matsubara summation yields a final form of the gap equation:

$\ln\left(\frac{T_{c}}{T_{c}^{0}}\right)+\Phi(\rho_{-},\tilde{\chi}_{0})+\Phi(\rho_{+},\tilde{\chi}_{0})-\frac{\tilde{{\boldsymbol{p}}}_{+}\cdot\tilde{{\boldsymbol{p}}}_{-}}{|\tilde{{\boldsymbol{p}}}_{+}||\tilde{{\boldsymbol{p}}}_{-}|}\left[\Phi(\rho_{-},\tilde{\chi}_{0})-\Phi(\rho_{+},\tilde{\chi}_{0})\right]=0,$
(27)

where $\tilde{\chi}_{0}=-\tilde{g}_{\rm orb}\mu_{B}B_{\parallel}/2\pi T_{c}$
($\tilde{g}_{\rm orb}\equiv g_{\rm orb}(k_{F}+k_{0,y})$ denotes a
characteristic scale for the orbital depairing),
$\tilde{{\boldsymbol{p}}}_{\pm}=(\pm g_{R}k_{F}/2+\mu_{B}B_{\parallel},\mp
g_{R}k_{F}/2,\pm g_{I}/2)$, and
$\rho_{\pm}=(|\tilde{{\boldsymbol{p}}}_{+}|\pm|\tilde{{\boldsymbol{p}}}_{-}|)/2\pi
T_{c}$. The function $\Phi(\rho,\tilde{\chi}_{0})$ is defined in terms of the
digamma function $\psi(z)$ as

$\Phi(\rho,\tilde{\chi}_{0})=\frac{1}{4}\left\\{\mathrm{Re}\left[\psi\left(\frac{1+i\rho}{2}+i\tilde{\chi}_{0}\right)-\psi\left(\frac{1}{2}\right)\right]+\mathrm{Re}\left[\psi\left(\frac{1+i\rho}{2}-i\tilde{\chi}_{0}\right)-\psi\left(\frac{1}{2}\right)\right]\right\\}\,.$
(28)

In the limit $\tilde{\chi}_{0}\to 0$, i.e., without orbital depairing, Eq.
(27) reduces to the form used in Ref. 23. This equation can be solved
numerically to obtain the relationship between the critical temperature
$T_{c}$ and the critical in-plane field $B_{c\parallel}$ of the
superconductor, given input SOC and orbital coupling parameters. As in the
main text we denote the limits of zero-field critical temperature and zero-
temperature critical field by $T_{c}^{0}$ and $B_{c\parallel}^{0}$,
respectively.

Extended Data Fig. 8a,b displays the dependence of the Pauli-limit violation
ratio on the Ising SOC ($g_{I}$), Rashba SOC ($g_{R}$) and orbital coupling
($\tilde{g}_{\rm orb}$) parameters. In the presence of a purely Ising-type SOC
($g_{R}=\tilde{g}_{\rm orb}=0$), the evolution of $B_{c\parallel}$ as a
function of $T_{c}$ and $g_{I}$ shows the characteristic low-temperature
divergence47 of $B_{c\parallel}$, which is due to the inability of the in-
plane field—perpendicular to the Ising spin quantization axis—to destroy the
resonance condition for spin-singlet pairing between electronic states at
${\boldsymbol{k}}$ and $-{\boldsymbol{k}}$. In other words, there is always a
non-zero density of electronic states available for pairing opposite spin-
components (albeit decreasing with $B_{\parallel}$), thus leading to
persistent superconductivity555Our linearized treatment can only capture
second-order transitions, and thus neglects a possible first-order transition
to a polarized normal state, in analogy with van Vleck paramagnetism. In this
framework, the Zeeman energy gain in the normal state scales as
$E_{z}\sim\left(\mu_{B}B\right)^{2}/\lambda_{I}$ to second order in
perturbation theory, instead of the standard form $\mu_{B}B$. This is due to
the spin projection being locked in the out-of-plane directions by Ising SOC.
Equating to the condensation energy $\Delta$ leads to the condition
$B_{c}\sim\sqrt{\Delta\lambda_{I}/\mu_{B}}$ or
$B_{c}/B_{p}\sim\sqrt{\lambda_{I}/\Delta}$ with the Pauli limiting field
$B_{p}=\Delta_{0}/\sqrt{2}\mu_{B}$, a scaling form often quoted in the study
of Ising superconductivity in TMDs 23; 24; 25 as $T\rightarrow 0$. In
contrast, when either Rashba SOC666In the case of Rashba SOC we recover the
$\sqrt{2}$ enhancement of the critical field compared to a spin-degenerate
metallic state, first predicted by Gork’ov and Rashba50 based on a microscopic
calculation of the in-plane spin susceptibility. or orbital decoupling is
added, the low-temperature divergence of $B_{c\parallel}$ is strongly
suppressed, alongside a reduction of the Pauli-limit violation at all
temperatures. This suppression occurs because both Rashba and orbital effects
lead to a non-trivial deformation of the Fermi pockets under an in-plane
field, which destroys the resonance conditions necessary for spin-singlet,
zero-momentum pairing (we neglect finite-momentum, FFLO-type pairing channels
in our analysis).

##  10 Modeling of the Pauli-limit violation data

We now describe various efforts at fitting the Pauli-limit violation data from
Fig. 4c and Extended Data Fig. 6f to our theoretical model. The difference
between these two data sets is that $B_{c\parallel}$ reported in Fig. 4c is
obtained by a phenomenological extrapolation to zero temperature, whereas
$B_{c\parallel}$ in Extended Data Fig. 6f is measured at a fixed base
temperature $T_{c}\sim 30$ mK.

In the following we fix the value of Ising SOC to $g_{I}=0.7$ meV as extracted
from quantum Hall measurements, and assume that it remains constant as a
function of doping (thus ignoring a potential interaction-induced enhancement
of its bare value). We first consider Rashba and orbital effects separately.
Model 1 (see Table below) considers only Rashba SOC ($\tilde{\mathrm{g}}_{\rm
orb}=0$), which because of its linear dependence on $k_{F}$ is expected to
scale as $g_{R}k_{F}\sim\sqrt{n_{\rm small}}$ with $n_{\rm small}$ the
electronic density in the small pocket. Due to the difficulty of extracting
the size of the small Fermi pockets directly from quantum oscillation data, we
take a phenomenological ansatz for the small pocket density. Motivated by the
quantum oscillation data for the ${\rm FP}(2,2)_{+}$ phase, we assume that the
normalized SdH frequency $f_{\nu}=n_{\rm small}/n$ ($n$ is the total
electronic density in the system) varies linearly with $n$ in the ${\rm
FP}(2,2)_{+}$ phase—that is, we take $f_{\nu}=an+b$ with constants $a$, $b$ to
be treated as fitting parameters. This assumption leads to the form
$g_{R}k_{F}\sim\sqrt{an^{2}+bn}$ for the energy scale associated with Rashba
coupling.

As shown in Extended Data Fig. 8c,d, Model 1 fits the PVR data well. However
we find that to account for the strong PVR dependence, the Rashba energy scale
$g_{R}k_{F}$ must vary strongly over the corresponding doping range. In the
current Model 1, this variation occurs through an order-of-magnitude increase
in the electronic density in the small pocket, which would require the small
pocket density to nearly vanish at the edge of the SC dome. Part of the
variation in $g_{R}k_{F}$ could also come from a doping dependence of the
Rashba SOC parameter $g_{R}$ itself, either due to interaction-induced
renormalization or through band structure effects (e.g., higher-order terms in
the low-energy projection that relates the bare $\lambda_{R}$ to $g_{R}$ in
the low-energy description, Eq. 24).

The same conclusion is obtained for the case of only orbital effects
($g_{R}=0$, see fits to the data in Extended Data Fig. 8c,d). A model with
only an orbital source of depairing (as introduced in the previous section)
can similarly account for the suppression of PVR as hole doping is increased.
In our fits, motivated by the form of Eq. (24), we assume for simplicity a
dependence of $\tilde{g}_{\mathrm{orb}}\sim a\sqrt{n}+b$ where the constant
offset qualitatively stems from the finite location of the locus of the
trigonally warped pockets with respect to the Dirac point. In Models 2, 3 and
4 (see Table at the end of this section) we assume: filling-dependent
$g_{R}k_{F}$ and constant orbital coupling $\tilde{g}_{\mathrm{orb}}\sim c$;
$g_{R}k_{F}=0$ and filling dependent $\tilde{g}_{\mathrm{orb}}$; and both
filling dependent $g_{R}k_{F}$ and $\tilde{g}_{\mathrm{orb}}$.

In summary, our theoretical modeling can account for the observed PVR
evolution through a strong doping-dependence of either Rashba SOC and/or
orbital depairing effects, both of which compete with Ising SOC. This is
reminiscent to the phenomenology of Ref. 23, where a doping-dependent PVR was
observed and attributed to increasing effects of Rashba SOC with carrier
density. However, the required doping dependence appears very large in view of
naive band-structure estimates. Consequently, band structure reconstruction
mediated by electron interactions in the ${\rm FP}(2,2)_{+}$ phase must be
significant for our scenario to capture the physics responsible for the
evolution of Pauli-limit violation—providing a guidance for further theory
modelling of superconductivity in BLG that is beyond the scope of this work. A
potential resolution of this issue could involve an interaction-induced
enhancement of Ising SOC—or the nucleation of another symmetry-broken phase
which increases the out-of-plane spin canting, see discussion below Eq.
(18)—in a doping dependent manner, particularly near the low hole doping
region of the ${\rm FP}(2,2)_{+}$ phase.

Model # |  $g_{I}$ (meV) |  $g_{R}k_{F}$ (meV) |  $\tilde{g}_{\rm orb}$ (1) |  $T=30$ mK | $T\to 0$
---|---|---|---|---|---
1 | $0.7$ | $\sqrt{an+bn^{2}}$ | $0$ |  | $a=-0.212$
---
$b=0.0359$
| $a=-0.137$
---
$b=0.0231$
2 | $0.7$ | $\sqrt{an+bn^{2}}$ | $c$ |  | $a=-0.225$
---
$b=0.0382$
$c=0.0848$
| $a=-0.136$
---
$b=0.0236$
$c=0.205$
3 | $0.7$ | 0 | $a\sqrt{n}+b$ |  | $a=1.81$
---
$b=-4.25$
| $a=1.79$
---
$b=-4.24$
4 | $0.7$ | $\sqrt{an+bn^{2}}$ | $c\sqrt{n}+d$ |  | $a=-0.259$
---
$b=0.0439$
$c=0.0122$
$d=0.0872$
| $a=-0.134$
---
$b=0.0227$
$c=0.0426$
$d=-0.0796$
SI Table 1: Models used for the fitting procedure. Here $n$ is the total
doping density in units of $\times 10^{11}~{}\text{cm}^{-2}$. Units of $a$,
$b$, $c$, $d$ fitting parameters are chosen to yield correct units of the
physical parameters $g_{I}$, $g_{R}k_{F}$, $\tilde{g}_{\rm orb}$. Their
filling dependence is shown in Extended Data Fig. 8e,f.