# Linear subspaces of minimal codimension in hypersurfaces

David Kazhdan  and  Alexander Polishchuk  Einstein Institute of Mathematics,
The Hebrew University of Jerusalem, Jerusalem 91904, Israel
<EMAIL_ADDRESS>Department of Mathematics, University of Oregon,
Eugene, OR 97403, USA; National Research University Higher School of
Economics; and Korea Institute for Advanced Study<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbf{k}$ be a perfect field and let $X\subset{\mathbb{P}}^{N}$ be a
hypersurface of degree $d$ defined over $\mathbf{k}$ and containing a linear
subspace $L$ defined over $\overline{\mathbf{k}}$ with
${\operatorname{codim}}_{{\mathbb{P}}^{N}}L=r$. We show that $X$ contains a
linear subspace $L_{0}$ defined over $\mathbf{k}$ with
${\operatorname{codim}}_{{\mathbb{P}}^{N}}L\leq dr$. We conjecture that the
intersection of all linear subspaces (over $\overline{\mathbf{k}}$) of minimal
codimension $r$ contained in $X$, has codimension bounded above only in terms
of $r$ and $d$. We prove this when either $d\leq 3$ or $r\leq 2$.

A.P. is partially supported by the NSF grant DMS-2001224, and within the
framework of the HSE University Basic Research Program and by the Russian
Academic Excellence Project ‘5-100’.

##  1\. Introduction

Let $f(x_{1},\ldots,x_{n})$ be a homogeneous polynomial of degree $d\geq 2$
over a field $\mathbf{k}$. Recall that the slice rank
${\operatorname{srk}}_{\mathbf{k}}(f)$ of $f$ is the minimal number $r$ such
that there exists a decomposition

$f=l_{1}f_{1}+\ldots+l_{r}f_{r},$

where $l_{i}$ are linear forms defined over $\mathbf{k}$.

The slice rank ${\operatorname{srk}}_{\mathbf{k}}(f)$ has a simple geometric
meaning: it is the minimal codimension in ${\mathbb{P}}^{n-1}$ of a linear
$\mathbf{k}$-subspace $P\subset{\mathbb{P}}^{n-1}$ contained in the projective
hypersurface $f=0$. Note that if
${\operatorname{srk}}_{\overline{\mathbf{k}}}(f)<n/2$ then this hypersurface
is necessarily singular.

It is clear that
${\operatorname{srk}}_{\mathbf{k}}(f)\geq{\operatorname{srk}}_{\overline{\mathbf{k}}}(f)$
and it is easy to find examples when
${\operatorname{srk}}_{\mathbf{k}}(f)>{\operatorname{srk}}_{\overline{\mathbf{k}}}(f)$.

One can ask for an upper estimate for ${\operatorname{srk}}_{\mathbf{k}}(f)$
in terms of ${\operatorname{srk}}_{\overline{\mathbf{k}}}(f)$. The first
result of this paper is presicely such an estimate, which we obtain by
adapting to homogeneous polynomials the theory of $G$-rank of tensors from the
work of Derksen [3].

A bit more generally, for a collection $f_{1},\ldots,f_{s}$ of homogeneous
polynomials of the same degree $d$, we set

${\operatorname{srk}}_{\mathbf{k}}(f_{1},\ldots,f_{s})=\inf_{(c_{1},\ldots,c_{s})\neq(0,\ldots,0)}{\operatorname{srk}}_{\mathbf{k}}(c_{1}f_{1}+\ldots+c_{s}f_{s}).$

Theorem A. Assume that the field $\mathbf{k}$ is perfect. Then for a
homogeneous polynomial $f$ over $\mathbf{k}$ of degree $d\geq 2$, one has

${\operatorname{srk}}_{\mathbf{k}}(f)\leq
d\cdot{\operatorname{srk}}_{\overline{\mathbf{k}}}(f),$
${\operatorname{srk}}_{\mathbf{k}}(f_{1},\ldots,f_{s})\leq d\cdot
s\cdot{\operatorname{srk}}_{\overline{\mathbf{k}}}(f_{1},\ldots,f_{s}).$

The inequality for a single $f$ is sharp for every degree $d$: if
$E/\mathbf{k}$ is a Galois extension of degree $d$ then the norm
$E\to\mathbf{k}$ is a polynomial of degree $d$ that has slice rank $d$ over
$\mathbf{k}$ and slice rank $1$ over $E$.

Note that Derksen’s theory of $G$-rank for not necessarily symmetric tensors,
applied to symmetric tensors implies only the inequality
${\operatorname{srk}}_{\mathbf{k}}(f)\leq\frac{d^{2}}{2}\cdot{\operatorname{srk}}_{\overline{\mathbf{k}}}(f)$.

Our second goal in this paper is to understand the inequality of Theorem A
more constructively. Geometrically, the statement is that if a hypersurface
$X\subset{\mathbb{P}}^{N}$ of degree $d$, defined over $\mathbf{k}$, contains
a linear subspace $L$ of codimension $r$ in ${\mathbb{P}}^{N}$, defined over
$\overline{\mathbf{k}}$, then $X$ contains a linear subspace $L_{0}$ of
codimension $\leq dr$ in ${\mathbb{P}}^{N}$, defined over $\mathbf{k}$. One
can ask for an explicit construction of $L_{0}$ from $L$ and its Galois
conjugates. The simplest answer would be that one can just take $L_{0}$ to be
the intersection of all the Galois conjugates of $L$. We conjecture that the
following stronger geometric statement holds.

Conjecture B. Let $X=(f=0)\subset{\mathbb{P}}^{N}$ be a hypersurface of degree
$d$ (over any ground field), and let $r$ be the minimal natural number such
that $X$ contains a linear subspace $L$ with
${\operatorname{codim}}_{{\mathbb{P}}^{N}}L=r$. Set

$L_{f}:=\cap_{L\subset X,{\operatorname{codim}}_{{\mathbb{P}}^{n}}L=r}L.$

Then there exists a function $c(r,d)$ such that
${\operatorname{codim}}L_{f}\leq c(r,d)$.

It is an easy exercise to check that the conjecture holds if $d=2$ or $r=1$
with $c(r,2)=2r$ and $c(1,d)=d$. We prove the following particular cases of
the weak form of the conjecture: when $d=3$ (and $r$ is arbitrary) and when
$r=2$ (and $d$ is arbitrary).

Theorem C. (i) Conjecture B holds for cubic hypersurfaces with

$c(r,3)=c(r):=\frac{1}{2}\bigl{(}\frac{(r+1)^{2}}{4}+r+3\bigr{)}\cdot\bigl{(}\frac{(r+1)^{2}}{4}+r\bigr{)}.$

(ii) Conjecture B holds for $r=2$ with

$c(2,d)=d^{2}+1.$

More precisely, for a polynomial $f$ of slice rank $2$ and degree $d$, either
${\operatorname{codim}}L_{f}\leq d^{2}-1$ or $f$ is a pullback from a space of
dimension $\leq d^{2}+1$.

One can ask how far are the estimates of Theorem C from being optimal. In the
case $d=3$ and $r=2$ we show that Conjecture B holds with $c(2,3)=6$ by giving
a partial classification of cubic hypersurfaces of rank $2$ (see Theorem 3.6).
Consider the cubic $f(x_{i},y_{jk})$, where $i=1,\ldots,n$, $1\leq j<k\leq n$
(so the number of variables is $n(n+1)/2$), given by

$f=\sum_{i<j}x_{i}x_{j}y_{ij}.$

It is reasonable to expect that the rank of $f$ is equal to $n-1$. Since for
every $i<j$, we have $f\in(y_{ij};x_{k}\ |\ k\neq i,k\neq j)$, this would
imply that ${\operatorname{codim}}L_{f}=n(n+1)/2$, which depends quadratically
on the rank. So it seems that the optimal bound $c(r,3)$ is at least quadratic
in $r$.

On the other hand, let us consider a polynomial $f$ in $n$ groups of variables

$(x_{1}(1),\ldots,x_{m}(1)),\ \ldots,(x_{1}(n),\ldots,x_{m}(n))$

given by

$f=\sum_{i=1}^{m}x_{1}(1)\ldots\widehat{x_{i}(1)}\ldots x_{m}(1)\cdot
x_{i}(2)\ldots x_{i}(n).$

Then $\deg(f)=m+n-2$ and it is easy to see that $f$ has rank $2$ and
${\operatorname{codim}}L_{f}=mn$. This shows that the optimal bound $c(2,d)$
grows quadratically in $d$.

Theorem C(i) implies that if $L$ is a linear subspace of minimal codimension
$r$ in ${\mathbb{P}}^{N}$, defined over a Galois extension of $\mathbf{k}$,
contained in a cubic hypersurface $X$ (defined over $\mathbf{k}$), then by
taking intersection of all Galois conjugates of $L$ we get a linear subspace
$L_{0}$ of codimension $\leq c(r)$, contained in $X$ and defined over
$\mathbf{k}$.

Since we don’t know the validity of Conjecture B for general $r$ and $d$, we
give a more complicated construction of constructing a linear subspace
$L_{0}\subset X$, defined over $\mathbf{k}$, starting from the Galois
conjugates of $L\subset X$ defined over a Galois extension of $\mathbf{k}$.
For this we introduce the following recursive definition, where for linear
subspaces $L_{1},\ldots,L_{s}\subset{\mathbb{P}}^{N}$ we denote by $\langle
L_{1},\ldots,L_{s}\rangle\subset{\mathbb{P}}^{N}$ their linear span.

######  Definition 1.1.

For a collection ${\mathcal{L}}=\\{L_{1},\ldots,L_{s}\\}$ of linear subspaces
of ${\mathbb{P}}^{N}$, we define a new collection of linear subspaces of
${\mathbb{P}}^{N}$ as follows. Let $L=\langle L_{1},\ldots,L_{s}\rangle$. For
each minimal subset $J\subset[1,s]$ such that $\langle L_{j}\ |\ j\in
J\rangle=L$, we set $L_{J}:=\cap_{j\in J}L_{j}$, and we denote by
${\mathcal{L}}^{(1)}$ the collection of all such subspaces $L_{J}$. We denote
by ${\mathcal{L}}^{(i)}$, $i\geq 1$, the collections of linear subspaces
obtained by iterating this construction.

Theorem D. Let $X\subset{\mathbb{P}}^{N}$ be a hypersurface of degree $d\geq
2$ and let ${\mathcal{L}}=(L_{1},\ldots,L_{s})$ be a collection of linear
subspaces contained in $X$, such that
${\operatorname{codim}}_{{\mathbb{P}}^{N}}L_{i}\leq r$, where $r\geq 2$. Then
for the linear subspace

$L_{0}:=\langle L\ |\ L\in{\mathcal{L}}^{(d-1)}\rangle,$

we have $L_{0}\subset X$ and

${\operatorname{codim}}_{{\mathbb{P}}^{N}}L_{0}\leq r^{2^{d-1}}.$

Applying the construction of Theorem D to the collection of all Galois
conjugates of a linear subspace of codimension $r\geq 2$ in
${\mathbb{P}}^{N}$, defined over some Galois extension of $\mathbf{k}$,
contained in a hypersurface $X$ (defined over $\mathbf{k}$), we get an
algorithm for producing a linear subspace of codimension $\leq r^{2^{d-1}}$ in
${\mathbb{P}}^{N}$, contained in $X$ and defined over $\mathbf{k}$.

One can ask whether the second inequality in Theorem A can also be explained
constructively. In other words, starting with an $s$-dimensional subspace $F$
of homogeneous polynomials of the same degree $d$, defined over a perfect
field $\mathbf{k}$, such that there exists a nonzero $f\in
F_{\overline{\mathbf{k}}}$ and subspace of linear forms of dimension $r$ over
$\overline{\mathbf{k}}$ such that $f\in(L)$, we want to produce an element
$f_{0}\in F\setminus 0$ and a subspace of linear forms $L_{0}$, both defined
over $\mathbf{k}$, such that $f_{0}\in(L_{0})$ and dimension of $L_{0}$ is
$\leq c(sr)$. In Remark 4.1 we show how to do this using the algorithm of
Theorem D for a single polynomial.

Our study is partially motivated by the desire to understand the related
notion of the Schmidt rank (also known as strength) of a homogeneous
polynomial (see [1], [2] and references therein), defined as the minimal
number $r$ such that $f$ admits a decomposition
$f=g_{1}h_{1}+\ldots+g_{r}h_{r}$, with $\deg(g_{i})$ and $\deg(h_{i})$ smaller
than $\deg(f)$. Similarly to Theorem A one can try to estimate the Schmidt
rank of a polynomial over a non-closed field in terms of its Schmidt rank over
an algebraic closure. In [4], we show how to do this for quartic polynomials.

Acknowledment. The second author is grateful to Nick Addington for introducing
him to Macaulay 2, which helped to find some examples of polynomials of small
rank with large ${\operatorname{codim}}L_{f}$.

##  2\. $G$-rank for homogeneous polynomials

Throughout this section we assume that the ground field $\mathbf{k}$ is
perfect.

###  2.1. Definition of the $G$-rank and the relation to the slice rank

Below we introduce an analog of $G$-rank for symmetric tensors, or
equivalently, for homogeneous polynomials, $r_{\mathbf{k}}^{G}(f)$ (where
$G=\operatorname{GL}_{n}$). We show that it enjoys similar properties to the
Derksen’s $G$-rank of a non-symmetric tensor studied in [3], in particular, it
does not change under algebraic extensions of perfect fields. We also
introduce the notion $r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})$ of a $G$-rank
for a collection of polynomials of the same degree.

Let $V$ be an $n$-dimensional space over $\mathbf{k}$. We consider the group
$G=\operatorname{GL}(V)\simeq\operatorname{GL}_{n}(\mathbf{k})$ acting
naturally on the space $S^{d}V$, and the induced action on
${\bigwedge}^{s}(S^{d}V)$.

We consider points of $G$ and of ${\bigwedge}^{s}S^{d}V$ with values in the
ring of formal power series $\mathbf{k}[\\![t]\\!]$. For a $\mathbf{k}$-vector
space $W$ and a vector $w\in W[\\![t]\\!]$, we denote by
${\operatorname{val}}_{t}(w)$ the minimal $m\geq 0$ such that $w\in
t^{m}W[\\![t]\\!]$.

For $f\in S^{d}V$ and $g(t)\in G(\mathbf{k}[\\![t]\\!])$ such that
${\operatorname{val}}_{t}(g(t)\cdot f)>0$, we set

$\mu(g(t),f)=d\cdot\frac{{\operatorname{val}}_{t}(\det(g(t)))}{{\operatorname{val}}_{t}(g(t)\cdot
f)}.$

The factor $d$ in front is a matter of convention: it makes the factor $d$
disappear in some of the statements below.

######  Definition 2.1.

(i) For nonzero $f\in S^{d}V$ we define its $G$-rank by

$r^{G}_{\mathbf{k}}(f)=\inf_{g(t)}\mu(g(t),f),$

where we take the infimum over all $g(t)\in G(\mathbf{k}[\\![t]\\!])$ such
that ${\operatorname{val}}_{t}(g(t)\cdot f)>0$.

(ii) More generally, for linearly independent $f_{1},\ldots,f_{s}\in S^{d}V$,
we define the $G$-rank by

$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})=\inf_{g(t)}\mu(g(t),f_{1},\ldots,f_{s}),$

where

$\mu(g(t),f_{1},\ldots,f_{s})=ds\cdot\frac{{\operatorname{val}}_{t}(\det(g(t)))}{{\operatorname{val}}_{t}(g(t)\cdot
f_{1}\wedge\ldots\wedge f_{s})},$

and the infinum is taken over $g(t)\in G(\mathbf{k}[\\![t]\\!])$ such that
${\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge f_{s})>0$.

The formula

${\operatorname{val}}_{t}(g(t)\cdot
f^{m})=m\cdot{\operatorname{val}}_{t}(g(t)\cdot f)$

immediately implies the following property.

######  Lemma 2.2.

For any $f\in S^{d}V$ and any $m\geq 1$ one has

$r^{G}_{\mathbf{k}}(f^{m})=r^{G}_{\mathbf{k}}(f).$

Here is the main result connecting the $G$-rank with the slice rank and also
with the Waring rank.

######  Theorem 2.3.

Assume the base field $\mathbf{k}$ is perfect.

(i) For a homogeneous polynomial $f$ of degree $d$ over $\mathbf{k}$ one has

${\operatorname{srk}}_{\mathbf{k}}(f)\leq r_{\mathbf{k}}^{G}(f)\leq
d\cdot{\operatorname{srk}}_{\mathbf{k}}(f).$

For a collection $f_{1},\ldots f_{s}$ of homogeneous polynomials of degree $d$
over $\mathbf{k}$ one has

${\operatorname{srk}}_{\mathbf{k}}(f_{1},\ldots,f_{s})\leq
r_{\mathbf{k}}^{G}(f_{1},\ldots,f_{s})\leq
ds\cdot{\operatorname{srk}}_{\mathbf{k}}(f_{1},\ldots,f_{s}).$

(ii) Suppose $m_{1},\ldots,m_{r}$ are divisors of $d$, and
$f_{1},\ldots,f_{r}$ are homogeneous polynomials of degrees
$\deg(f_{i})=d/m_{i}$. Then

$r_{\mathbf{k}}^{G}(f_{1}^{m_{1}}+\ldots+f_{r}^{m_{r}})\leq
r_{\mathbf{k}}^{G}(f_{1})+\ldots+r_{\mathbf{k}}^{G}(f_{r}).$

In particular,

$r_{\mathbf{k}}^{G}(f)\leq w_{\mathbf{k}}(f),$

where $w_{\mathbf{k}}(f)$ is the Waring rank of $f$, i.e., the minimal number
$r$ such that

$f=l_{1}^{d}+\ldots+l_{r}^{d},$

where $l_{i}$ are linear forms defined over $\mathbf{k}$.

The proof will be given in Sec. 2.3 after some preparations. The argument is
very close to the one in [3].

####  2.1.1. Relation to the GIT stability

Let $W$ be a finite dimensional algebraic representation of
$G=\operatorname{GL}(V)$ over $\mathbf{k}$. Recall that a point $w\in W$ is
called $G$-semistable if the orbit closure $\overline{G\cdot v}$ does not
contain $0$. Recall that Kempf’s $\mathbf{k}$-rational version of the Hilbert-
Mumford criterion (see [5]) states (assuming $\mathbf{k}$ is perfect) that if
$w$ is not $G$-semistable then there exists a $1$-parameter subgroup
$\lambda:{\mathbb{G}}_{m}\to G$ defined over $\mathbf{k}$ such that
$\lim_{t\to 0}\lambda(t)\cdot w=0$. Here $\lambda$ has form
$g\cdot\operatorname{diag}(t^{\lambda_{1}},\ldots,t^{\lambda_{n}})\cdot
g^{-1}$ for some $g\in G(\mathbf{k})$ and
$\lambda_{i}\in\operatorname{\mathbb{Z}}$.

######  Proposition 2.4.

For integers $p\geq 0$ and $q>0$, let us consider the $G$-representation

$W=({\bigwedge}^{s}S^{d}V)^{\otimes p}\otimes{\det}^{-dsq}\oplus V^{n}.$

Let $u\in V^{n}$ be a fixed element of rank $n$. Then we have
$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})\geq\frac{p}{q}$ if and only if
$w=((f_{1}\wedge\ldots\wedge f_{s})^{\otimes p}\otimes 1,u)$ is
$G$-semistable.

###### Proof.

By Hilbert-Mumford-Kempf’s criterion, if $w$ is not $G$-semistable then there
exists a $1$-parameter subgroup $\lambda:{\mathbb{G}}_{m}\to G$ over
$\mathbf{k}$, such that $\lim_{t\to 0}\lambda(t)\cdot w=(0,0)$. In particular,
we have $\lim_{t\to 0}\lambda(t)\cdot u=0$, so $\lambda(t)\in
G(\mathbf{k}[t])$, and

${\operatorname{val}}_{t}(\lambda(t)\cdot(f_{1}\wedge\ldots\wedge
f_{s})^{\otimes p}\otimes 1)=p\cdot{\operatorname{val}}_{t}(\lambda(t)\cdot
f_{1}\wedge\ldots\wedge
f_{s})-dsq\cdot{\operatorname{val}}_{t}\det(\lambda(t))>0,$

which implies that ${\operatorname{val}}_{t}(\lambda(t)\cdot
f_{1}\wedge\ldots\wedge f_{s})>0$ and

$\mu(\lambda(t),f_{1},\ldots,f_{s})<\frac{p}{q}.$

Hence, $r^{G}(f_{1},\ldots,f_{s})<\frac{p}{q}$.

Conversely, assume there exists $g(t)\in G(\mathbf{k}[\\![t]\\!])$ such that
${\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge f_{s})>0$ and
$\mu(g(t),f_{1},\ldots,f_{s})<\frac{p}{q}$, i.e.,

${\operatorname{val}}_{t}(g(t)\cdot(f_{1}\wedge\ldots\wedge f_{s})^{\otimes
p}\otimes 1)>0.$

Truncating $g(t)$ at high enough order in $t$, we can assume that $g(t)\in
G(\mathbf{k}[t])$. Then the fact that $\lim_{t\to 0}g(t)\cdot w=(0,g(0)\cdot
u)$ implies that $(0,g(0)\cdot u)$ lies in the closure of the $G$-orbit of
$w$. Since $0$ lies in the closure of the $G$-orbit of $g(0)\cdot u$, we see
that $(0,0)$ lies in the closure of $G\cdot w$, so $w$ is not $G$-semistable.
∎

As a consequence of Proposition 2.4, in the definition of
$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})$ it is enough to take $g(t)$ to be a
$1$-parameter subgroup of $G$ defined over $\mathbf{k}$. Also, since
$G$-semistability does not change under the base field extension, we deduce
the following

######  Corollary 2.5.

Let $\overline{\mathbf{k}}$ be an algebraic closure of $\mathbf{k}$. Then one
has

$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})=r^{G}_{\overline{\mathbf{k}}}(f_{1},\ldots,f_{s}).$

Let $T\subset G$ denote the maximal torus, i.e., the group of diagonal
matrices with respect to a $\mathbf{k}$-basis $(e_{i})$ of $V$. Replacing $G$
everywhere by $T$ we get a notion of $T$-rank,
$r^{T}_{\mathbf{k}}(f_{1},\ldots,f_{s})$. From Hilbert-Mumford-Kempf criterion
we get

$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})=\inf_{g\in
G(\mathbf{k})}r^{T}_{\mathbf{k}}(g\cdot(f_{1},\ldots,f_{s})).$

The reason we introduced the factor $ds$ in the definition of
$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})$ is so as to have the following
normalization property.

######  Lemma 2.6.

One has $r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})\geq 1$.

###### Proof.

It is enough to check that for any $g(t)\in T(\mathbf{k}[\\![t]\\!])$ and any
distinct monomials $M_{1},\ldots,M_{s}$ of $(e_{i})$ in $S^{d}V$, one has

${\operatorname{val}}_{t}(g(t)\cdot M_{1}\wedge\ldots\wedge M_{s})\leq
ds\cdot{\operatorname{val}}_{t}(\det(g(t))).$

Let $c_{1},\ldots,c_{n}\geq 0$ be the valuations of the diagonal entries of
$g(t)$, so that ${\operatorname{val}}_{t}(\det(g(t)))=c_{1}+\ldots+c_{n}$.
Then for a monomial $M=e_{1}^{a_{1}}\ldots e_{n}^{a_{n}}$, we have

${\operatorname{val}}_{t}(g(t)\cdot
M)=a_{1}c_{1}+\ldots+a_{n}c_{n}\leq(a_{1}+\ldots+a_{n})(c_{1}+\ldots+c_{n})=d(c_{1}+\ldots+c_{n}).$

Hence, ${\operatorname{val}}_{t}(g(t)\cdot M_{1}\wedge\ldots\wedge M_{s})\leq
ds$, which gives the required inequality. ∎

###  2.2. Triangle inequality

######  Proposition 2.7.

For $f_{1},f_{2}\in S^{d}V$ one has $r^{G}_{\mathbf{k}}(f_{1}+f_{2})\leq
r^{G}_{\mathbf{k}}(f_{1})+r^{G}_{\mathbf{k}}(f_{2})$.

###### Proof.

This is proved exactly as [3, Prop. 3.6]. Starting with $g_{1}(t),g_{2}(t)\in
G(\mathbf{k}[\\![t]\\!])$ such that ${\operatorname{val}}_{t}(g_{i}(t)\cdot
f_{i})>0$, one has to produce $u(t)\in G(\mathbf{k}[\\![t]\\!])$ with
${\operatorname{val}}_{t}(u(t)\cdot(f_{1}+f_{2}))>0$ and

$\mu(u(t),f_{1}+f_{2})\leq\mu(g_{1}(t),f_{1})+\mu(g_{2}(t),f_{2}).$

Making changes of variables $t\mapsto t^{i}$ if necessary, we can assume that

${\operatorname{val}}_{t}(g_{1}(t)\cdot
f_{1})={\operatorname{val}}_{t}(g_{2}(t)\cdot f_{2})=s>0.$

By [3, Lem. 3.5], there exists $u(t)\in G(\mathbf{k}[\\![t]\\!])$ such that
$u(t)=u_{1}(t)g_{1}(t)=u_{2}(t)g_{2}(t)$ with $u_{i}(t)\in
G(\mathbf{k}[\\![t]\\!])$ and

${\operatorname{val}}_{t}(\det u(t))\leq{\operatorname{val}}_{t}(\det
g_{1}(t))+{\operatorname{val}}_{t}(\det g_{2}(t)).$

Then

${\operatorname{val}}_{t}(u(t)\cdot(f_{1}+f_{2}))\geq\min({\operatorname{val}}_{t}(u_{1}(t)g_{1}(t)\cdot
f_{1}),{\operatorname{val}}_{t}(u_{2}(t)g_{2}(t)\cdot f_{2}))\geq s,$

and

$\displaystyle\frac{1}{d}\mu(u(t),f_{1}+f_{2})=\frac{{\operatorname{val}}_{t}(\det
u(t))}{{\operatorname{val}}_{t}(u(t)\cdot(f_{1}+f_{2}))}\leq$
$\displaystyle\frac{{\operatorname{val}}_{t}(\det
g_{1}(t))+{\operatorname{val}}_{t}(\det
g_{2}(t))}{s}=\frac{1}{d}(\mu(g_{1}(t),f_{1})+\mu(g_{2}(t),f_{2})).$

∎

###  2.3. Relation to the slice rank and to the sums of powers

######  Proposition 2.8.

(i) Let $f=v^{d}$ for some $v\in V\setminus 0$. Then
$r^{G}_{\mathbf{k}}(f)=1$.

(ii) If $f$ is of slice rank $r$ then $r^{G}_{\mathbf{k}}(f)\leq dr$.

(iii) If there exists a nontrivial linear combination
$c_{1}f_{1}+\ldots+c_{s}f_{s}$ that has slice rank $r$ then
$r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})\leq dsr$.

###### Proof.

(i) By Lemma 2.6, $r^{G}_{\mathbf{k}}(f)\geq 1$, so it is enough to find
$g(t)\in G(\mathbf{k}[t])$ such that $\mu(g(t),v^{d})=1$. We can assume that
$v=e_{1}$, and take

$g(t)=\operatorname{diag}(t,1,\ldots,1).$

Then ${\operatorname{val}}_{t}(g(t)\cdot e_{1}^{d})=d$ and
${\operatorname{val}}_{t}(\det(g))=1$. (Alternatively, we can use Lemma 2.2 to
reduce to the easy case $d=1$.)

(ii) We can assume that $f=e_{1}\cdot f_{1}+\ldots+e_{r}\cdot f_{r}$. Then for
$g(t)=\operatorname{diag}(\underbrace{t,\ldots,t}_{r},1,\ldots,1)$, we have
${\operatorname{val}}_{t}(g(t)\cdot f)\geq 1$, while
${\operatorname{val}}_{t}(\deg(g))=r$, so

$\mu(g(t),f)\leq d\cdot\frac{r}{{\operatorname{val}}_{t}(g(t)\cdot f)}\leq
dr.$

(iii) If this is the case then $f_{1}\wedge\ldots\wedge f_{s}$ has form
$(e_{1}h_{1}+\ldots+e_{r}h_{r})\wedge\ldots$, hence, for the same $g(t)$ as in
(ii), we have ${\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge
f_{s})\geq 1$. ∎

######  Proposition 2.9.

One has

${\operatorname{srk}}_{\mathbf{k}}(f_{1},\ldots,f_{s})\leq
r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s}).$

###### Proof.

Suppose $r^{G}_{\mathbf{k}}(f_{1},\ldots,f_{s})<r$. Then there exists a
$1$-parameter subgroup $g(t)$ such that

$r\cdot{\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge
f_{s})>ds\cdot{\operatorname{val}}_{t}(\det(g(t))).$

We can assume that $g(t)$ is diagonal with respect to some basis
$(e_{1},\ldots,e_{n})$ of $V$. Now consider the set

$S:=\\{i\in[1,n]\ |\ {\operatorname{val}}_{t}(g(t)\cdot
e_{i})\geq\frac{{\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge
f_{s})}{ds}\\}.$

Note that

${\operatorname{val}}_{t}(\deg(g(t)))\geq\sum_{i\in
S}{\operatorname{val}}_{t}(g(t)\cdot
e_{i})\geq|S|\cdot\frac{{\operatorname{val}}_{t}(g(t)\cdot
f_{1}\wedge\ldots\wedge f_{s})}{ds},$

hence,

$|S|<r.$

We claim that there exists a nontrivial linear combination
$f=c_{1}f_{1}+\ldots+c_{s}f_{s}$ such that all the monomials appearing in $f$
are divisible by some $e_{i}$ with $i\in S$. Indeed, otherwise, the projection

$\langle
f_{1},\ldots,f_{s}\rangle\to\mathbf{k}[e_{1},\ldots,e_{n}]\to\mathbf{k}[e_{1},\ldots,e_{n}]/(e_{i}\
|\ i\in S)\simeq\mathbf{k}[e_{i}\ |i\not\in S]$

is injective, so there exist $s$ distinct monomials $M_{1},\ldots,M_{s}$ of
degree $d$ in $\mathbf{k}[e_{i}\ |i\not\in S]$ such that
$M_{1}\wedge\ldots\wedge M_{s}$ appears with a nonzero coefficient in
$f_{1}\wedge\ldots\wedge f_{s}$. But then by the choice of $S$,

${\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge
f_{s})\leq{\operatorname{val}}_{t}(g(t)\cdot M_{1}\wedge\ldots\wedge
M_{s})<{\operatorname{val}}_{t}(g(t)\cdot f_{1}\wedge\ldots\wedge f_{s})$

which is a contradiction, proving our claim. Now for the obtained linear
combination $f$ we have

${\operatorname{srk}}_{\mathbf{k}}(f)\leq|S|<r.$

∎

###### Proof of Theorem 2.3.

(i) This follows from Proposition 2.8(iii) and Proposition 2.9.

(ii) This follows from Proposition 2.7 (the triangle inequality), Lemma 2.2
and Proposition 2.8(i) (for the part concerning the Waring rank). ∎

###### Proof of Theorem A.

We combine Theorem 2.3(i) with Corollary 2.5:

${\operatorname{srk}}_{\mathbf{k}}(f_{1},\ldots,f_{s})\leq
r_{\mathbf{k}}^{G}(f_{1},\ldots,f_{s})=r_{\overline{\mathbf{k}}}^{G}(f_{1},\ldots,f_{s})\leq
ds\cdot{\operatorname{srk}}_{\overline{\mathbf{k}}}(f_{1},\ldots,f_{s}).$

∎

###  2.4. Example of a calculation of $G$-rank

As we have seen before, for any linear form form $l$ one has
$r_{\mathbf{k}}^{G}(l^{d})=1$. Here is the next simplest case.

######  Proposition 2.10.

Assume that $n=2$. Then for any $m>0$, one has

$r_{\mathbf{k}}^{G}(x_{1}^{2m}x_{2}^{m})=\frac{3}{2}.$

###### Proof.

By Lemma 2.2, it is enough to prove that

$r_{\mathbf{k}}^{G}(x_{1}^{2}x_{2})=\frac{3}{2}.$

Considering $g(t)=\operatorname{diag}(t,1)$, we immediately see that
$r_{\mathbf{k}}^{G}(x_{1}^{2}x_{2})\leq 3/2$.

Now consider any

$g=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)\in
G(\mathbf{k}[\\![t]\\!]).$

It is enough to prove that $\mu(g,x_{1}^{2}x_{2})\leq 3/2$. We have

$g\cdot x_{1}^{2}x_{2}=a^{2}c\cdot x_{1}^{3}+a(ad+2bc)\cdot
x_{1}^{2}x_{2}+b(bc+2ad)\cdot x_{1}x_{2}^{2}+b^{2}d\cdot x_{2}^{3}.$

Let us abbreviate $v(\cdot)={\operatorname{val}}_{t}(\cdot)$, etc. Set
$s:=v(g\cdot x_{1}^{2}x_{2})$. Then we have

$2v(a)+v(c)\geq s,\ v(a)+v(ad+2bc)\geq s,\ v(b)+v(bc+2ad)\geq s,\
2v(b)+v(d)\geq s.$

We consider three cases.

Case $v(ad)>v(bc)$.

Then we have $v(\det(g))=v(bc)$ and $v(bc+2ad)=v(bc)$. Hence, from the above
inequalities we get $v(b)+v(bc)\geq s$, hence, $2v(bc)\geq s$, so
$v(\det(g))=v(bc)\geq s/2$, and so $\mu(g,x_{1}^{2}x_{2})\geq 3/2$.

Case $v(ad)<v(bc)$.

Then we have $v(\det(g))=v(ad)$ and $v(ad+2bc)=v(ad)$. Hence, $2v(ad)\geq
v(a)+v(ad)\geq s$, and we again get $v(\det(g))\geq s/2$.

Case $v(ad)=v(bc)$.

Set $t=v(ad)=v(bc)$. Then we have $v(\det(g))\geq t$. Now by the above
inequalities,

$4t=2v(ad)+2v(bc)\geq(2v(a)+v(c))+(2v(b)+v(d))\geq 2s,$

which again implies $v(\det(g))\geq s/2$. ∎

##  3\. Linear subspaces of minimal codimension in cubics

###  3.1. Some general observations

Let $f\in\mathbf{k}[V]$ be a nonzero homogeneous polynomial of slice rank $r$,
and let $X\subset{\mathbb{P}}V$ be the corresponding projective hypersurface.
We are interested in the intersection

$L_{f}:=\cap_{L\subset
X,{\operatorname{codim}}_{{\mathbb{P}}V}L=r}L\subset{\mathbb{P}}V.$

Recall that we are looking for an estimate for the codimension of $L_{f}$. The
case $r=1$ is straightforward:

######  Lemma 3.1.

Let $f$ be a homogeneous polynomial of degree $d$ and slice rank $1$. Then
there are most $d$ hyperplanes contained in $X$, so
${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq 3$.

Since the slice rank is determined in terms of ideals
$(P)\subset\mathbf{k}[V]$ generated by subspaces $P$ of linear forms, we
record some easy observations about such ideals.

######  Lemma 3.2.

Let $A\subset B$ be an extension of commutative rings, such that $B$ is flat
as $A$-algebra. Then for any pair of ideals $J_{1},J_{2}\subset A$, one has

$(J_{1}\cdot B)\cap(J_{2}\cdot B)=(J_{1}\cap J_{2})\cdot B.$

In particular, for a collection of linear subspaces $P_{i}\subset W$,
$i=1,\ldots,s$, where $W\subset V^{*}$ is a subspace, we have

$P_{1}\mathbf{k}[V]\cap\ldots P_{s}\mathbf{k}[V]=(P_{1}S(W)\cap\ldots\cap
P_{s}S(W))\cdot\mathbf{k}[V].$

###### Proof.

Since for any ideal $J\subset A$ the natural map $J\otimes_{A}B\to J\cdot B$
is an isomorphism in this case, the assertion follows by applying the exact
functor $?\otimes_{A}B$ to the exact sequence

$0\to J_{1}\cap J_{2}\to J_{1}\oplus J_{2}\to A.$

For the last statement we apply this to the flat extension of rings
$S(W)\subset S(V^{*})=\mathbf{k}[V]$. ∎

######  Lemma 3.3.

Let $P_{1},\ldots,P_{s}\subset V^{*}$ be subspaces such that the ideal
$(P_{1})^{a_{1}}\cap\ldots\cap(P_{s})^{a_{s}}$ contains no nonzero homogeneous
polynomials of degree $m$, for some powers $a_{i}\geq 1$. Then we have an
inclusion of ideals in $\mathbf{k}[V]$,

$(P_{1})^{a_{1}}\cap\ldots\cap(P_{s})^{a_{s}}\subset(W)^{m+1}.$

where $W=P_{1}+\ldots+P_{s}$. In particular, if $P_{1}\cap\ldots\cap P_{s}=0$
then

$(P_{1})^{m}\cap\ldots\cap(P_{s})^{m}\subset(W)^{m+1}.$

###### Proof.

Applying Lemma 3.2 to the extension of rings $S(W)\subset
S(V^{*})=\mathbf{k}[V]$, we reduce to the case when $W=V^{*}$. But then the
first statement reduces to the fact that if the ideal
$I=(P_{1})^{a_{1}}\cap\ldots\cap(P_{s})^{a_{s}}$ does not contain polynomials
of degree $m$ then $I\subset(x_{1},\ldots,x_{n})^{m+1}$.

To prove the second statement we need to check that
$(P_{1})^{m}\cap\ldots\cap(P_{s})^{m}$ does not contain any homogeneous
polynomials of degree $\leq m$. This is clear in degrees $<m$ and in degree
$m$ follows from the statement that

$0=S^{m}(P_{1}\cap\ldots\cap P_{s})=S^{m}(P_{1})\cap\ldots\cap
S^{m}(P_{s})\subset S^{m}W,$

since $P_{1}\cap\ldots\cap P_{s}=0$. ∎

We say that a polynomial $f\in\mathbf{k}[V]=S(V^{*})$ is a pullback from a
space of dimension $m$ if there exists a linear subspace $W\subset V^{*}$ of
dimension $m$ such that $f\in S(W)\subset S(V^{*})$. In this case, if
$f\in(P)$, where $P\subset V^{*}$ is a subspace of linear forms, then
$f\in(W\cap P)$. In particular, the slice rank of $f$ in $S(V^{*})$ can be
calculated within $S(W)$.

###  3.2. Proof of Theorem C(i)

Theorem C(i) is a consequence of the following more precise theorem.

######  Theorem 3.4.

Let $f$ be a cubic of rank $r$, $X\subset{\mathbb{P}}V$ the corresponding
hypersurface. Set

$c(r):=\frac{1}{2}\bigl{(}\frac{(r+1)^{2}}{4}+r+3\bigr{)}\cdot\bigl{(}\frac{(r+1)^{2}}{4}+r\bigr{)}.$

Then

  * •

either all linear subspaces $L\subset X$ with
${\operatorname{codim}}_{{\mathbb{P}}V}L=r$ are contained in a fixed
hyperplane,

  * •

or $f$ is a pullback from a space of dimension $c(r)$.

In either case ${\operatorname{codim}}L_{f}\leq c(r)$.

######  Lemma 3.5.

Let $P_{1},\ldots,P_{s}\subset V^{*}$ be an irredundant collection of
subspaces such that $P_{1}\cap\ldots\cap P_{s}=0$ (i.e., the intersection of
any proper subcollection is nonzero). Assume that $\dim P_{i}\leq r$ for every
$i$. Then

$\dim(P_{1}+\ldots+P_{s})\leq r+\frac{(r+1)^{2}}{4}.$

###### Proof.

Let $a$ be the minimal dimension of intersections $P_{i}\cap P_{j}$. Then we
claim that $s\leq a+2$. Indeed, without loss of generality we can assume that
$\dim P_{1}\cap P_{2}=a$. Then for each $i\geq 2$ we should have

$\dim P_{1}\cap P_{2}\cap\ldots\cap P_{i}\leq a+2-i,$

due to irredundancy of the collection, which proves the claim for $i=s$.

On the other hand, since $\dim P_{i}/(P_{i}\cap P_{1})\leq r-a$ for $i>1$, we
get that

$N:=\dim(P_{1}+\ldots+P_{s})\leq r+(s-1)(r-a)\leq r+(a+1)(r-a)\leq
r+\frac{(r+1)^{2}}{4}.$

∎

###### Proof of Theorem 3.4.

We use induction on $r$. For $r=1$ the assertion is clear. Assume $r>1$ and
the assertion holds for $r-1$. Let ${\mathcal{P}}_{f}$ denote the set of
$r$-dimensional subspaces $P\subset V^{*}$ such that $f|_{P^{\perp}}=0$, or
equivalently, $f\subset(P)$.

If all $P\in{\mathcal{P}}_{f}$ contain the same line $(v^{*})$ then we can
apply the induction assumption to the restriction of $f$ to the hyperplane
$v^{*}=0$ in $V$, which has slice rank $r-1$. Then the induction assumption
implies that

${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq c(r-1)+1\leq c(r).$

Otherwise, there exist $P_{1},\ldots,P_{s}\in{\mathcal{P}}_{f}$ such that
$P_{1}\cap\ldots\cap P_{s}=0$. Choosing a minimal such collection of subspaces
and using Lemma 3.5, we get

$N:=\dim(P_{1}+\ldots+P_{s})\leq r+\frac{(r+1)^{2}}{4}.$

Now by Lemma 3.3, $f$ belongs to $(W)\cdot(W)$, where $W=P_{1}+\ldots+P_{s}$.
Hence, $f$ can be written in the form

$f=\sum_{1\leq i\leq j\leq N}w_{i}w_{j}l_{ij},$

for some linear forms $l_{ij}$, where $(w_{i})$ is a basis of $W$. Hence, $f$
is a pullback from a space of dimension $\leq\frac{N(N+1)}{2}+N\leq c(r)$. ∎

###  3.3. Cubics of slice rank $2$

Here we study in more detail the case of cubics of slice rank $2$, proving in
this case Conjecture B with $c(2,3)=6$ and partially classifying such cubics.

######  Theorem 3.6.

Let $f$ be a cubic of rank $2$. Then

  * •

either all $L\subset X$ with ${\operatorname{codim}}_{{\mathbb{P}}V}=2$ are
contained in a fixed hyperplane, or

  * •

$f$ is a pullback from a $6$-dimensional space, or

  * •

$f$ can be written in the form
$f=x_{1}y_{1}z_{1}+x_{1}y_{2}z_{2}+x_{2}y_{1}z_{3}$, where
$x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},z_{3}$ are linearly independent, or

  * •

$f$ is a pullback from an $8$-dimensional space and
${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq 4$, or

  * •

$f$ is a pullback from a $9$-dimensional space and
${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq 3$.

In either case ${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq 6$.

From now on we fix a cubic $f\in\mathbf{k}[V]$ of slice rank $2$. As in the
proof of Theorem 3.4 we denote by ${\mathcal{P}}_{f}$ the set of
$2$-dimensional subspaces $P\subset V^{*}$ such that $f|_{P^{\perp}}=0$, or
equivalently, $f\subset(P)$, where $(P)\subset\mathbf{k}[V]$ denotes the ideal
generated by $P$.

The following result is well known but we include the (simple) proof for
reader’s convenience.

######  Lemma 3.7.

Let $\SS$ be a set of $2$-dimensional subspaces in $V^{*}$ such that for any
$P_{1},P_{2}\in\SS$ we have $P_{1}\cap P_{2}\neq 0$. Then either there exists
a line $L\subset V^{*}$ such that $L\subset P$ for all $P\in\SS$, or there
exists a $3$-dimensional subspace $W\subset V^{*}$ such that $P\subset W$ for
all $P\in\SS$.

###### Proof.

We can think of $\SS$ as a family of projective lines in the projective space
such that any two intersect. Our statement is that either they all pass
through one point, or they are contained in a plane. Indeed, assume they do
not all pass through one point. Pick a pair of lines $\ell_{1},\ell_{2}$
intersecting at a point $p$. There exists a line $\ell_{3}$, not passing
through $p$. Then $\ell_{1},\ell_{2},\ell_{3}$ form a triangle in a plane. Now
given any other line $\ell$ from $\SS$, we can pick a vertex of the triangle
such that $\ell$ does not pass through it. Say, assume $\ell$ does not pass
through $p$. Then $\ell\cap\ell_{1}$ and $\ell\cap\ell_{2}$ are two distinct
points of $\ell$, so $\ell$ is contained in the plane of the triangle. ∎

######  Lemma 3.8.

Assume that for any pair $P_{1},P_{2}\in{\mathcal{P}}_{f}$ we have $P_{1}\cap
P_{2}\neq 0$. Then either there exists a nonzero linear form $v^{*}\in V^{*}$,
such that $v^{*}\in P$ for all $P\in{\mathcal{P}}_{f}$, in which case
${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq 4$, or $f$ is a pullback from
a $9$-dimensional space and ${\operatorname{codim}}_{{\mathbb{P}}V}L_{f}\leq
3$.

###### Proof.

By Lemma 3.7, either all planes in ${\mathcal{P}}_{f}$ span at most
$3$-dimensional subspace $W\subset V^{*}$, or there exists a nonzero linear
form $v^{*}\in V^{*}$ such that $v^{*}\in P$ for all $P\in{\mathcal{P}}_{f}$.
In the latter case let us consider the restriction $\widetilde{f}$ of our
cubic to the hyperplane $H_{v^{*}}\subset V$. Then $\widetilde{f}$ has rank
$1$ and ${\mathcal{P}}_{f}$ can be identified with
${\mathcal{P}}_{\widetilde{f}}$. So by Lemma 3.1, $L_{f}$ has codimension $3$
in $H_{v^{*}}$, hence, it has codimension $4$ in $V$.

Now let us consider the case when all planes in ${\mathcal{P}}_{f}$ are
contained in a $3$-dimensional subspace $W$, and have zero intersection. Then
by Lemma 3.3, $f\in(W)^{2}$. Hence, as in the proof of Theorem 3.4, we deduce
that $f$ depends on $\leq 9$ variables. ∎

######  Lemma 3.9.

Assume there exists linearly independent linear forms
$x_{1},x_{2},y_{1},y_{2}\in V^{*}$ such that
${\operatorname{span}}(x_{1},x_{2})\in{\mathcal{P}}_{f}$ and
${\operatorname{span}}(y_{1},y_{2})\in{\mathcal{P}}_{f}$. Then $f$ is a
pullback from an $8$-dimensional space, and one of the following possibilities
hold:

  1. (1)

$f$ is a pullback from a $6$-dimensional space;

  2. (2)

for all $P\in{\mathcal{P}}_{f}$ one has
$P\subset{\operatorname{span}}(x_{1},x_{2},y_{1},y_{2})$;

  3. (3)

$f$ can be written in the form
$f=x_{1}y_{1}z_{1}+x_{1}y_{2}z_{2}+x_{2}y_{1}z_{3}$, where
$x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},z_{3}$ are linearly independent.

###### Proof.

Note that we can write

$f=x_{1}y_{1}l_{11}+x_{1}y_{2}l_{12}+x_{2}y_{1}l_{21}+x_{2}y_{2}l_{22},$

for some linear forms $l_{ij}\in V^{*}$. This immediately implies that $f$
depends on $\leq 8$ variables.

Let $P={\operatorname{span}}(l_{1},l_{2})$ be in ${\mathcal{P}}_{f}$. First,
we claim that if $P\cap{\operatorname{span}}(x_{1},x_{2})=0$ and
$P\cap{\operatorname{span}}(y_{1},y_{2})=0$ then either
$P\subset{\operatorname{span}}(x_{1},x_{2},y_{1},y_{2})$ or $f$ is a pullback
from a $6$-dimensional space. Indeed, assume that $P$ is not contained in
${\operatorname{span}}(x_{1},x_{2},y_{1},y_{2})$. First, we observe that for
generic $x\in{\operatorname{span}}(x_{1},x_{2})$ and generic
$y\in{\operatorname{span}}(y_{1},y_{2})$ we should have
$P\cap{\operatorname{span}}(x,y_{1},y_{2})=0$ and
$P\cap{\operatorname{span}}(y,x_{1},x_{2})=0$. Indeed, otherwise we could pick
generic $x,x^{\prime}\in{\operatorname{span}}(x_{1},x_{2})$ such that there
exist nonzero vectors $v\in P\cap{\operatorname{span}}(x,y_{1},y_{2})$ and
$v^{\prime}\in P\cap{\operatorname{span}}(x^{\prime},y_{1},y_{2})$. But then,
since $P\cap{\operatorname{span}}(y_{1},y_{2})=0$, we would have that $v$ and
$v^{\prime}$ are linearly independent, and so
$P={\operatorname{span}}(v,v^{\prime})\subset(x_{1},x_{2},y_{1},y_{2})$.
Hence, changing bases ${\operatorname{span}}(x_{1},x_{2})$ and
${\operatorname{span}}(y_{1},y_{2})$ if necessary, we can assume that

$P\cap{\operatorname{span}}(x_{1},y_{1},y_{2})=P\cap{\operatorname{span}}(x_{2},y_{1},y_{2})=P\cap{\operatorname{span}}(y_{1},x_{1},x_{2})=P\cap{\operatorname{span}}(y_{2},x_{1},x_{2})=0.$

Now the fact that $f\in(P)$ implies that

$x_{1}(y_{1}l_{11}+y_{2}l_{12})\in(x_{2},P).$

Since $x_{1}\not\in(x_{2},P)$, we get that
$y_{1}l_{11}+y_{2}l_{12}\in(x_{2},P)$. Hence,

$y_{1}l_{11}\in(x_{2},y_{2},P).$

We know that $y_{1}\not\in(x_{2},y_{2},P)$ since otherwise we would get a
nonzero intersection $P\cap(x_{2},y_{1},y_{2})$. Hence
$l_{11}\in(x_{2},y_{2},P)$. Similarly, we get $l_{12}\in(x_{2},y_{1},P)$,
$l_{21}\in(x_{1},y_{2},P)$, and $l_{22}\in(x_{1},y_{1},P)$. But this implies
that $f$ is a pull-back from a $6$-dimensional space.

It remains to consider the case when there exists $P$ in ${\mathcal{P}}_{f}$,
such that $P\cap{\operatorname{span}}(y_{1},y_{2})=0$ and
$P\cap{\operatorname{span}}(x_{1},x_{2})={\operatorname{span}}(x_{1})$. Then
the condition $f\in(P)$ gives

$x_{2}(y_{1}l_{21}+y_{2}l_{22})\in(P).$

Hence, $y_{1}l_{21}+y_{2}l_{22}\in(P)$, which implies that

$y_{1}l_{21}\in(y_{2},P).$

Since $y_{1}\not\in(y_{2},P)$, we get $l_{21}\in(y_{2},P)$. Similarly, we get
$l_{22}\in(y_{1},P)$. Let $P={\operatorname{span}}(x_{2},l)$, where $l\in
V^{*}$. Then we can write

$l_{21}=a_{1}x_{1}+b_{1}y_{2}+c_{1}l,\ \ l_{22}=a_{2}x_{1}+b_{2}y_{1}+c_{2}l,$

so we can rewrite $f$ in the form

$f=x_{1}y_{1}(l_{11}+a_{1}x_{2})+x_{1}y_{2}(l_{12}+a_{2}x_{2})+x_{2}(c_{1}y_{1}+c_{2}y_{2})l+(b_{1}+b_{2})x_{2}y_{1}y_{2}.$

The condition $f\in(x_{1},l)$ gives
$(b_{1}+b_{2})x_{2}y_{1}y_{2}\in(x_{1},l)$, which is possible only if
$b_{1}+b_{2}=0$. This easily implies that either $f$ is a pullback from a
$6$-dimensional space, or can be written in the form (3). ∎

###### Proof of Theorem 3.6.

Taking into account Lemmas 3.8 and 3.9, it remains to prove that in the
situation of Lemma 3.9 one has ${\operatorname{codim}}L_{f}\leq 6$. This is
clear in cases (1) and (2). In case (3), it is easy to check that
${\mathcal{P}}_{f}$ consists of $4$ elements:

$(x_{1},x_{2}),(y_{1},y_{2}),(x_{1},z_{3}),(y_{1},z_{2}).$

The corresponding intersection has codimension $6$. ∎

##  4\. Hypersurfaces of higher degree

###  4.1. Proof of Theorem C(ii)

We use induction on $d\geq 1$. The case $d=1$ is clear, so assume that $d\geq
2$ and the assertion holds for degrees $<d$. Assume that
$\dim\sum_{P\in{\mathcal{P}}_{f}}P>d^{2}-1$ (otherwise we are done), and let
$\\{P_{1},\ldots,P_{n}\\}$ be a minimal subset of ${\mathcal{P}}_{f}$ such
that

$\dim\sum_{i=1}^{n}P_{i}>d^{2}-1.$

Note that by minimality, $\dim\sum_{i=1}^{n-1}P_{i}\leq d^{2}-1$, so

$\dim\sum_{i=1}^{n}P_{i}\leq d^{2}+1.$

Claim. There are no nonzero homogeneous polynomials of degree $d-1$ in the
ideal $(P_{1})\cap\ldots\cap(P_{n})$.

Indeed, suppose $g\in(P_{1})\cap\ldots\cap(P_{n})$ is such a polynomial. We
have one of the two cases:

Case 1. $g=l_{1}\ldots l_{k}\cdot h$, where $\deg l_{i}=1$, $0\leq k<d-2$,
${\operatorname{srk}}(h)\geq 2$.

Case 2. $g=l_{1}\ldots l_{d-1}$, where $\deg l_{i}=1$.

Let us consider Case 1 first. Since each $(P_{i})$ is a prime ideal, we should
have a decomposition

$\\{1,\ldots,n\\}=S_{1}\cup\ldots\cup S_{k}\cup S,$

where $l_{j}\in P_{i}$ for all $i\in S_{j}$ and $h\in(P_{i})$ for $i\in S$
(and $S=\emptyset$ if ${\operatorname{srk}}h>2$).

Let us fix $j$ such that $S_{j}\neq\emptyset$. Then $f\mod(l_{i})$ has slice
rank $1$, hence $\dim\sum_{i\in S_{j}}P_{i}/(l_{i})\leq d$ (by Lemma 3.1). In
other words,

$\dim\sum_{i\in S_{j}}P_{i}\leq d+1.$

On the other hand, assuming that $S\neq\emptyset$ and applying the induction
hypothesis to $h$, we get

$\dim\sum_{i\in S}P_{i}\leq(d-1-k)^{2}+1.$

Hence, we obtain

$\dim\sum_{i=1}^{n}P_{i}\leq k(d+1)+(d-1-k)^{2}+1\leq d^{2}-1,$

which is a contradiction.

Similarly, in Case 2 we get

$\dim\sum_{i=1}^{n}P_{i}\leq(d-1)(d+1)=d^{2}-1,$

which is a contradiction. This proves the Claim.

Combining the Claim with Lemma 3.3, we get the inclusion

$f\in(P_{1})\cap\ldots\cap(P_{n})\subset(P_{1}+\ldots+P_{n})^{d}.$

Hence, $f$ is a pullback from a space of dimension $\leq d^{2}+1$. This
finishes the proof.

###  4.2. Proof of Theorem D

Let us dualize the recursive procedure described in Definition 1.1. For a
collection ${\mathcal{P}}=(P_{1},\ldots,P_{s})$ of subspaces of $V^{*}$ we set
$P^{(1)}=\cap_{i=1}^{s}P_{i}$, and for each minimal subset $J\subset[1,s]$
such that $\cap_{j\in J}P_{j}=P^{(1)}$, we set $P_{J}:=\sum_{j\in J}P_{j}$. We
denote by ${\mathcal{P}}^{(1)}$ the collection of all subspaces $P_{J}$ of
$V^{*}$ obtained in this way. Iterating this procedure we get collections of
subspaces ${\mathcal{P}}^{(i)}$ for $i\geq 0$, where
${\mathcal{P}}^{(0)}={\mathcal{P}}$. Let us also set $P^{(0)}=0$ and for
$i\geq 0$,

$P^{(i+1)}:=\cap_{P\in{\mathcal{P}}^{(i)}}P.$

Note that $P^{(i)}\subset P^{(i+1)}$.

Step 1. If $\dim P_{i}\leq r$ for all $i$ then $\dim P_{J}\leq r^{2}$. Indeed,
let $a=\dim P^{(1)}$. Then $\dim P_{i}/P^{(1)}\leq r-a$ and applying Lemma 3.5
we see that for every minimal subset $J$ with $\cap_{j\in J}P_{j}=P^{(1)}$,
one has

$\dim P_{J}=a+\dim P_{J}/P^{(1)}\leq a+(r-a)+\frac{(r-a+1)^{2}}{4}\leq
r+\frac{(r+1)^{2}}{4}.$

Since

$\lfloor r+\frac{(r+1)^{2}}{4}\rfloor\leq r^{2}$

for $r\geq 2$, the assertion follows.

Step 2. Suppose $f$ is a homogeneous polynomials such that $f\in(P_{i})$ for
$i=1,\ldots,s$. Let us prove by induction on $i\geq 0$ that

$f\in(P^{(i)})+(P)^{i+1}$

for any $P\in{\mathcal{P}}^{(i)}$. Indeed, for $i=0$ this is true by
assumption. Assume that $i>0$ and the assertion holds for $i-1$. Let us apply
Lemma 3.3 to a collection of subspaces
$\\{Q_{1},\ldots,Q_{p}\\}\subset{\mathcal{P}}^{(i-1)}$ such that
$Q_{1}\cap\ldots\cap Q_{p}=P^{(i)}$, or rather to the corresponding subspaces
$\overline{Q}_{i}=Q_{i}/P^{(i)}$ of $V^{*}/P^{(i)}$. We get the inclusion of
ideals

$(\overline{Q}_{1})^{i}\cap\ldots\cap(\overline{Q}_{p})^{i}\subset(\sum\overline{Q}_{j})^{i+1}$

in the symmetric algebra of $V^{*}/P^{(i)}$. Let us consider the polynomial
$\overline{f}=f\mod(P^{(i)})$ in this algebra. By assumption,
$\overline{f}\in(\overline{Q}_{j})^{i}$ for $j=1,\ldots,p$. Hence, we deduce
that $\overline{f}\in(\sum\overline{Q}_{j})^{(i+1)}$, i.e.,

$f\in(P^{(i)})+(\sum Q_{j})^{(i+1)}.$

Since every subspace in ${\mathcal{P}}^{(i)}$ has form $\sum Q_{j}$, with
$(Q_{1},\ldots,Q_{p})$ as above, this proves the induction step.

Step 3. For $i=d$, since $f$ is homogeneous of degree $d$, the result of the
previous step gives

$f\in(P^{(d)}).$

Recall that $P^{(d)}$ is the intersection of all subspaces in
${\mathcal{P}}^{(d-1)}$. Iterating the result of Step 1, we see that the
dimension of any subspace in ${\mathcal{P}}^{(d-1)}$, and hence of $P^{(d)}$,
is $\leq r^{2^{d-1}}$. This ends the proof of Theorem D.

######  Remark 4.1.

Suppose we have an $s$-dimensional subspace $F$ of homogeneous polynomials of
the same degree $d$, defined over $\mathbf{k}$, such that there exists a
nonzero $f\in F_{\overline{\mathbf{k}}}$ and subspace of linear forms of
dimension $r$ over $\overline{\mathbf{k}}$ such that $f\in(L)$. One can ask
how to produce an element $f_{0}\in F\setminus 0$ and a subspace of linear
forms $L_{0}$, both defined over $\mathbf{k}$, such that $f_{0}\in(L_{0})$ and
dimension of $L_{0}$ is $\leq c(sr)$ (by Theorem A, we know that such an
element exists).

Let $F_{0}\subset F$ denote the subspace spanned by all the Galois conjugates
of $f$. Then $F_{0}$ is defined over $\mathbf{k}$. As $f_{0}$ we will take any
nonzero element of $F_{0}$.

Since $\dim F_{0}\leq\dim F\leq s$, we can choose a set of elements of the
Galois group $\sigma_{1},\ldots,\sigma_{s}$, such that
$(\sigma_{1}f,\ldots,\sigma_{s}f)$ span $F_{0}$. Hence, $f_{0}$ is a linear
combination of $(\sigma_{1}f,\ldots,\sigma_{s}f)$, and so,

$f_{0}\in(\sigma_{1}L+\ldots+\sigma_{s}L).$

Now applying our algorithm from Theorem D for $f_{0}$, we find a subspace
$L_{0}$ of dimension $\leq c(sr)$ defined over $\mathbf{k}$, with
$f_{0}\in(L_{0})$.

## References

  * [1] K. Adiprasito, D. Kazhdan, T. Ziegler, On the Schmidt and analytic ranks for trilinear forms, arXiv:2102.03659.
  * [2] E. Ballico, A. Bik, A. Oneto, E. Ventura, Strength and slice rank of forms are genericaly equal, arXiv:2102.11549.
  * [3] H. Derksen, The $G$-stable rank for tensors, arXiv:2002.08435.
  * [4] D. Kazhdan, A. Polishchuk, Schmidt rank of quartics over perfect fields, arXiv:2110.10244.
  * [5] G. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), 299-316.


# Datasets for Multilingual Answer Sentence Selection

Matteo Gabburo1 , Stefano Campese1 , Federico Agostini2,3 , Alessandro
Moschitti4
1University of Trento , 2Polytechnic University of Turin, 3University of
Padua, 4Amazon Alexa AI
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Answer Sentence Selection (AS2) is a critical task for designing effective
retrieval-based Question Answering (QA) systems. Most advancements in AS2
focus on English due to the scarcity of annotated datasets for other
languages. This lack of resources prevents the training of effective AS2
models in different languages, creating a performance gap between QA systems
in English and other locales. In this paper, we introduce new high-quality
datasets for AS2 in five European languages (French, German, Italian,
Portuguese, and Spanish), obtained through supervised Automatic Machine
Translation (AMT) of existing English AS2 datasets such as ASNQ, WikiQA, and
TREC-QA using a Large Language Model (LLM). We evaluated our approach and the
quality of the translated datasets through multiple experiments with different
Transformer architectures. The results indicate that our datasets are pivotal
in producing robust and powerful multilingual AS2 models, significantly
contributing to closing the performance gap between English and other
languages.

Datasets for Multilingual Answer Sentence Selection

Matteo Gabburo1 , Stefano Campese1 , Federico Agostini2,3 , Alessandro
Moschitti4 1University of Trento , 2Polytechnic University of Turin,
3University of Padua, 4Amazon Alexa AI
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1 Introduction

Answer Sentence Selection (AS2) represents a crucial component in many QA
systems in both academic and industrial settings. The role of this component
is to select the correct answer for a given question among a pool of candidate
sentences. While in recent years significant progress has been made in
developing models and datasets for AS2 (Wang et al., 2007; Yang et al., 2015;
Garg et al., 2020; Di Liello et al., 2022; Gupta et al., 2023), most of these
are designed and evaluated in English. By contrast, less attention has been
paid to other medium resource languages, such as French, German, Italian,
Portuguese, and Spanish, for which researchers struggle to obtain adequate
amounts of quality data to train their models. Recently, Machine Translation
(MT) has proven to be an effective approach to address the challenges of low-
resource language QA systems (Kumar et al., 2021; Ranathunga et al., 2023;
Gupta et al., 2023).

For the AS2 task, researchers have released a plethora of AS2 datasets in
English, such as ASNQ (Garg et al., 2020), WikiQA (Yang et al., 2015), and
TREC-QA (Wang et al., 2007), but there remains a gap for lower-resource
languages that still needs to be filled.

In this work, we contribute to this research area by introducing three new
large multilingual AS2 corpora named mASNQ, mWikiQA, and mTREC-QA for the most
common European languages, comprising over 100 million question-answer pairs.
We prepared these datasets by translating existing datasets (ASNQ, WikiQA, and
TREC-QA) into five European languages (French, German, Italian, Portuguese,
and Spanish) using a recent state-of-the-art translation model (Team et al.,
2022). To validate the effectiveness of our approach, we trained several
models using the mASNQ111https://huggingface.co/datasets/matteogabburo/mASNQ,
mWikiQA222https://huggingface.co/datasets/matteogabburo/mWikiQA, and mTREC-
QA333https://huggingface.co/datasets/matteogabburo/mTRECQA datasets and
evaluated their performance. Our results demonstrate that these new datasets
can be reliably used to train robust rankers for lower resource languages,
yielding higher performance levels than those other competitors achieve. This
contribution helps to reduce the language barrier and provides valuable assets
for researchers working in low-resource languages.

##  2 Related Work

#### Multilingual Models:

The development of multilingual models has seen significant progress due to
the necessity of solving multilingual NLP tasks and cross-lingual
applications. mBERT (Devlin et al., 2019), an extension of the original BERT
model, can handle tasks across multiple languages. XLM-RoBERTa (Conneau et
al., 2019), trained on 100 languages, and mDeBERTa (He et al., 2021b), a
variant of DebertaV3, have shown remarkable improvements in cross-lingual
tasks. Similarly, mT5 (Xue et al., 2021), a multilingual variant of T5, and
BLOOM (Scao et al., 2023), trained on the ROOTS corpus, exemplify advancements
in multilingual models. Despite these efforts, multilingual models often
underperform compared to their English versions due to the lower availability
of training data (Gupta et al., 2023).

#### Translation Models:

State-of-the-art Machine Translation (MT) models have demonstrated remarkable
capabilities. OPUS-MT (Tiedemann and Thottingal, 2020), a set of translation
tools, supports both bilingual and multilingual translations. The T5 model
(Raffel et al., 2020), originally designed for various generative NLP tasks,
is widely used for MT. The NLLB model (Team et al., 2022), trained on
professionally translated datasets, supports translations between over 200
languages, facilitating broader support for low-resource languages.

#### Machine-Translated Datasets:

MT has been widely used to address the lack of resources for multilingual AS2,
showing promising results in the QA domain (Vu and Moschitti, 2021a; Kumar et
al., 2021; Ranathunga et al., 2023). The itSQuAD dataset (Croce et al., 2018),
the Spanish SQuAD (Carrino et al., 2019), and XQuAD (Dumitrescu et al., 2021)
are examples of datasets translated via MT, used to build QA systems in
different languages. The MLQA dataset (Lewis et al., 2019), mMARCO (Bonifacio
et al., 2021), and Mintaka QA dataset (Sen et al., 2022) further highlight the
success of machine-translated datasets in QA. Xtr-WikiQA and TyDi-AS2 (Gupta
et al., 2023) are recent additions that extend AS2 datasets to multiple
languages.

###  2.1 Answer Sentence Selection (AS2)

The AS2 task involves selecting the correct sentence from a pool of candidates
to answer a given question. Early models like Severyn and Moschitti (2016)
used separate embeddings for questions and answers, followed by convolutional
layers. Garg et al. (2020) implemented Transformer-based models with an
intermediate fine-tuning step, creating the ASNQ corpus from the Natural
Questions dataset (Kwiatkowski et al., 2019). Contextual information has been
shown to enhance AS2 models (Tan et al., 2018; Lauriola and Moschitti, 2021a;
Campese et al., 2023). The translation of English AS2 data into target
languages has been explored, demonstrating the potential for reducing the
complexity of creating multilingual QA systems (Vu and Moschitti, 2021b).
Recently, Cross-Lingual Knowledge Distillation (CLKD) (Gupta et al., 2023) has
shown impressive results for low-resource languages, although the quality of
machine translations remains a critical factor.

##  3 AS2 Translated Datasets

For dataset translation, we use the largest variant of the NLLB model
(NLLB-200-3.3B), which has 3.3 billion parameters. We consider three datasets:
TREC-QA, WikiQA, and ASNQ. For each one, we translate both the questions and
the answers. Our translation process can be described as a _two-step_
procedure. In the first step, we utilize the NLLB model to translate the
source datasets into five languages: French, German, Italian, Portuguese, and
Spanish. In the second step, we employ two techniques to evaluate the quality
of the translations, identifying poor translations, and correcting any
misleading sentences.

Firstly, we use a cross-language semantic similarity model released by Reimers
and Gurevych (2020)444https://huggingface.co/sentence-transformers/paraphrase-
multilingual-mpnet-base-v2 to assess the quality of the translated sentences.
This model compares the semantic similarity between the original sentences in
English and their translated versions in the target language. By measuring the
similarity, we can identify bad translations that deviate significantly from
the original sentences in English, due to the presence of errors. Secondly, we
target these errors by applying a set of heuristics to correct misleading
sentences. These heuristics are designed to correct errors, improve clarity,
remove non-original text, and enhance the overall quality of the translated
datasets.

###  3.1 Datasets

In this work, we considered and translated three datasets for answer sentence
selection (AS2) in $5$ different locales:

mTREC-QA, originates from TREC-QA (Wang et al., 2007), which is created from
the TREC 8 to TREC 13 QA tracks. TREC 8-12 constitutes the training set, while
TREC 13 questions are set aside for development and testing. We used the
_Clean_ setting, meaning that questions without an answer, or with only
correct or incorrect answer-sentence candidates are removed.

mWikiQA is the translated version of WikiQA (Yang et al., 2015). It contains
3047 questions sampled from Bing query logs; candidate answer sentences are
extracted from Wikipedia, and then manually labeled to assess whether it is a
correct answer. Some sentences do not have a correct answer (_all -_), or have
only correct answers (_all +_). We trained using _no all -_ mode and tested in
the _clean_ setting (without both _all +_ and _all -_).

mASNQ comes from ASNQ (Garg et al., 2020) which is an AS2 dataset created by
adapting the Natural Question (Kwiatkowski et al., 2019) corpus from Machine
Reading (MR) to the AS2 task. We replicated this passage using the scripts
provided by Lauriola and Moschitti (2021b).

We summarize the statistics of these datasets in Appendix F.

###  3.2 Removing Translation Artifacts

Despite the good quality of the translation, the dataset still presents some
inconsistencies and artifacts. We identified four major classes of translation
artifacts: (i) Meaning mismatch between the original and the translated
sentences, (ii) The addition of not necessary suffixes and prefixes, (iii) The
difficulty in interpreting and translating numerical strings, (iv) Out-of-
topic translations of partial contexts. We provide some examples of these
translation artifacts in Table 1.

Language | Split | Examples
---|---|---
ITA | Original | ISSN 0362 - 4331
Translated | ISSN 0362 - 4331 \- Non lo so.
DEU | Original | Kusumanjali Prakashan .
Translated | Ich bin ein guter Mensch.
FRA | Original | Jump up to : “ Safe Haven ( 2013 )
Translated | Sauter sur refuge

Table 1: Examples of translation artifacts. The artifacts are highlighted in
red. Notice that (i) "Non lo so" is an Italian sentence which translated in
English means "I don’t know", (ii) "Ich bin ein guter Mensch" means "I am a
good person" in German, and (iii) the maening of the "Sauter sur refuge" in
French is "Jump on refuge".

To tackle these issues, we apply some heuristics to improve the dataset
quality by designing a simple human-centered pipeline to mitigate these
artifacts.

In our approach, we first compute the similarity score between every
translated example in the dataset and the corresponding original text. Then we
filter out translated examples below a similarity threshold of $0.8$ and, on
the remaining set, we compute the most common $1k$ n-grams with $n$ ranging
from $4$ to $9$. Second, we manually inspect these extracted n-grams,
identifying and removing artifact patterns that could distort the data.
Subsequently, we systematically remove occurrences of those problematic
artifacts from the translated dataset. To further improve this operation, we
also identify the examples where the original sentence and the $75\%$ of the
not-blank characters are numbers, and if the similarity score is low (under
$0.8$) we replace the translated sentences with the original one.

###  3.3 Semantic Similarities

To assess the quality of the translations and to quantify the benefit given by
our heuristics, we evaluate the semantic similarity between the original
sentences and their translated versions. For each question-answer pair of each
dataset, we compare the original sentences in English with their translated
version in the target language. The overall similarity measure between the
originals and the translated sentences in each dataset is computed considering
the mean of the semantic similarity scores across all the question-answer
pairs. This average score indicates how closely the translated sentences align
with their original counterparts in terms of semantic meaning. In Appendix F
we report the comparison between the similarity scores of the translated
dataset and the original one.

##  4 Experiments

Language | Transfer | Adapt | mWikiQA | mTrecQA | Xtr-WikiQA
---|---|---|---|---|---
MAP | P@1 | MAP | P@1 | MAP | P@1
ENG |  | ✓ | 0.796 ($\pm$ 0.011) | 0.691 ($\pm$ 0.015) | 0.866 ($\pm$ 0.015) | 0.853 ($\pm$ 0.031) | 0.796 ($\pm$ 0.011) | 0.691 ($\pm$ 0.015)
✓ | ✓ |  0.874 ($\pm$ 0.007) |  0.813 ($\pm$ 0.017) |  0.892 ($\pm$ 0.007) |  0.871 ($\pm$ 0.019) |  0.874 ($\pm$ 0.007) |  0.813 ($\pm$ 0.017)
DEU |  | ✓ | 0.770 ($\pm$ 0.024) | 0.657 ($\pm$ 0.031) | 0.870 ($\pm$ 0.011) | 0.871 ($\pm$ 0.016) | 0.779 ($\pm$ 0.012) | 0.669 ($\pm$ 0.015)
✓ | ✓ |  0.853 ($\pm$ 0.005) |  0.767 ($\pm$ 0.006) |  0.904 ($\pm$ 0.006) |  0.903 ($\pm$ 0.017) |  0.868 ($\pm$ 0.005) |  0.800 ($\pm$ 0.011)
FRA |  | ✓ | 0.769 ($\pm$ 0.012) | 0.662 ($\pm$ 0.018) | 0.872 ($\pm$ 0.007) | 0.859 ($\pm$ 0.013) | 0.760 ($\pm$ 0.009) | 0.646 ($\pm$ 0.012)
✓ | ✓ |  0.836 ($\pm$ 0.006) |  0.752 ($\pm$ 0.012) |  0.891 ($\pm$ 0.010) |  0.874 ($\pm$ 0.029) |  0.844 ($\pm$ 0.005) |  0.778 ($\pm$ 0.014)
ITA |  | ✓ | 0.768 ($\pm$ 0.019) | 0.660 ($\pm$ 0.026) | 0.855 ($\pm$ 0.024) | 0.844 ($\pm$ 0.049) | 0.761 ($\pm$ 0.021) | 0.657 ($\pm$ 0.027)
✓ | ✓ |  0.828 ($\pm$ 0.004) |  0.749 ($\pm$ 0.008) |  0.870 ($\pm$ 0.006) |  0.871 ($\pm$ 0.016) |  0.820 ($\pm$ 0.012) |  0.742 ($\pm$ 0.027)
POR |  | ✓ | 0.798 ($\pm$ 0.011) | 0.704 ($\pm$ 0.019) | 0.855 ($\pm$ 0.021) | 0.704 ($\pm$ 0.019) | 0.780 ($\pm$ 0.011) | 0.684 ($\pm$ 0.020)
✓ | ✓ |  0.853 ($\pm$ 0.018) |  0.781 ($\pm$ 0.029) |  0.874 ($\pm$ 0.009) |  0.781 ($\pm$ 0.029) |  0.849 ($\pm$ 0.023) |  0.775 ($\pm$ 0.042)
SPA |  | ✓ | 0.795 ($\pm$ 0.009) | 0.691 ($\pm$ 0.016) | 0.882 ($\pm$ 0.013) | 0.900 ($\pm$ 0.016) | 0.786 ($\pm$ 0.015) | 0.691 ($\pm$ 0.024)
✓ | ✓ |  0.847 ($\pm$ 0.013) |  0.768 ($\pm$ 0.020) |  0.898 ($\pm$ 0.004) |  0.929 ($\pm$ 0.019) |  0.859 ($\pm$ 0.010) |  0.790 ($\pm$ 0.020)

Table 2: Performance comparison of XLM-RoBERTa on mTREC-QA, mWikiQA, and Xtr-
WikiQA (zero-shot from the model trained on mWikiQA). The transfer step is
done on mASNQ, while the Adaptation is on mTREC-QA and mWikiQA. Results in
terms of MAP and P@1, for various language and model configurations. The
experiments on the English split represent the models trained and tested on
the original, not translated versions of ASNQ, WikiQA and TREC-QA.

In this section, we measure the benefits of our datasets applied to existing
multilingual models. With these experiments, we aim to verify and prove the
effectiveness of our contributions. Specifically, we want to show that the
translated data could be used to train state-of-the-art AS2 models in multiple
languages. For each considered language, we finetune existing multilingual
transformer models on both the original and our translated datasets. We
measure the performance by using information retrieval metrics like Mean
Average Precision (MAP) and the Precision at 1 (P@1). These metrics allow us
to compare the results with the English baselines trained on the original
datasets and to measure the performance improvement given by the ASNQ transfer
step on different languages.

To verify these hypotheses, we consider an existing multilanguage pre-trained
cross-encoder transformer model, which is XLM-RoBERTa
base555https://huggingface.co/xlm-roberta-base, and BERT-
multilingual666https://huggingface.co/bert-base-multilingual-cased. Following
the TANDA approach Garg et al. (2020), we perform a two-stage training for
each model. Precisely, this technique consists of a two-stage training
paradigm, where the first training stage, named _transfer step_ , involves
training the models on ASNQ to teach them to recognize and solve the AS2
tasks. In the second step, named _adaptation step_ , the transferred models
are fine-tuned on the final target AS2 datasets. In our setting, we apply this
paradigm by first training and doing a separate transfer step on each language
of mASNQ. Secondly, we finetune the obtained models on mWikiQA and mTREC-QA.

To have a comprehensive perspective of how our approach behaves, we measure
the performance on three test sets from (i) mWikiQA, (ii) mTREC-QA, and (iii)
Xtr-WikiQA777https://huggingface.co/datasets/AmazonScience/xtr-wiki_qa.

With the first two datasets, we aim to show the performance of our models in a
controlled environment where the translation pipeline and the heuristics are
the same as those used on mASNQ. The third dataset, instead, allows us to
prove that (i) our approach is robust in a zero-shot setting, and (ii) can be
extended to datasets translated using different pipelines.

To perform our experiments, we test XLM-RoBERTa on ASNQ and mASNQ datasets
with specific parameters: batch size of $1024$, Adam optimizer with a learning
rate of $5e-6$, precision set to $32$, and $10$ training epochs. For mWikiQA
and mTREC-QA datasets, the batch size was $32$, Adam optimizer with a learning
rate of $5e-6$, precision set to 16-mixed, and $40$ training epochs with early
stopping. We select the best model maximizing the mean average precision (MAP)
on the development set. The same parameters were used for training the
Multilingual BERT architecture. All experiments utilized 8 NVIDIA V100 $32$ GB
GPUs.

For space reasons, we propose additional experiments using different
multilingual models in Appendix B.

###  4.1 Results

In this section, we present the experimental results of our approaches on
three different AS2 datasets: mWikiQA, mTREC-QA, and the existing Xtr-WikiQA
dataset (Gupta et al., 2023). Table 2 provides an overview of the performance
achieved by our models. First, we observe that our models achieve performance
levels comparable to those of English models. This finding is particularly
noticeable when considering the Portuguese language across all datasets. When
evaluating the models on Xtr-WikiQA, which can be considered as a zero-shot
scenario, as the models are trained on mWikiQA and tested on Xtr-WikiQA, we
find that our approaches demonstrate robustness even when dealing with
datasets translated using a different translation pipeline (Xtr-WikiQA is
translated using Amazon Translate). The results obtained on Xtr-WikiQA
validate the effectiveness of our procedures in handling such translation
variations.

However, we also observe some negative results. Specifically, our experiments
highlight challenges specific to the French language. The XLM-RoBERTa model
performance in this context is notably subpar, which aligns with earlier
findings documented in the relevant literature.

Language | Model | MAP | P@1
---|---|---|---
ENG | ✱ mmarco-mMiniLMv2 | 0.812 | 0.722
✱ bert-multilingual-msmarco | 0.798 | 0.714
xlm-roberta-base | 0.855 | 0.794
bert-multilingual | 0.814 | 0.724
DEU | ✱ mmarco-mMiniLMv2 | 0.797 | 0.700
✱ bert-multilingual-msmarco | 0.759 | 0.663
xlm-roberta-base | 0.844 | 0.770
bert-multilingual | 0.834 | 0.753
FRA | ✱ mmarco-mMiniLMv2 | 0.782 | 0.675
✱ bert-multilingual-msmarco | 0.734 | 0.621
xlm-roberta-base | 0.813 | 0.720
bert-multilingual | 0.863 | 0.807
ITA | ✱ mmarco-mMiniLMv2 | 0.778 | 0.671
✱ bert-multilingual-msmarco | 0.735 | 0.634
xlm-roberta-base | 0.830 | 0.741
bert-multilingual | 0.846 | 0.765
POR | ✱ mmarco-mMiniLMv2 | 0.809 | 0.724
✱ bert-multilingual-msmarco | 0.755 | 0.646
xlm-roberta-base | 0.840 | 0.761
bert-multilingual | 0.841 | 0.761
SPA | ✱ mmarco-mMiniLMv2 | 0.791 | 0.691
✱ bert-multilingual-msmarco | 0.753 | 0.650
xlm-roberta-base | 0.832 | 0.737
bert-multilingual | 0.853 | 0.774

Table 3: Performance comparison of XLM-RoBERTa base model in a zero-shot
setting on the Xtr-WikiQA task. Models trained on mASNQ dataset, denoted by ✱,
outperform those trained on other datasets like mMARCO and MSMARCO. Moreover,
BERT-multilingual consistently performs better than XLM-RoBERTa in various
languages (Italian, Portuguese, Spanish), indicating the robustness and
competitiveness of the approach on AS2 datasets.

In addition, we present a comparison of various models on Xtr-WikiQA in a
zero-shot setting in Table 3. These models have been trained on mASNQ and are
evaluated against existing models trained on well-known and extensive passage
reranking datasets. We find that models trained on mASNQ for the Xtr-WikiQA
task outperform models trained on other datasets such as mMARCO and MSMARCO.
This observation suggests that mASNQ is a more suitable dataset for AS2
compared to mMARCO and MSMARCO. Moreover, when comparing the performance of
BERT-multilingual and XLM-RoBERTa, we find that, on average, BERT-multilingual
performs better. This finding is evident when analyzing the results across
different languages, including Italian, Portuguese, and Spanish. Overall, our
results demonstrate the effectiveness and robustness of our approaches on AS2
datasets, showcasing competitive performance and the superiority of certain
training configurations and models over others.

##  5 Ablation Studies

We conducted several experiments to estimate the benefits provided by our
multilingual datasets, assessing their impact on different aspects of model
performance.

#### Cross-Lingual:

Models trained on the mASNQ dataset consistently outperformed those trained on
ASNQ, demonstrating higher MAP and P@1 scores across all languages. This
confirms the effectiveness of mASNQ in enhancing cross-lingual model
performance (Appendix C).

#### Ranks Correlation:

Models trained on mASNQ and mWikiQA showed strong positive correlations in
their ranking outputs compared to those trained on ASNQ and WikiQA. This
indicates consistent translation quality and robust model performance
(Appendix D).

#### Passage Ranking:

Models trained on mMARCO outperformed those trained on MSMARCO, emphasizing
the significant advantages provided by adapting models trained on our
multilingual datasets for various tasks (Appendix E).

##  6 Conclusion

Our study tackles the language barrier in QA systems by focusing on European
languages such as Italian, German, Portuguese, Spanish, and French. We
introduced new large multilingual AS2 datasets (mASNQ, mWikiQA, and mTREC-QA)
by translating existing English AS2 datasets using a state-of-the-art
translation model. This approach provides valuable resources for lower-
resource languages. Our extensive experiments demonstrated the effectiveness
of these datasets in training robust AS2 rankers across various languages,
achieving performance comparable to English datasets. This contributes
significantly to reducing the language barrier, making AS2 more accessible and
effective across different linguistic contexts. To support further research,
we will release the new models and multilingual AS2 datasets to the research
community. We hope our work inspires future studies to address language
diversity challenges in QA, leading to more inclusive and effective solutions
for global users.

## Limitations

This paper focuses on five European languages (Italian, German, Portuguese,
Spanish, and French). This could represent a limitation since we limit the
applicability of the findings to other languages. Another possible limitation
is that the accuracy and quality of machine translation can affect the
performance of trained models by introducing errors and inconsistencies,
compromising dataset reliability. Moreover, biases present in the original
English data might be transferred to the translated datasets, potentially
resulting in skewed or unrepresentative training examples for specific
languages. Finally, we reserve for future analysis on larger and more powerful
pre-trained multilingual language models (e.g., XLM-RoBERTa large, and
mDeBERTa).

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##  Appendix A ASNQ additional results

Table 4 presents the performance of the XLM-RoBERTa model trained on the
development set of the multilingual ASNQ dataset. The performance of XLM-
RoBERTa on the original ASNQ development set is also reported. The results
indicate that the English baseline outperforms the models trained in other
languages, as expected, while the performance of the models trained in
different languages is consistent and relatively close. These results
highlight that the use of our translated datasets can improve the performance
in terms of MAP, P@1, MRR, and NDCG metrics across multiple languages.

Language | With Translated Data | Without Translated Data
---|---|---
MAP | P@1 | MRR | NDCG | MAP | P@1 | MRR | NDCG
ENG | 0.870 ($\pm$ 0.007) | 0.812 ($\pm$ 0.017) | 0.745 ($\pm$ 0.011) | 0.831 ($\pm$ 0.015) | 0.870 ($\pm$ 0.007) | 0.812 ($\pm$ 0.017) | 0.745 ($\pm$ 0.011) | 0.831 ($\pm$ 0.015)
DEU | 0.850 ($\pm$ 0.005) | 0.768 ($\pm$ 0.006) | 0.710 ($\pm$ 0.008) | 0.811 ($\pm$ 0.009) | 0.845 ($\pm$ 0.005) | 0.762 ($\pm$ 0.010) | 0.705 ($\pm$ 0.010) | 0.806 ($\pm$ 0.012)
FRA | 0.835 ($\pm$ 0.006) | 0.750 ($\pm$ 0.012) | 0.692 ($\pm$ 0.009) | 0.798 ($\pm$ 0.011) | 0.830 ($\pm$ 0.010) | 0.743 ($\pm$ 0.017) | 0.689 ($\pm$ 0.014) | 0.793 ($\pm$ 0.017)
ITA | 0.842 ($\pm$ 0.004) | 0.755 ($\pm$ 0.008) | 0.699 ($\pm$ 0.009) | 0.804 ($\pm$ 0.010) | 0.835 ($\pm$ 0.003) | 0.748 ($\pm$ 0.011) | 0.692 ($\pm$ 0.011) | 0.798 ($\pm$ 0.012)
POR | 0.853 ($\pm$ 0.018) | 0.781 ($\pm$ 0.029) | 0.715 ($\pm$ 0.021) | 0.822 ($\pm$ 0.023) | 0.848 ($\pm$ 0.011) | 0.777 ($\pm$ 0.020) | 0.712 ($\pm$ 0.015) | 0.818 ($\pm$ 0.017)
SPA | 0.847 ($\pm$ 0.013) | 0.768 ($\pm$ 0.020) | 0.705 ($\pm$ 0.016) | 0.812 ($\pm$ 0.018) | 0.840 ($\pm$ 0.012) | 0.760 ($\pm$ 0.024) | 0.700 ($\pm$ 0.018) | 0.805 ($\pm$ 0.019)

Table 4: Performance comparison of XLM-RoBERTa on the multilingual ASNQ
dataset with and without translated data. Results are reported in terms of
MAP, P@1, MRR, and NDCG metrics.

##  Appendix B Results using better multilingual models

In this section, we present the results of our experiments using the newly
created multilingual Answer Sentence Selection (AS2) datasets. The goal is to
evaluate the performance of our approach using mDeBERTa across different
languages and settings. We consider three main tables that provide a
comprehensive overview of the results.

Table 5 presents the performance of mDeBERTa on the mASNQ dataset, covering
multiple languages (DEU, FRA, ITA, SPA, POR). The metrics reported include
Mean Average Precision (MAP), Mean Reciprocal Rank (MRR), Normalized
Discounted Cumulative Gain (NDCG), and Precision at 1 (P@1). These results
highlight the effectiveness of our multilingual datasets, showcasing the
robustness and consistency of the model across different languages.

Approach | MAP | MRR | NDCG | P1
---|---|---|---|---
ASNQ | 0.677 | 0.743 | 0.707 | 0.638
mASNQDEU | 0.633 | 0.702 | 0.662 | 0.590
mASNQFRA | 0.629 | 0.700 | 0.657 | 0.591
mASNQITA | 0.639 | 0.708 | 0.670 | 0.597
mASNQSPA | 0.632 | 0.703 | 0.663 | 0.589
mASNQPOR | 0.635 | 0.706 | 0.664 | 0.595
Table 5: Results of mDeberta on mASNQ

Table 6 compares the performance of mDeBERTa-v3-base when transferred on mASNQ
and ASNQ, and subsequently tested on mWikiQA. The results are presented in
terms of MAP and P@1 for each language considered (ITA, DEU, SPA, POR, FRA).
This table demonstrates the improvements achieved by utilizing the mASNQ
dataset, with notable gains in performance across all languages.

| ASNQ | mASNQ
---|---|---
| MAP | P@1 | MAP | P@1
ITA | 0.868 | 0.801 | 0.884 | 0.821
DEU | 0.882 | 0.827 | 0.885 | 0.821
SPA | 0.875 | 0.811 | 0.886 | 0.829
POR | 0.882 | 0.825 | 0.883 | 0.830
FRA | 0.851 | 0.766 | 0.884 | 0.819
Table 6: Results of mDeBERTa-v3-base transferred on mASNQ and ASNQ and tested
on mWikiQA.

Table 7 presents a detailed performance comparison of mDeBERTa on three
datasets: mWikiQA, mTREC-QA, and Xtr-WikiQA (zero-shot from the model trained
on mWikiQA). The results are reported in terms of MAP and P@1 for various
language and model configurations. This table illustrates the benefits of the
transfer step on mASNQ and the adaptation step on mTREC-QA and mWikiQA, with
the mDeBERTa models consistently achieving high performance across all tasks
and languages.

Language | Transfer | Adapt | mWikiQA | mTREC-QA | Xtr-WikiQA
---|---|---|---|---|---
MAP | P@1 | MAP | P@1 | MAP | P@1
ENG |  | ✓ | 0.872 ($\pm$ 0.008) | 0.801 ($\pm$ 0.013) | 0.905 ($\pm$ 0.004) | 0.909 ($\pm$ 0.022) | 0.872 ($\pm$ 0.008) | 0.801 ($\pm$ 0.013)
✓ | ✓ |  0.901 ($\pm$ 0.006) |  0.854 ($\pm$ 0.008) |  0.921 ($\pm$ 0.003) |  0.950 ($\pm$ 0.008) |  0.901 ($\pm$ 0.006) |  0.854 ($\pm$ 0.008)
DEU |  | ✓ | 0.847 ($\pm$ 0.006) | 0.765 ($\pm$ 0.006) | 0.898 ($\pm$ 0.011) | 0.912 ($\pm$ 0.023) | 0.847 ($\pm$ 0.006) | 0.765 ($\pm$ 0.006)
✓ | ✓ |  0.888 ($\pm$ 0.007) |  0.837 ($\pm$ 0.013) |  0.925 ($\pm$ 0.008) |  0.944 ($\pm$ 0.016) |  0.888 ($\pm$ 0.007) |  0.837 ($\pm$ 0.013)
FRA |  | ✓ | 0.848 ($\pm$ 0.010) | 0.778 ($\pm$ 0.018) | 0.900 ($\pm$ 0.008) | 0.924 ($\pm$ 0.016) | 0.848 ($\pm$ 0.010) | 0.778 ($\pm$ 0.018)
✓ | ✓ |  0.894 ($\pm$ 0.003) |  0.848 ($\pm$ 0.003) |  0.918 ($\pm$ 0.004) |  0.932 ($\pm$ 0.008) |  0.894 ($\pm$ 0.003) |  0.848 ($\pm$ 0.003)
ITA |  | ✓ | 0.851 ($\pm$ 0.010) | 0.774 ($\pm$ 0.011) | 0.890 ($\pm$ 0.009) | 0.900 ($\pm$ 0.016) | 0.851 ($\pm$ 0.010) | 0.774 ($\pm$ 0.011)
✓ | ✓ |  0.885 ($\pm$ 0.006) |  0.822 ($\pm$ 0.010) |  0.919 ($\pm$ 0.005) |  0.938 ($\pm$ 0.012) |  0.885 ($\pm$ 0.006) |  0.822 ($\pm$ 0.010)
POR |  | ✓ | 0.846 ($\pm$ 0.015) | 0.765 ($\pm$ 0.023) | 0.896 ($\pm$ 0.013) | 0.900 ($\pm$ 0.019) | 0.846 ($\pm$ 0.015) | 0.765 ($\pm$ 0.023)
✓ | ✓ |  0.889 ($\pm$ 0.005) |  0.842 ($\pm$ 0.007) |  0.925 ($\pm$ 0.004) |  0.953 ($\pm$ 0.012) |  0.889 ($\pm$ 0.005) |  0.842 ($\pm$ 0.007
SPA |  | ✓ | 0.857 ($\pm$ 0.010) | 0.781 ($\pm$ 0.016) | 0.902 ($\pm$ 0.012) | 0.921 ($\pm$ 0.022) | 0.857 ($\pm$ 0.010) | 0.781 ($\pm$ 0.016)
✓ | ✓ |  0.879 ($\pm$ 0.007) |  0.819 ($\pm$ 0.011) |  0.915 ($\pm$ 0.007) |  0.924 ($\pm$ 0.019) |  0.879 ($\pm$ 0.007) |  0.819 ($\pm$ 0.011)

Table 7: Performance comparison of mDeBERTa on mWikiQA, mTREC-QA, and Xtr-
WikiQA (zero-shot from the model trained on mWikiQA). The transfer step is
done on mASNQ, while the adaptation is on mTREC-QA and mWikiQA. Results in
terms of MAP and P@1, for various language and model configurations.

The results in these tables provide comprehensive insights into the
effectiveness of our multilingual datasets and the benefits of the proposed
transfer and adaptation steps. These findings underline the importance of
high-quality multilingual datasets in improving the performance of AS2 models
across diverse languages, demonstrating the robustness and generalizability of
our approach.

##  Appendix C Ablation: Cross-Lingual

This ablation aims to determine the advantages of using the mASNQ dataset to
train state-of-the-art answer ranking models on languages different from
English. To achieve this, we compare the performance of cross-lingual models
trained on ASNQ and WikiQA with models that were first trained on mASNQ and
mWikiQA, across the different languages that compose mWikiQA.

Table 8 compares the performance of models trained only on the original
versions of ASNQ and WikiQA with the performance of the same architecture
(XLM-RoBERTa base) but trained on our multilingual datasets. To achieve this
goal, we measure the performance of each model across all the different test
sets of mWikiQA and across their languages. For the evaluation, we considered
two proxy measures to understand the quality of the models: Mean Average
Precision (MAP) and Precision at 1 (P@1). The results show that the models
achieve higher MAP and P@1 scores when trained on mASNQ compared to ASNQ,
indicating that training on the mASNQ dataset improves the performance of
multilingual models in cross-lingual tasks. Across all languages, the models
trained on mASNQ consistently outperform the models trained on ASNQ. This
suggests that the mASNQ dataset can guarantee a performance boost for non-
English target datasets, confirming our hypotheses.

Language | ASNQ | mASNQ
---|---|---
MAP | P@1 | MAP | P@1
DEU | 0.814 | 0.705 | 0.839 | 0.755
FRA | 0.819 | 0.717 | 0.793 | 0.671
ITA | 0.819 | 0.715 | 0.839 | 0.726
POR | 0.822 | 0.722 | 0.842 | 0.755
SPA | 0.830 | 0.738 | 0.835 | 0.751

Table 8: Comparison of XLM-RoBERTa base transferred on mASNQ and ASNQ and
tested on mWikiQA in a cross-lingual setting.

##  Appendix D Ablation: Ranks Correlation

This study compares the ranking outputs of two sets of models, analyzing the
correlation between their rankings. The first set comprehends models trained
on mASNQ and mWikiQA and then tested on the mWikiQA test set, while the second
set contains models trained on ASNQ and WikiQA and evaluated on the original
English WikiQA test set.

We design this experiment in order to compare the rank provided for each
question $q_{Eng}^{i}$ of the original English dataset (WikiQA), with the
semantically equivalent question $q_{T}^{i}$ and its rank for each language
$T$ in mWikiQA. To measure the performance, we compute three correlation
metrics to properly evaluate the correlation between the rankings of each pair
of questions $\\{q_{Eng}^{i},q_{T}^{i}\\}$; in this way, we allow determining
the level of agreement between the two models’ ranking outputs, providing
insights into the potential differences between them. Specifically, we
consider XLM-RoBERTa base and compute the Kendall, Spearman, and Pearson
correlation metrics on mWikiQA and mTREC-QA.

The results in Table 9 show a strong positive correlation between the
performance of models trained in English and tested in English, and the models
trained in other languages (using mASNQ, mWikiQA, and mTREC-QA). This
correlation is evident across all evaluation metrics, with Kendall
correlations ranging from $0.694$ to $0.720$, Spearman correlations ranging
from $0.802$ to $0.824$, and Pearson correlations ranging from $0.872$ to
$0.908$ for the mASNQ$\rightarrow$mWikiQA task. The high correlation values,
ranging from $0.547$ to $0.733$, across all languages for the
mASNQ$\rightarrow$mTREC-QA task further support this notion. The Kendall,
Spearman, and Pearson correlations show consistently high values, indicating
that the translation quality and model performance are consistently strong.
The results of the analysis demonstrate (i) the effectiveness of the
translation process for mASNQ and (ii) the strong performance of the models.

mASNQ$\rightarrow$mWikiQA
---
Language | Kendall | Spearman | Pearson
DEU | 0.720 | 0.820 | 0.908
FRA | 0.694 | 0.802 | 0.872
ITA | 0.713 | 0.817 | 0.903
POR | 0.710 | 0.824 | 0.908
SPA | 0.698 | 0.807 | 0.902

mASNQ$\rightarrow$mTREC-QA
---
Language | Kendall | Spearman | Pearson
DEU | 0.547 | 0.666 | 0.713
FRA | 0.566 | 0.663 | 0.709
ITA | 0.513 | 0.629 | 0.695
POR | 0.567 | 0.669 | 0.733
SPA | 0.587 | 0.688 | 0.728

Table 9: Kendall, Spearman and Pearson correlation computed between the ranks
originated from model trained the original ASNQ and mASNQ. The reported values
are computed using XLM-RoBERTa base models transferred on ASNQ and mASNQ and
then finetuned on mWikiQA and mTREC-QA.

##  Appendix E Ablation: Passage Ranking

To further evaluate the robustness of our datasets, we also perform several
experiments on a different task: Passage Reranking (PR). Passage Reranking is
an Information Retrieval (IR) task that consists of reordering a set of
retrieved passages for a given query. For this reason, we consider a well-
known dataset named mMARCO (Bonifacio et al., 2021), well known in the
multilingual IR community. Specifically, we select a random language among the
ones considered in the previous experiments, and we train several multi-
language models. In detail, we split the original Italian dataset into train,
validation, and test splits (Tab. 11).

We compare the results obtained by our approaches with two models: the first
is a multilingual BERT trained on the English MSMARCO, while the second model
is trained on our train split. In Table 10, we present the results of this
comparison. They clearly show that our models trained on the mMARCO dataset
outperform the model trained on MSMARCO (e.g., $0.687$ vs $0.682$ in terms of
MAP).

Although the improvement is modest, it becomes significant due to the large
size of the mMARCO test set. These findings highlight the advantages our
datasets offer for tasks beyond AS2. Even with a marginal improvement, it is
evident that adapting a model trained on our multilingual datasets can yield
further performance enhancements.

Dataset | Transfer | Adapt | mMARCO
---|---|---|---
MAP | P@1
MSMARCO | ✓ |  | 0.631 | 0.502
mMARCO |  | ✓ | 0.682 | 0.553
mASNQ$\xrightarrow{}$mMARCO | ✓ | ✓ | 0.687 | 0.559

Table 10: Comparison of BERT-multilingual performance on
mMARCO${}_{\text{ITA}}$ test set. We train the two baselines respectively on
the English MSMARCO and the mMARCO Italian split. The models trained on mASNQ
and adapted to mMARCO consistently improve the two presented baselines,
showing that the transfer step on mASNQ is helpful in this domain.

##  Appendix F Datasets

In Table 11, we provide the datasets we described in Section 3.

Dataset | Split | #Question | #QA Pairs
---|---|---|---
mASNQ | Train | 57240 | 20377168
Validation | 2672 | 930062
mWikiQA | Train | 2118 | 20356
Validation | 296 | 2731
Validation (++) | 126 | 1130
Validation (clean) | 122 | 1126
Test | 633 | 6160
Test (++) | 243 | 2350
Test (clean) | 237 | 2340
mTREC-QA | Train | 1227 | 53282
Validation | 65 | 1117
Test | 68 | 1441

Table 11: Dataset statistics for mASNQ, mWikiQA, and mTREC-QA for each
language. The datasets have the same statistics in their original version, and
considering all the languages, the corpora comprehend more than 100M examples.
Notice that for mWikiQA we report also the statistics of the clean and the no-
all-negatives (++) splits.

In addition, in Table 12, we report the semantic similarity between ASNQ and
mASNQ to support the translation quality further.

| Language | Similarity
---|---|---
Dev | DEU | $0.869\pm 0.178\rightarrow 0.991\pm 0.003$
FRA | $0.838\pm 0.223\rightarrow 0.990\pm 0.003$
ITA | $0.913\pm 0.115\rightarrow 0.990\pm 0.001$
POR | $0.915\pm 0.117\rightarrow 0.992\pm 0.003$
SPA | $0.857\pm 0.211\rightarrow 0.990\pm 0.001$
Train | DEU | $0.871\pm 0.175\rightarrow 0.923\pm 0.110$
FRA | $0.841\pm 0.218\rightarrow 0.923\pm 0.101$
ITA | $0.915\pm 0.112\rightarrow 0.940\pm 0.081$
POR | $0.915\pm 0.115\rightarrow 0.941\pm 0.082$
SPA | $0.860\pm 0.206\rightarrow 0.932\pm 0.090$

Table 12: Similarities between ASNQ and mASNQ. On the left of the arrow
($\rightarrow$) the similarity reached after the initial translation is
reported; on the right side, there is the similarity score after the
application of the heuristics.