# On rank in algebraic closure

Amichai Lampert and Tamar Ziegler

###### Abstract.

Let ${\mathbf{k}}$ be a field and $Q\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ a
form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf{k}}$-Schmidt rank
$\textnormal{rk}_{\mathbf{k}}(Q)$ of $Q$ is the minimal $r$ such that
$Q=\sum_{i=1}^{r}R_{i}S_{i}$ with
$R_{i},S_{i}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ forms of degree $<d$. When
${\mathbf{k}}$ is algebraically closed and $\textnormal{char}({\mathbf{k}})$
doesn’t divide $d,$ this rank is closely related to
$\textnormal{codim}_{\mathbb{A}^{s}}(\nabla Q(x)=0)$ \- also known as the
Birch rank of $Q.$ When ${\mathbf{k}}$ is a number field, a finite field or a
function field, we give polynomial bounds for
$\textnormal{rk}_{\mathbf{k}}(Q)$ in terms of
$\textnormal{rk}_{\bar{\mathbf{k}}}(Q)$ where $\bar{\mathbf{k}}$ is the
algebraic closure of ${\mathbf{k}}.$ Prior to this work no such bound (even
ineffective) was known for $d>4$. This result has immediate consequences for
counting integer points (when ${\mathbf{k}}$ is a number field) or prime
points (when ${\mathbf{k}}=\mathbb{Q}$) of the variety $(Q=0)$ assuming
$\textnormal{rk}_{\mathbf{k}}(Q)$ is large.

The authors were supported by ERC grant ErgComNum 682150, and ISF grant
2112/20.

##  1\. Introduction

In [16], Schmidt made the following definition:

######  Definition 1.1.

Let ${\mathbf{k}}$ be a field and $Q\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ a
form, i.e. homogeneous polynomial. The Schmidt rank
$\textnormal{rk}_{\mathbf{k}}(Q)$ is the smallest natural number $r$ such that
$Q=\sum_{i=1}^{r}R_{i}S_{i}$ with
$R_{i},S_{i}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ forms of positive degree. If
$Q$ is any degree $d$ polynomial, we define $\textnormal{rk}_{\mathbf{k}}(Q)$
as the rank of its degree $d$ part.

Note that for $l\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ linear and non-constant,
$\textnormal{rk}_{\mathbf{k}}(l)=\infty.$ Schmidt introduced this quantity
when ${\mathbf{k}}=\mathbb{Q}$ and $Q$ is homogeneous and proved that if
$\textnormal{rk}_{\mathbb{Q}}(Q)$ is sufficiently large then a certain local-
global rule applies for counting the number of integer points in the set
$\\{x\in{\mathbb{Z}}^{s},|x|\leq P:Q(x)=0\\}.$ He also showed that
$\textnormal{rk}_{\mathbb{C}}(Q)$ is essentially equivalent to
$\textnormal{codim}_{{\mathbb{C}}^{s}}(\nabla Q(x)=0),$ known as the Birch
rank of $Q$ (we will define Birch rank more generally in the next section).
Note that it follows immediately from the definition that
$\textnormal{rk}_{\mathbb{C}}(Q)\leq\textnormal{rk}_{\mathbb{Q}}(Q).$ The
object of this paper is to prove a polynomial bound in the opposite direction,
for ${\mathbf{k}}=\mathbb{Q}$ and several other fields. For perfect fields
${\mathbf{k}},$ inequalities bounding $\textnormal{rk}_{{\mathbf{k}}}(Q)$ in
terms of $\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q)$ are known for $d=2,3$ [5]
and for $d=4$ there is an ineffective bound when
$\textnormal{char}({\mathbf{k}})\neq 2$ [10]. The bound we prove is quite
interesting in its own right, but also has a number of immediate applications
to Diophantine equations, since many results which use the Hardy-Littlewood
circle method assume the Birch rank is large. Our result allows us to relax
this assumption to large Schmidt rank over the base field.

We describe some direct applications of our result in the next section -
counting prime solutions to Diophantine equations, counting integer solutions
to polynomial equations over number fields, and an analogue for function
fields.

######  Definition 1.2.

We call the following fields _admissible_ : number fields, finite fields and
finite separable extensions of $\mathbb{F}_{q}(t).$

We prove the following:

######  Theorem 1.3 (Schmidt rank and field extensions).

Let ${\mathbf{k}}$ be an admissible field, $\bar{{\mathbf{k}}}$ its algebraic
closure, and $Q\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ a polynomial of degree
$d>1$ such that $\textnormal{char}({\mathbf{k}})>d$ or
$\textnormal{char}({\mathbf{k}})=0.$ Then there exist constants
$A=A({\mathbf{k}},d),B=B({\mathbf{k}},d)$ such that

$\textnormal{rk}_{\mathbf{k}}(Q)\leq
A[\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q)+1]^{B}.$

We will soon give explicit values for the constants $A,B.$ Our result
generalizes to collections of forms of the same degree, with an appropriate
definition of rank.

######  Definition 1.4.

Let $Q_{1},\ldots,Q_{n}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ be homogeneous
polynomials all of degree $d>1.$ Their collective Schmidt rank is defined to
be

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n}):=\min_{0\neq
a\in{\mathbf{k}}^{n}}\textnormal{rk}_{\mathbf{k}}\left(\sum_{i=1}^{n}a_{i}\cdot
Q_{i}\right).$

If $Q_{1},\ldots,Q_{n}$ are all of degree $d$ but not necessarily homogeneous,
we define $\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})$ in terms of their
degree $d$ parts.

######  Theorem 1.5 (Schmidt rank and field extensions).

Let ${\mathbf{k}}$ be an admissible field, $\bar{{\mathbf{k}}}$ its algebraic
closure, and $Q_{1},\ldots,Q_{n}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ a
collection of polynomials of common degree $d>1$ such that
$\textnormal{char}({\mathbf{k}})>d$ or $\textnormal{char}({\mathbf{k}})=0.$
Then there exist constants $A=A({\mathbf{k}},d),B=B({\mathbf{k}},d)$ such that
if ${\mathbf{k}}$ is a perfect field we have

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\leq
A[n\cdot\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})+1]^{B},$

and if ${\mathbf{k}}$ is an imperfect field we have

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\leq
A\left[n^{1+1/d}\cdot(\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})+1)\right]^{B}.$

We can take $A({\mathbf{k}},d),B({\mathbf{k}},d)$ as follows:

  1. (1)

$A=4^{d-1}d\binom{d}{\lfloor d/2\rfloor}^{d},\ B=d$ when
${\mathbf{k}}=\mathbb{Q}.$

  2. (2)

$A=2^{2^{O(d^{2})}},\ B=2^{2^{O(d^{2})}}$ when ${\mathbf{k}}$ is a finite
field.

  3. (3)

$A=2^{d-1}\binom{d}{\lfloor d/2\rfloor}^{d},\ B=d$ when ${\mathbf{k}}$ is a
finite field and $|{\mathbf{k}}|\geq
F(\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n}),n,d).$

  4. (4)

$A=2^{d-1}(d-1)\binom{d}{\lfloor d/2\rfloor}^{2d},\ B=2d$ when ${\mathbf{k}}$
is a number field or a finite separable extension of $\mathbb{F}_{q}(t).$

We mention a conjecture stated in [1]

######  Conjecture 1.6.

Let $d>1$ and let ${\mathbf{k}}$ be a field with $char({\mathbf{k}})>d$ or
$char({\mathbf{k}})=0$ There exists a constant $C(d)$ such that for any
polynomial $Q\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ of degree $d$ we have

$\textnormal{rk}_{\mathbf{k}}(Q)\leq
C\cdot\textnormal{rk}_{\bar{\mathbf{k}}}(Q).$

This conjecture is known for $d=2,3$ when ${\mathbf{k}}$ is a perfect field
[5].

The paper is organized as follows: In section 2 we give some number theoretic
applications of our results. In section 3 we explain how to deduce them from
an analogous theorem about multilinear polynomials. In section 4 we prove
theorem 1.5 for finite fields and for $\mathbb{Q}$ (the result for
$\mathbb{Q}$ relies on the finite fields result), and in section 5 we prove it
for number fields and for finite separable extensions of $\mathbb{F}_{q}(t).$

###  1.1. Acknowledgements

The authors thank the anonymous referee for a careful reading and helpful
suggestions.

##  2\. Applications

In [2], Birch defined a geometric notion of rank which is closely related to
the singular locus of the variety defined by a collection of polynomials. In
many applications of the Hardy-Littlewood circle method, it’s necessary to
assume that the Birch rank of the polynomials we are interested in is large.

######  Definition 2.1.

Let $Q\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ be a form of degree $d.$ The Birch
rank of $Q$ is
$\textnormal{rk}_{B}(Q)=\textnormal{codim}_{\mathbb{A}^{s}}(\nabla Q(x)=0).$
Given a collection $Q_{1},\ldots,Q_{n}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ of
forms of degree $d,$ we define $S(Q_{1},\ldots,Q_{n})$ to be the variety where
the matrix of partial derivatives $(\partial Q_{i}/\partial x_{j})$ has rank
$<n.$ The Birch rank of the collection is
$\textnormal{rk}_{B}(Q_{1},\ldots,Q_{n})=\textnormal{codim}_{\mathbb{A}^{s}}S(Q_{1},\ldots,Q_{n}).$
For a general collection of polynomials
$Q_{1},\ldots,Q_{n}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ of degree $d,$ we
define their Birch rank in terms of the homogeneous degree $d$ parts. As a
matter of convention, set $\textnormal{rk}_{B}(\emptyset)=s.$

It’s not difficult to show that there is an inequality

$\textnormal{rk}_{B}(Q_{1},\ldots,Q_{n})\leq
2\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n}).$

Let us discuss for a moment what happens with quadratics. For a single
quadratic $Q$, there is an elementary inequality in the opposite direction
$\textnormal{rk}_{\mathbf{k}}(Q)\leq\textnormal{rk}_{B}(Q).$ If ${\mathbf{k}}$
is algebraically closed, we can take the union of the Birch singular loci of
non-trivial linear combinations $c_{1}Q_{1}+\ldots+c_{n}Q_{n},$ which
correspond to points of $\mathbb{P}^{n-1},$ and apply the above inequality to
deduce:

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\leq\textnormal{rk}_{B}(Q_{1},\ldots,Q_{n})+n-1.$

As a consequence, when $\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\geq
2n-1$ we deduce that $X=\\{Q_{1}=\ldots=Q_{n}=0\\}$ is a complete intersection
of codimension $n.$ For suppose $Y\subset X$ has codimension $<n.$ Since
$\textnormal{rk}_{B}(Q_{1},\ldots,Q_{n})\geq n,$ there exists a point $y\in Y$
such that the Jacobian matrix has rank $n$ at $y.$ But then the tangent space
at $y$ has codimension $n,$ contradicting the fact that the local codimension
at $y$ is $<n.$

Kazhdan, Polishchuk and the first author [9] extended results of Schmidt to
show that a similar inequality holds also for higher degrees. Cohen and
Moshkovitz [3] proved a similar result for a single multi-linear polynomial.

######  Theorem 2.2 (Theorem 1.4 in [9]).

Assume that $d\geq 2,$ ${\mathbf{k}}$ is algebraically closed, and
$\textnormal{char}({\mathbf{k}})$ doesn’t divide $d.$ Then we have

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\leq(d-1)\left[\textnormal{rk}_{B}(Q_{1},\ldots,Q_{n})+n-1\right].$

In particular, if
$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\geq(d-1)(2n-1)$ then
$(Q_{1}=\ldots=Q_{n}=0)$ is a complete intersection of codimension $n.$

######  Corollary 2.3.

Let ${\mathbf{k}}$ be an admissible field, and
$Q_{1},\ldots,Q_{n}\in{\mathbf{k}}[x_{1},\ldots,x_{s}]$ forms of degree $d\geq
2,$ such that $\textnormal{char}({\mathbf{k}})>d$ or
$\textnormal{char}({\mathbf{k}})=0.$ Then there exist constants
$A^{\prime}=A^{\prime}({\mathbf{k}},d),B^{\prime}=B^{\prime}({\mathbf{k}},d)$
such that if $\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\geq
A^{\prime}(nr)^{B^{\prime}}$ then $\textnormal{rk}_{B}(Q_{1},\ldots,Q_{n})\geq
r.$ In particular, if $r\geq n$ then $(Q_{1}=\ldots=Q_{n}=0)$ is a complete
intersection of codimension $n.$

###### Proof.

If $A^{\prime},B^{\prime}$ are taken sufficiently large then by theorem 1.5 we
obtain

$\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})\geq(d-1)(r+n-1).$

Plugging this into theorem 2.2 proves the corollary. ∎

As we mentioned above, many applications of the Hardy-Littlewood circle method
require that the Birch rank of the collection of forms be large. The above
corollary which follows immediately from theorem 1.5 allows us to replace this
condition by requiring the Schmidt rank to be large. We now give some examples
of this.

###  2.1. Prime solutions

Yamagishi [19], extending a result of Cook and Magyar [4], proved a local to
global rule holds for counting prime solutions to systems of integral
polynomial equations with sufficiently large Birch rank. Cook and Magyar
conjectured that it’s sufficient to assume that the Schmidt rank over
$\mathbb{Q}$ is large. The conjecture was proved for polynomials of degree
$\leq 3$ in [18]. As a consequence of our results, we can prove their
conjecture for polynomials of any degree. Let $D\geq 1$, and let
$\bar{Q}=(\bar{Q}_{1},\ldots,\bar{Q}_{D})$ be a system of polynomials in
$\mathbb{Z}[x_{1},\ldots,x_{s}]$, where
$\bar{Q}_{d}=(Q_{d,1},\ldots,Q_{d,t_{d}})$ is the subsystem of degree $d$
polynomials of $\bar{Q}$ $(1\leq d\leq D)$. Let $X=\\{\bar{Q}=0\\}$ be the
affine variety defined by $\bar{Q}.$ Following Cook and Magyar, we use a
different notion of rank for systems of linear polynomials. If
$l_{1},\ldots,l_{n}$ are linear polynomials with degree one homogeneous parts
$L_{1},\ldots,L_{n}$ we define $r(l_{1},\ldots,l_{n})$ to be the minimum
number of non-zero coefficients in a non-trivial linear combination

$\lambda_{1}L_{1}+\ldots+\lambda_{n}L_{n},$

where
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Q}^{n}\backslash\\{\mathbf{0}\\}.$
Let $\Lambda$ be the von Mangoldt function, Given $x=(x_{1},\ldots,x_{s})$, we
let

$\Lambda(x)=\Lambda(x_{1})\ldots\Lambda(x_{s}).$

Set

$\mathcal{M}_{\bar{Q}}(N):=\sum_{x\in[0,N]^{s}}\Lambda(x)\ 1_{X}(x).$

As an immediate consequence of combining corollary 2.3 with Yamagishi’s result
we obtain:

######  Theorem 2.4.

There exists a constant $\chi=\chi(D,t_{1},\ldots,t_{D})$ such that if
$r(\bar{Q}_{1})\geq\chi$ and
$\textnormal{rk}_{\mathbb{Q}}(\bar{Q}_{d})\geq\chi$ for each $2\leq d\leq D$
with $t_{d}>0$ then there exist constants $\mathcal{C}=\mathcal{C}(\bar{Q})$,
and $c>0$ such that

$\mathcal{M}_{\bar{Q}}(N)=\mathcal{C}(\bar{Q})\
N^{s-\sum_{d=1}^{D}dt_{d}}+O\left(\frac{N^{s-\sum_{d=1}^{D}dt_{d}}}{(\log
N)^{c}}\right).$

Furthermore, if the system of equations $\bar{Q}=0$ has a non-singular
solution in $\mathbb{Z}_{p}^{\times}$, the units of $p$-adic integers, for
each prime $p$, and the system of homgeneous equations given by the leading
forms has a non-singular real point in $(0,1)^{s}$, then

$\mathcal{C}(\bar{Q})>0.$

###  2.2. Number fields

Frei and Madritsch [7], extending an earlier result of Skinner [17], proved a
local to global rule for counting integral solutions to a system of polynomial
equations over a number field, assuming the Birch rank is sufficiently large.
Let $K$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\bar{Q}$ be a
system of polynomials in $s$ variables over the ring of integers $\mathcal{O}$
of $K$ with maximal degree $D.$ We assume the polynomials to be ordered by
their degrees, that is, for each $d\in\\{1,\ldots,D\\}$, we are given
polynomials

$\bar{Q}_{d}=(Q_{d,1},\ldots,Q_{d,t_{d}})\in\mathcal{O}_{K}[x_{1},\ldots,x_{s}]$

of degree $d$, where $t_{d}\geq 0$ with $t_{D}\geq 1$. Fix an integral ideal
$\eta$ of $\mathcal{O}$ and a ${\mathbb{Z}}$-basis
$\omega_{1},\ldots,\omega_{n}$ of $\eta$. We will also consider
$\omega_{1},\ldots,\omega_{n}$ as an ${\mathbb{R}}$-basis of
$V:=K\otimes_{\mathbb{Q}}{\mathbb{R}}$. By a _box $\mathcal{B}$ aligned to the
basis_, we mean the set of all $x=(x_{1},\ldots,x_{s})\in V^{s}$, where each
$x_{i}$ has the form $x_{i,1}\omega_{1}+\cdots+x_{i,n}\omega_{n}$, such that
the coordinates $(x_{i,j})_{i,j}\in{\mathbb{R}}^{ns}$ lie in a given box
$B\subseteq[-1,1]^{ns}$ with sides aligned to the coordinate axes of
${\mathbb{R}}^{ns}$. Given such a box $\mathcal{B}$, we define the counting
function

$N(P):=\left|\\{x\in\eta^{s}\cap P\mathcal{B}:Q_{d,i}(x)=0\textnormal{ for all
}1\leq d\leq D,1\leq i\leq t_{d}\\}\right|.$

Let

$\Delta:=\\{1\leq d\leq D:t_{d}\geq 1\\}.$

Let $\mathcal{D}_{0}:=0$ and, for $1\leq d\leq D$,

$\mathcal{D}_{d}:=t_{1}+2t_{2}+\cdots+dt_{d},$

so that $\mathcal{D}:=\mathcal{D}_{D}$ is the sum of the degrees of all the
polynomials $Q_{d,i}$. Set

$\displaystyle u_{d}$
$\displaystyle:=\sum_{k=d}^{D}2^{k-1}(k-1)t_{k}\quad\textnormal{ for }\quad
1\leq d\leq D+1,$ $\displaystyle s_{0}(d)$
$\displaystyle:=\mathcal{D}_{d}(2^{d-1}+u_{d+1})+u_{d+1}+\sum_{j=d+1}^{D}u_{j}t_{j}$
$\displaystyle s_{0}$
$\displaystyle:=\max\\{s_{0}(d):d\in\Delta\cup\\{0\\}\\}.$

By combining corollary 2.3 with the result of Frei and Madritsch we obtain:

######  Theorem 2.5.

There exist constants $A(d),B(d)$ such that if for all $d\in\Delta$ we have
$\textnormal{rk}_{K}(\bar{Q}_{d})>A(t_{d}s_{0})^{B}$ then there is a positive
$\delta$ with

$N(P)=\mathfrak{S}\mathfrak{J}\cdot
P^{n(s-\mathcal{D})}+O(P^{n(s-\mathcal{D})-\delta})$

for $P\to\infty$. Here, $\mathfrak{S}$ is the usual singular series and
$\mathfrak{J}$ is the usual singular integral.

###  2.3. Function fields

For function fields, Lee proved a local to global rule for systems of
polynomials with high Birch rank [13]. Suppose
${\mathbf{k}}=\mathbb{F}_{q}(t)$ where $q$ is a prime power and write
$\mathbf{A}=\mathbb{F}_{q}[t].$ Let
$\bar{Q}=(Q_{1},\ldots,Q_{n})\in\mathbf{A}[x_{1},\ldots,x_{s}]$ be forms of
degree $d.$ Let $|\cdot|$ be the absolute value on ${\mathbf{k}}$ given by
$|f/g|=q^{\deg f-\deg g}.$ For $x\in{\mathbf{k}}^{s}$ we define
$\|x\|_{\infty}=\max_{i\in[s]}|x_{i}|.$ For $P>0$ set

$N(P)=\\#\\{x\in\mathbf{A}^{s}:\|x\|_{\infty}<q^{P},Q_{i}(x)=0\ \forall
i\in[n]\\}.$

Corollary 2.3, together with Lee’s result yields:

######  Theorem 2.6.

There exist constants $A(d),B(d))$ such that the following holds: Assume
$\textnormal{char}({\mathbf{k}})>d,$ $\textnormal{rk}(\bar{Q})>An^{B},$ and
that $\\{\bar{Q}=0\\}$ contains a non-singular point in the completion of
${\mathbf{k}}$ at every place. Then we have

$N(P)=\mu q^{P(s-nd)}+O(q^{P(s-nd)-\delta})$

where $\mu,\delta>0.$

##  3\. Reduction to multilinear polynomials and universality

In this section we reduce theorem 1.5 to a statement about multilinear forms.
Let $V$ be a finite dimensional vector space over ${\mathbf{k}}$ and
$Q\in{\mathbf{k}}[V]$ a form of degree $d.$ Suppose either
$\textnormal{char}({\mathbf{k}})=0$ or $\textnormal{char}({\mathbf{k}})>d.$
Then there is an associated multilinear form $\tilde{Q}:V^{d}\to{\mathbf{k}}$
given by
$\tilde{Q}({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d})=\triangle_{{\mathbf{x}}_{1}}\ldots\triangle_{{\mathbf{x}}_{d}}Q$
which satisfies $Q(x)=\frac{1}{d!}\tilde{Q}(x,\ldots,x).$ For multilinear
forms such as this there is a finer notion of rank.

######  Definition 3.1.

Let ${\mathbf{k}}$ be any field, $V_{1},\ldots,V_{d}$ finite dimensional
vector spaces over $k,$ and $P:V_{1}\times\ldots\times V_{d}\to{\mathbf{k}}$ a
multilinear form. We say that $P$ has _partition rank_ $1$ if there exists a
partition $[d]=I\sqcup J$ with $I,J\neq\emptyset$ and multilinear forms
$R:\prod_{i\in I}V_{i}\to k,S:\prod_{j\in J}V_{j}\to k$ such that

$P({\mathbf{x}})=R(({\mathbf{x}}_{i})_{i\in I})\cdot
S(({\mathbf{x}}_{j})_{j\in J}).$

In general the partition rank $\textnormal{prk}_{\mathbf{k}}(P)$ is the
smallest natural number $r$ such that $P=P_{1}+\ldots+P_{r}$ and $P_{i}$ has
partition rank $1$ for all $1\leq i\leq r.$

The next claim states that the partition rank of $\tilde{Q}$ is essentially
equivalent to the rank of $Q.$

######  Claim 3.2.

Let $V$ be a finite dimensional vector space over ${\mathbf{k}}$ and
$Q\in{\mathbf{k}}[V]$ a form of degree $d.$ Suppose either
$\textnormal{char}({\mathbf{k}})=0$ or $\textnormal{char}({\mathbf{k}})>d.$
then

$\textnormal{rk}_{\mathbf{k}}(Q)\leq\textnormal{prk}_{\mathbf{k}}(\tilde{Q})\leq\binom{d}{\lfloor
d/2\rfloor}\textnormal{rk}_{\mathbf{k}}(Q).$

###### Proof.

For the left inequality, suppose
$\tilde{Q}({\mathbf{x}})=\sum_{i=1}^{r}R({\mathbf{x}})\cdot S({\mathbf{x}})$
and plug in

$Q(x)=\frac{1}{d!}\tilde{Q}(x,\ldots,x)=\frac{1}{d!}\sum_{i=1}^{r}R_{i}(x,\ldots,x)\cdot
S_{i}(x,\ldots,x).$

For the right inequality, suppose $Q(x)=\sum_{i=1}^{r}R_{i}(x)S_{i}(x)$ and
take derivatives

$\tilde{Q}({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d})=\sum_{i=1}^{r}\sum_{J\subset[d],|J|=\deg
R_{i}}\tilde{R_{i}}(({\mathbf{x}}_{j})_{j\in
J})\tilde{S_{i}}(({\mathbf{x}}_{\mathbf{k}})_{k\in[d]\setminus J}).$

∎

Partition rank is also defined for collections of multi-linear forms.

######  Definition 3.3.

Let $P_{1},\ldots,P_{n}:V_{1}\times\ldots\times V_{d}\to{\mathbf{k}}$ be
multilinear forms. Their partition rank is

$\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n})=\min_{0\neq
a\in{\mathbf{k}}^{n}}\textnormal{prk}_{\mathbf{k}}\left(\sum_{i=1}^{n}a_{i}\cdot
P_{i}\right).$

By claim 3.2, we can deduce theorem 1.5 from a corresponding theorem for
multi-linear forms.

######  Theorem 3.4 (theorem 1.5 for multi-linear forms).

Let ${\mathbf{k}}$ be an admissible field, $V_{1},\ldots,V_{d}$ finite
dimensional vector spaces over ${\mathbf{k}},$ and
$P_{1},\ldots,P_{n}:V_{1}\times\ldots\times V_{d}\to{\mathbf{k}}$ multi-linear
forms. There exist constants
$\tilde{A}({\mathbf{k}},d),\tilde{B}({\mathbf{k}},d)$ such that if
${\mathbf{k}}$ is a perfect field we have

$\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n})\leq\tilde{A}[n\cdot\textnormal{prk}_{\bar{{\mathbf{k}}}}(P_{1},\ldots,P_{n})+1]^{\tilde{B}},$

and if ${\mathbf{k}}$ is an imperfect field we have

$\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n})\leq\tilde{A}[n^{1+1/d}\cdot(\textnormal{prk}_{\bar{{\mathbf{k}}}}(P_{1},\ldots,P_{n})+1)]^{\tilde{B}}.$

###### Proof of theorem 1.5 assuming theorem 3.4.

We will show that theorem 1.5 holds with $A=\tilde{A}\binom{d}{\lfloor
d/2\rfloor}^{\tilde{B}}$ and $B=\tilde{B}.$ Let ${\mathbf{k}}$ be an
admissible field and let $Q_{1},\ldots,Q_{n}$ be degree $d$ polynomials with
$\textnormal{char}({\mathbf{k}})>d$ or $\textnormal{char}({\mathbf{k}})=0.$
Let $\tilde{Q_{1}},\ldots,\tilde{Q_{n}}$ be the associated multi-linear forms.
If ${\mathbf{k}}$ is perfect then by claim 3.2 and theorem 3.4

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\leq\textnormal{prk}_{\mathbf{k}}(\tilde{Q_{1}},\ldots,\tilde{Q_{n}})\leq\tilde{A}[n\cdot\textnormal{prk}_{\bar{{\mathbf{k}}}}(\tilde{Q_{1}},\ldots,\tilde{Q_{n}})+1]^{\tilde{B}}\\\
\leq\tilde{A}\left[n\cdot\binom{d}{\lfloor
d/2\rfloor}\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})+1\right]^{\tilde{B}}\leq\tilde{A}\binom{d}{\lfloor
d/2\rfloor}^{\tilde{B}}\left[n\cdot\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})+1\right]^{\tilde{B}}.$

If ${\mathbf{k}}$ is imperfect then again by claim 3.2 and theorem 3.4

$\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n})\leq\textnormal{prk}_{\mathbf{k}}(\tilde{Q_{1}},\ldots,\tilde{Q_{n}})\leq\tilde{A}[n^{1+1/d}\cdot(\textnormal{prk}_{\bar{{\mathbf{k}}}}(\tilde{Q_{1}},\ldots,\tilde{Q_{n}})+1)]^{\tilde{B}}\\\
\leq\tilde{A}[n^{1+1/d}\cdot(\binom{d}{\lfloor
d/2\rfloor}\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})+1)]^{\tilde{B}}\leq\tilde{A}\binom{d}{\lfloor
d/2\rfloor}^{\tilde{B}}\left[n^{1+1/d}\cdot(\textnormal{rk}_{\bar{{\mathbf{k}}}}(Q_{1},\ldots,Q_{n})+1)\right]^{\tilde{B}}.$

∎

We prove theorem 3.4 for all of the above fields (except $\mathbb{Q}$) via a
universality result, which may be of independent interest. This generalizes a
universality result of Kazhdan and Ziegler appearing in [12].

######  Theorem 3.5 (Universality).

Let ${\mathbf{k}}$ be an admissible field and
$R_{1},\ldots,R_{n}:({\mathbf{k}}^{t})^{d}\to{\mathbf{k}}$ multilinear forms
of degree $d.$ There exist constants $C({\mathbf{k}},d),D({\mathbf{k}},d)$
such that whenever $P_{1},\ldots,P_{n}:V_{1}\times\ldots\times
V_{d}\to{\mathbf{k}}$ are multilinear forms with
$\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n})>C(nt^{d})^{D},$ there exist
linear maps $T_{i}:{\mathbf{k}}^{t}\to V_{i}$ such that
$P_{j}\circ(T_{1}\times\ldots\times T_{d})=R_{j}.$

We now explain how theorem 3.4 follows from theorem 3.5. Theorem 3.5 will be
proved in the following sections.

###### Proof of theorem 3.4 assuming theorem 3.5.

We will show that theorem 3.4 holds with $\tilde{A}=C,\tilde{B}=dD.$ Supposing

$\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n})>C(n^{1+1/d}\bar{r})^{dD},$

we will show that
$\textnormal{prk}_{\bar{{\mathbf{k}}}}(P_{1},\ldots,P_{n})\geq\bar{r}.$ By
theorem 3.5 with $t=n\bar{r},$ there exist linear maps
$T_{i}:{\mathbf{k}}^{t}\to V_{i}$ such that

$P_{j}\circ
T=\sum_{k=(j-1)\bar{r}+1}^{j\bar{r}}\prod_{i=1}^{d}{\mathbf{x}}_{i}(k)=:D_{j}.$

Clearly
$\textnormal{prk}_{\bar{\mathbf{k}}}(P_{1},\ldots,P_{n})\geq\textnormal{prk}_{\bar{\mathbf{k}}}(D_{1},\ldots,D_{n}).$
To show that the right hand side equals $\bar{r},$ we need the following
simple claim:

######  Claim 3.6.

Let ${\mathbf{k}}$ be a field and $P:V_{1}\times\ldots\times
V_{d}\to{\mathbf{k}}$ a multilinear form. The variety

$Z_{P}:=\\{(v_{2},\ldots,v_{d}):P(\cdot,v_{2},\ldots,v_{d})\equiv 0\\}$

satisfies

$\textnormal{codim}_{V_{2}\times\ldots\times
V_{d}}Z_{P}\leq\textnormal{prk}_{\mathbf{k}}(P).$

###### Proof.

Suppose $P=\sum_{k=1}^{r}R_{k}(({\mathbf{x}}_{i})_{i\in I_{k}})\cdot
S_{k}(({\mathbf{x}}_{j})_{j\in[d]\setminus I_{k}})$ and assume without loss of
generality that $1\notin I_{k}$ for all $k.$ Then the variety
$\\{R_{1}=\ldots=R_{r}=0\\}$ has codimension $\leq r$ and is contained in
$Z_{P}.$ ∎

######  Lemma 3.7.

$\textnormal{prk}_{\bar{\mathbf{k}}}(D_{1},\ldots,D_{n})=\bar{r}.$

###### Proof.

Obviously the partition rank is $\leq\bar{r}.$ On the other hand, let
$D=\sum_{j=1}^{n}a_{j}D_{j}$ be a non-trivial linear combination, and assume
without loss of generality that $a_{1},\ldots,a_{m}\neq 0$ and
$a_{m+1}=\ldots=a_{n}=0$ for some $m\geq 1.$ We can decompose $Z_{D}$
explicitly,

$Z_{D}=\\{\prod_{i=2}^{d}{\mathbf{x}}_{i}(k)=0:\forall
k\in[m\bar{r}]\\}=\bigcup_{\sigma:[m\bar{r}]\to[2,d]}\\{{\mathbf{x}}_{\sigma(j)}(j)=0:\forall
j\in[m\bar{r}]\\}.$

$Z_{D}$ is a finite union of linear subspaces of codimension $m\bar{r},$ so
$\textnormal{codim}_{V_{2}\times\ldots\times V_{d}}Z_{D}=m\bar{r}\geq\bar{r}.$
By the previous claim, we get
$\textnormal{prk}_{\bar{\mathbf{k}}}(D_{1},\ldots,D_{n})\geq\bar{r}.$ ∎

This completes the proof of theorem 3.4 for all fields in the case $n=1$ and
for imperfect fields in the case of general $n.$ To obtain our improved bounds
for perfect fields and general $n$ we use the following argument which we
learned from A. Polishchuk.

###### Proof of theorem 3.4 for perfect fields assuming the case $n=1$.

Let

$\bar{r}=\textnormal{prk}_{\bar{{\mathbf{k}}}}(P_{1},\ldots,P_{n}),\mathcal{P}=\textnormal{sp}_{\bar{{\mathbf{k}}}}(P_{1},\ldots,P_{n})$

Choose $0\neq P\in\mathcal{P}$ with
$\textnormal{rk}_{\bar{{\mathbf{k}}}}(P)=\bar{r}.$ Let
$\mathcal{P}^{\prime}\subset\mathcal{P}$ be the
${\bar{{\mathbf{k}}}}$-subspace spanned by $P$ and its Galois conjugates. By
Galois descent,
$\mathcal{P}^{\prime}=\textnormal{sp}_{\bar{{\mathbf{k}}}}(P^{\prime}_{1},\ldots,P^{\prime}_{m})$
for some polynomials
$P^{\prime}_{1},\ldots,P^{\prime}_{m}\in\textnormal{sp}_{\mathbf{k}}(P_{1},\ldots,P_{n}).$
Let
$\mathcal{P^{\prime}}({\mathbf{k}}):=\textnormal{sp}_{\mathbf{k}}(P^{\prime}_{1},\ldots,P^{\prime}_{m})$
and choose $0\neq P^{\prime}\in\mathcal{P^{\prime}}({\mathbf{k}}).$ Since
$\dim\mathcal{P^{\prime}}\leq n,$ there are Galois conjugates $\sigma_{1}\cdot
P,\ldots,\sigma_{n}\cdot P$ such that
$P^{\prime}\in\textnormal{sp}_{\bar{{\mathbf{k}}}}(\sigma_{1}\cdot
P,\ldots,\sigma_{n}\cdot P).$ By our assumption on $P$ this means
$\textnormal{rk}_{\bar{{\mathbf{k}}}}(P^{\prime})\leq n\bar{r},$ which by
theorem 3.4 implies

$\textnormal{rk}_{\mathbf{k}}(P^{\prime})\leq\tilde{A}(n\bar{r}+1)^{\tilde{B}}.$

∎

∎

We now begin the proof of theorem 3.5, which will be completed in subsequent
sections. Assume without loss of generality that $V_{i}={\mathbf{k}}^{s_{i}}.$
Then linear maps $T_{i}:{\mathbf{k}}^{t}\to V_{i}$ are of the form
$T_{i}(x)=A_{i}\cdot x$ where $A_{i}\in\mathcal{M}_{s_{i}\times
t}({\mathbf{k}}).$ For $j_{1},\ldots,j_{d},l$ the coefficient of
$y^{1}_{j_{1}}\cdot\ldots\cdot y^{d}_{j_{d}}$ in
$P_{l}\circ(T_{1}\times\ldots\times T_{d})$ is given by a multilinear form

$P^{l}_{j_{1},\ldots,j_{d}}:\mathcal{M}_{s_{1}\times
t}({\mathbf{k}})\times\ldots\times\mathcal{M}_{s_{d}\times
t}({\mathbf{k}})\to{\mathbf{k}},$

namely
$P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})=\left(P_{l}\circ(A_{1}\times\ldots\times
A_{d})\right)(e_{j_{1}},\ldots,e_{j_{d}}).$

######  Claim 3.8.

The collection of multi-linear forms thus obtained satisfies

$\textnormal{prk}_{\mathbf{k}}\left((P^{l}_{j_{1},\ldots,j_{d}})_{l,j_{1},\ldots,j_{d}}\right)=\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n}).$

###### Proof.

For a multilinear form
$P=\sum_{k_{1},\ldots,k_{d}}a_{k_{1},\ldots,k_{d}}\prod_{i\in[d]}{\mathbf{x}}_{i}(k_{i})$
and indices $j_{1},\ldots,j_{d}$

$P_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})=\sum_{k_{1},\ldots,k_{d}}a_{k_{1},\ldots,k_{d}}\prod_{i\in[d]}A_{i}(k_{i},j_{i}).$

Therefore, $P_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})$ is just $P$ with
relabeled variables, meaning they have the same rank. So for any
$c_{1},\ldots,c_{n}$ and fixed $j_{1},\ldots,j_{d}$ we have

$\textnormal{prk}_{\mathbf{k}}(\sum_{l}c_{l}P^{l}_{j_{1},\ldots,j_{d}})=\textnormal{prk}_{\mathbf{k}}(\sum_{l}c_{l}P_{l}).$

Now let
$P^{\prime}=\sum_{l,j_{1},\ldots,j_{d}}c^{l}_{j_{1},\ldots,j_{d}}P^{l}_{j_{1},\ldots,j_{d}}$
be a non-trivial linear combination with, say, $c^{l}_{1,\ldots,1}\neq 0$ for
some $l.$ Restricting to the subspace

$U=\\{A_{i}(k,j)=0:\forall i,k\ \forall j\neq 1\\}$

gives
$P^{\prime}\restriction_{U}=\sum_{l}c^{l}_{1,\ldots,1}P^{l}_{1,\ldots,1}$ so

$\textnormal{prk}_{\mathbf{k}}(P^{\prime})\geq\textnormal{prk}_{\mathbf{k}}(\sum_{l}c^{l}_{1,\ldots,1}P^{l}_{1,\ldots,1})=\textnormal{prk}_{\mathbf{k}}(\sum_{l}c^{l}_{1,\ldots,1}P_{l})\geq\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n}).$

∎

The desired equalities $P_{l}\circ(A_{1}\times\ldots\times A_{d})=R_{l}$ are
in fact $nt^{d}$ equations of the form

(1)
$P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})=r^{l}_{j_{1},\ldots,j_{d}}$

and we have just proved that the collection of multi-linear forms appearing
has partition rank $\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n}).$ Our
goal in the next sections is to show that such a system of equations has a
${\mathbf{k}}$-solution.

##  4\. Proof of results for finite fields and for $\mathbb{Q}$

###  4.1. Proof for finite fields:

We start by defining bias.

######  Definition 4.1.

Let $V$ be a finite dimensional vector space over a finite field
${\mathbf{k}}$ and $f:V\to{\mathbf{k}}$ some function. The bias of $f$ is

$\textnormal{bias}(f)=\left|{\mathbb{E}}_{x\in V}\chi(f(x))\right|$

where $\chi:k\to{\mathbb{C}}$ is some non-trivial character, and
${\mathbb{E}}_{x\in X}$ denotes $\frac{1}{|X|}\sum_{x\in X}$ for a finite set
$X.$

The bias of $f$ depends also on the choice of character $\chi$ but this won’t
matter in what follows. The key to proving theorem 3.5 is the following
theorem relating rank to bias for multilinear polynomials. The first part is
due to Milicevic [15], with a similar result proved independently by Janzer
[8], and the second to Cohen-Moshkovitz [3].

######  Theorem 4.2.

Let ${\mathbf{k}}$ be a finite field and $P:V_{1}\times\ldots\times
V_{d}\to{\mathbf{k}}$ a multilinear form. Assume either of the following
conditions holds for some $r\geq 1$:

  1. (1)

$\textnormal{prk}_{\mathbf{k}}(P)\geq\alpha r^{\beta},$ for constants
$\alpha(d)=2^{d^{2^{O(d^{2})}}},\beta(d)=2^{2^{O(d^{2})}}.$

  2. (2)

$|{\mathbf{k}}|\geq F(r,d)$ and $\textnormal{prk}_{\mathbf{k}}(P)\geq
2^{d-1}r.$

Then $\textnormal{bias}(P)\leq|{\mathbf{k}}|^{-r}.$

The proof of theorem 3.5 for finite fields, and hence of theorems 3.4 and 1.5,
now follows by a routine Fourier-analytic argument.

###### Proof.

We will show that theorem 3.5 holds for finite fields with
$C=2^{d^{2^{O(d^{2})}}},D=2^{2^{O(d^{2})}}$ or with $C=2^{d-1},D=1$ under the
additional assumption that $|{\mathbf{k}}|\geq F(nt^{d},d).$ Recall that we
want to show the existence of a ${\mathbf{k}}$-solution to the system of
$nt^{d}$ multilinear equations (1)

$P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})=r^{l}_{j_{1},\ldots,j_{d}}.$

We showed in claim 3.8 that

$\textnormal{prk}_{\mathbf{k}}\left(P^{l}_{j_{1},\ldots,j_{d}}\right)_{l,j_{1},\ldots,j_{d}}=\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n}).$

Applying theorem $\ref{bias-rank},$ we get that any nontrivial linear
combination satisfies

$\textnormal{bias}\left(\sum_{l,j_{1},\ldots,j_{d}}\alpha^{l}_{j_{1},\ldots,j_{d}}P^{l}_{j_{1},\ldots,j_{d}}\right)\leq|{\mathbf{k}}|^{-nt^{d}}.$

We can therefore estimate the number of solutions

$\displaystyle{\mathbb{E}}_{A_{1},\ldots,A_{d}}\prod_{l,j_{1},\ldots,j_{d}}1_{P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})=r^{l}_{j_{1},\ldots,j_{d}}}=$
$\displaystyle{\mathbb{E}}_{\alpha^{l}_{j_{1},\ldots,j_{d}}}{\mathbb{E}}_{A_{1},\ldots,A_{d}}\chi(\sum_{l,j_{1},\ldots,j_{d}}\alpha^{l}_{j_{1},\ldots,j_{d}}(P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})-r^{l}_{j_{1},\ldots,j_{d}}))=$
$\displaystyle|{\mathbf{k}}|^{-nt^{d}}\left(1+\sum_{\alpha^{l}_{j_{1},\ldots,j_{d}}\neq
0}{\mathbb{E}}_{A_{1},\ldots,A_{d}}\chi(\sum_{l,j_{1},\ldots,j_{d}}\alpha^{l}_{j_{1},\ldots,j_{d}}(P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})-r^{l}_{j_{1},\ldots,j_{d}}))\right)\geq$
$\displaystyle|{\mathbf{k}}|^{-nt^{d}}\left(1-\sum_{\alpha^{l}_{j_{1},\ldots,j_{d}}\neq
0}\textnormal{bias}(\sum_{l,j_{1},\ldots,j_{d}}\alpha^{l}_{j_{1},\ldots,j_{d}}P^{l}_{j_{1},\ldots,j_{d}})\right)>0.$

As we saw in the previous section, this proves theorem 3.4 with the constants

  1. (1)

$\tilde{A}=C=2^{d^{2^{O(d^{2})}}},\ \tilde{B}=dD=2^{2^{O(d^{2})}}.$

  2. (2)

$\tilde{A}=C=2^{d-1},\ \tilde{B}=dD=d$ if $|{\mathbf{k}}|\geq
F(\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n}),n,d),$

and theorem 1.5 with the constants

  1. (1)

$A=\tilde{A}\binom{d}{\lfloor d/2\rfloor}^{\tilde{B}}=2^{2^{O(d^{2})}},\
B=\tilde{B}=2^{2^{O(d^{2})}}.$

  2. (2)

$A=\tilde{A}\binom{d}{\lfloor d/2\rfloor}^{\tilde{B}}=2^{d-1}\binom{d}{\lfloor
d/2\rfloor}^{d},\ B=\tilde{B}=d$ if $|{\mathbf{k}}|\geq
F(\textnormal{rk}_{\mathbf{k}}(Q_{1},\ldots,Q_{n}),n,d).$

∎

###  4.2. Proof for $\mathbb{Q}$:

Now we turn to proving theorem 3.4 for the rationals. We prove it in the case
$n=1.$ The case of general $n$ follows by the argument given for perfect
fields in the previous section. Let
$P:\mathbb{Q}^{s_{1}}\times\ldots\times\mathbb{Q}^{s_{d}}\to\mathbb{Q}$ be a
multi-linear form, and write $s=s_{1}+\ldots+s_{d}.$ Assume without loss of
generality that $P$ has integer coefficients. We will show that for all but
finitely many primes $p,$ the reduction $P\mod p$ has partition rank which is
bounded from below in terms of $\textnormal{prk}_{\mathbb{Q}}(P).$

######  Proposition 4.3.

Let $P:\mathbb{Q}^{s_{1}}\times\ldots\times\mathbb{Q}^{s_{d}}\to\mathbb{Q}$ be
a multi-linear form with integer coefficients. Then for all sufficiently large
primes $p$ we have

$\textnormal{prk}_{\mathbb{Q}}(P)\leq
2^{d-1}d\cdot\textnormal{prk}_{\mathbb{F}_{p}}(P\mod p).$

We now complete the proof of theorem 3.4 and theorem 1.5 for $\mathbb{Q}$
assuming proposition 4.3.

###### Proof.

Suppose $\textnormal{prk}_{\mathbb{Q}}(P)\geq 4^{d-1}d\cdot\bar{r}^{d}.$ By
proposition 4.3 we get that for sufficiently large primes $p,$ we have
$\textnormal{prk}_{\mathbb{F}_{p}}(P\mod p)\geq 2^{d-1}\cdot\bar{r}^{d}.$ By
theorem 3.4 for sufficiently large finite fields, we conclude that for $p$
sufficiently large we have $\textnormal{prk}_{\bar{\mathbb{F}_{p}}}(P\mod
p)\geq\bar{r}.$ By a model-theoretic result (corollary 2.2.10 in [14]), we
deduce that $\textnormal{prk}_{\bar{\mathbb{Q}}}(P)\geq\bar{r}.$ This
completes the proof of theorem 3.4 with

$\tilde{A}=4^{d-1}d,\tilde{B}=d.$

By the argument given in the last section this proves theorem 1.5 with

$A=\tilde{A}\binom{d}{\lfloor
d/2\rfloor}^{\tilde{B}}=4^{d-1}d\binom{d}{\lfloor d/2\rfloor}^{d},\
B=\tilde{B}=d.$

∎

In order to prove proposition 4.3, we will need a simple scaling result for
solutions to systems of multilinear equations.

######  Lemma 4.4.

Let
$\bar{Q}=Q_{1},\ldots,Q_{n}:{\mathbb{Z}}^{s_{1}}\times\ldots\times{\mathbb{Z}}^{s_{d}}\to
G$ be a collection of multilinear maps where $G$ is some abelian group. For a
positive integer $R$ let

$\displaystyle N_{R}(\bar{Q})$
$\displaystyle=|\\{x\in{\mathbb{Z}}^{s}:Q_{i}(x)=0\ \forall i\in[n],0\leq
x_{j}<R\ \forall j\in[s]\\}|$ $\displaystyle N^{\prime}_{R}(\bar{Q})$
$\displaystyle=|\\{x\in{\mathbb{Z}}^{s}:Q_{i}(x)=0\ \forall
i\in[n],-R<x_{j}<R\ \forall j\in[s]\\}|.$

Then for any positive integer $L$ we have

$N_{LR}(\bar{Q})\leq L^{s}N^{\prime}_{R}(\bar{Q}),$

where $s=s_{1}+\ldots+s_{d}.$

###### Proof.

By induction on $d.$ For the base case $d=1,$ first note that for any
$c_{i}\in G$ the number of solutions to $Q_{i}(x)=c_{i}$ in $[0,R)^{s}$ is
bounded above by $N^{\prime}_{R}(\bar{Q}).$ This follows by substracting a
single solution from every one of the others. Writing $\bar{R}=(R,\ldots,R),$
we have

$\displaystyle N_{LR}(\bar{Q})$ $\displaystyle=\sum_{0\leq
x_{i}<LR}1_{Q_{i}(x)=0}=\sum_{\omega\in\\{0,1,\ldots,L-1\\}^{s}}\sum_{0\leq
x_{i}<R}1_{Q_{i}(x+\omega\cdot\bar{R})=0}$
$\displaystyle=\sum_{\omega\in\\{0,1,\ldots,L-1\\}^{s}}\sum_{0\leq
x_{i}<R}1_{Q_{i}(x)=-Q_{i}(\omega\cdot\bar{R})}\leq\sum_{\omega\in\\{0,1,\ldots,L-1\\}^{s}}N^{\prime}_{R}(\bar{Q})=L^{s}N^{\prime}_{R}(\bar{Q}).$

For general $d,$ we have

$\displaystyle N_{LR}(\bar{Q})$
$\displaystyle=\sum_{0\leq{\mathbf{x}}_{1}(i)<LR}N_{LR}(\bar{Q}({\mathbf{x}}_{1},\cdot))\leq\sum_{0\leq{\mathbf{x}}_{1}(i)<LR}L^{s_{2}+\ldots+s_{d}}N^{\prime}_{R}(\bar{Q}({\mathbf{x}}_{1},\cdot))$
$\displaystyle=L^{s_{2}+\ldots+s_{d}}\sum_{\|{\mathbf{x}}_{2}\|_{\infty},\cdots,\|{\mathbf{x}}_{d}\|_{\infty}<R}N_{LR}(\bar{Q}(\cdot,{\mathbf{x}}_{2},\cdots,{\mathbf{x}}_{d}))$
$\displaystyle\leq
L^{s}\sum_{\|{\mathbf{x}}_{2}\|_{\infty},\cdots,\|{\mathbf{x}}_{d}\|_{\infty}<R}N^{\prime}_{R}(\bar{Q}(\cdot,{\mathbf{x}}_{2},\cdots,{\mathbf{x}}_{d}))=L^{s}N^{\prime}_{R}(\bar{Q}),$

where the first inequality follows from the inductive hypothesis and the
second one follows from the base case $d=1.$ ∎

###### Proof of proposition 4.3.

Let $s=s_{1}+\ldots+s_{d-1}.$ Suppose $p$ is a prime and
$\textnormal{prk}_{\mathbb{F}_{p}}(P\mod p)=r.$ By the inequality
$\textnormal{bias}(P\mod p)\geq p^{-r},$ proved in [11], we have

$|\\{{\mathbf{x}}\in{\mathbb{Z}}^{s}:P({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},\cdot)=0\mod
p\\}|\geq p^{s-r}.$

Given $0<\eta<1,$ we can apply lemma 4.4 to the collection of multilinear maps

$\bar{Q}=Q_{1},\ldots,Q_{s_{d}}:{\mathbb{Z}}^{s_{1}}\times\ldots\times{\mathbb{Z}}^{s_{d-1}}\to\mathbb{F}_{p},$

where

$Q_{i}({\mathbf{x}})=P({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},e_{i})\mod
p,$

to get

$N_{p}(\bar{Q})\leq\lceil p^{1-\eta}\rceil^{s}N^{\prime}_{\lceil
p^{\eta}\rceil}(\bar{Q})\ll_{s}p^{s(1-\eta)}N^{\prime}_{\lceil
p^{\eta}\rceil}(\bar{Q}),$

so

$N^{\prime}_{\lceil
p^{\eta}\rceil}(\bar{Q})\gg_{s}p^{\eta(s-r/\eta)}\gg_{s,\eta,r}\lceil
p^{\eta}\rceil^{s-r/\eta}.$

For $\eta=\frac{1}{d}$ and $\|{\mathbf{x}}\|_{\infty}<\lceil p^{\eta}\rceil$
we have $|P({\mathbf{x}},e_{i})|\ll p^{\frac{d-1}{d}}$ so for sufficiently
large $p,$

$P({\mathbf{x}},e_{i})=0\mod p\implies P({\mathbf{x}},e_{i})=0.$

Therefore,

$|\\{{\mathbf{x}}\in{\mathbb{Z}}^{s}:P({\mathbf{x}},\cdot)=0,\|{\mathbf{x}}\|_{\infty}<\lceil
p^{1/d}\rceil\\}|\gg_{s,r}\lceil p^{1/d}\rceil^{s-dr}.$

By corollary 5.7, which will be proved in the next section, this implies
$\textnormal{prk}_{\mathbb{Q}}(P)\leq 2^{d-1}dr.$

∎

##  5\. Proof for number fields and finite separable extensions of
$\mathbb{F}_{q}(t)$

The obstruction to the solution to a system of equations given by multi-linear
forms is the Birch singular locus.

######  Definition 5.1.

Let $P:V_{1}\times\ldots\times V_{d}\to{\mathbf{k}}$ be a multilinear form of
degree $d>1.$ The Birch singular locus is the subvariety

$V_{1}\times\ldots\times V_{d-1}\supset
Z_{P}=\\{({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1}):P({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},\cdot)=0\\}.$

For a collection $\mathcal{P}=(P_{1},\ldots,P_{n})$ the Birch singular locus
is

$Z_{\mathcal{P}}=\\{({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1}):P_{1}({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},\cdot),\ldots,P_{n}({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},\cdot)\textnormal{
are linearly dependent}\\}.$

Kazhdan, Polishchuk and the first author [9] extended a result of Schmidt [16]
and showed a relationship between the Birch singular locus and the Schmidt
rank for a multilinear form $P$ over a general field ${\mathbf{k}}.$

######  Theorem 5.2.

Let $g_{\mathbf{k}}(P)$ denote the codimension in $V_{1}\times\ldots\times
V_{d-1}$ of the Zariski closure of $Z_{P}({\mathbf{k}}).$ Then
$\textnormal{prk}_{\mathbf{k}}(P)\leq 2^{d-1}\cdot g_{\mathbf{k}}(P).$

By adapting an argument that Schmidt gave for $\mathbb{Q}$ in [16], this
extends to collections of multilinear forms over some fields.

######  Proposition 5.3.

Let ${\mathbf{k}}$ be a number field or a finite separable extension of
$\mathbb{F}_{q}(t),$ and

$\mathcal{P}=(P_{1},\ldots,P_{n}):{\mathbf{k}}^{s_{1}}\times\ldots\times{\mathbf{k}}^{s_{d}}\to{\mathbf{k}}$

a collection of multilinear forms. Suppose for some
$c_{1},\ldots,c_{n}\in{\mathbf{k}}$ the system of equations $(P_{i}=c_{i})$
has no solution in
${\mathbf{k}}^{s_{1}}\times\ldots\times{\mathbf{k}}^{s_{d}}.$ Then

$\textnormal{prk}_{\mathbf{k}}(\mathcal{P})\leq 2^{d-1}(d-1)n^{2}.$

To prove proposition 5.3 we need an effective version of Noether
normalization.

######  Lemma 5.4 (Noether normalization).

Let ${\mathbf{k}}$ be an algebraically closed field and let
$V\subset{\mathbf{k}}^{s}$ be an algebraic set of codimension $t$ defined by
$m$ homogenenous equations of degree $\leq n.$ Look at matrices
$M\in\mathcal{M}_{(s-t)\times s}({\mathbf{k}}).$ There exists a constant
$C(s,m,n)$ (independent of $V$!) such that for any set $U\subset{\mathbf{k}}$
of size $|U|\geq C$ we can find a matrix $M$ with entries in $U$ for which the
map $M:V\to{\mathbf{k}}^{s-t}$ has finite fibers of size bounded by $C.$

###### Proof of lemma.

Just follow the proof of linear Noether normalization, for example in [6].
Note that we need to choose the entries of $M$ to avoid solutions of certain
polynomial equations of degree $\leq n.$ ∎

This lemma gives us a useful corollary when we have a pseudo-norm at our
disposal. It will be convenient to use a slightly non-standard definition.

######  Definition 5.5.

Let $\mathbf{A}$ be a ring. We call a function
$\phi:\mathbf{A}\to{\mathbb{Z}}^{\geq 0}$ a _pseudo-norm_ if it satisfies the
following conditions:

  1. (1)

$\phi(x+y)\leq\phi(x)+\phi(y).$

  2. (2)

$\phi(xy)\ll\phi(x)\phi(y)$ where the implied constant is independent of
$x,y.$

Given $(\mathbf{A},\phi)$ as above and $R>0$ we write
$B_{R}(\mathbf{A}):=\\{a\in\mathbf{A}:\phi(a)\leq R\\}$ for the ball of radius
$R.$ We say $(\mathbf{A},\phi)$ has _linear growth_ if for all $0<R<\infty$
the set $B_{R}(\mathbf{A})$ is finite and satisfies
$|B_{CR}(\mathbf{A})|\ll_{C}|B_{R}(\mathbf{A})|$ for any constant $C.$

######  Lemma 5.6 (Point counting and dimension).

Let $\mathbf{A}$ be an infinite integral domain equipped with a pseudo-norm
$\phi$ such that $(\mathbf{A},\phi)$ has linear growth. Let ${\mathbf{k}}$ be
the fraction field of $\mathbf{A}$ and $V\subset\bar{{\mathbf{k}}}^{s}$ an
algebraic set defined over $\mathbf{A}$ by $m$ homogeneous equations of degree
$n.$ For $R>0$ set

$N_{R}=|V\cap(B_{R}(\mathbf{A}))^{s}|.$

Given $D>0,$ there exists $R_{0}(s,m,n,D)$ (but independent of $V$!) such that
if for some $R>R_{0}$ we have $N_{R}\geq D|B_{R}(\mathbf{A})|^{s-t}$ then
$\textnormal{codim}(V)\leq t.$

###### Proof of lemma 5.6.

Suppose, to get a contradiction, that $\textnormal{codim}(V)=t^{\prime}>t.$ By
infinitude of $\mathbf{A},$ we can apply lemma 5.4 to get a constant
$C=C(s,m,n)$ and a matrix $M\in\mathcal{M}_{(s-t^{\prime})\times
s}(\mathbf{A})$ with entries in $B_{C}(\mathbf{A})$ such that
$M:V\to\bar{{\mathbf{k}}}^{s-t^{\prime}}$ has fibers of size $\leq C.$ This
allows us to bound $N_{R}$

$N_{R}\leq C\cdot|M((B_{R}(\mathbf{A}))^{s})|\leq
C\cdot|B_{C^{\prime}R}(\mathbf{A})|^{s-t^{\prime}}\ll_{s,m,n}|B_{R}(\mathbf{A})|^{s-t-1},$

where $C^{\prime}=C^{\prime}(s,m,n).$ The second and third inequalities use
the fact that $\phi$ is a pseudo-norm and that $(\mathbf{A},\phi)$ has linear
growth, respectively. If $N_{R}\geq D|B_{R}(\mathbf{A})|^{s-t}$ then we
combine with the above inequality to get

$D\leq D^{\prime}|B_{R}(\mathbf{A})|^{-1}$

where $D^{\prime}=D^{\prime}(s,m,n)>0.$ By infinitude of $\mathbf{A},$ this is
impossible for$R\geq R_{0}(s,m,n,D)$, so we must have
$\textnormal{codim}(V)\leq t.$ ∎

######  Corollary 5.7.

Let $\mathbf{A}$ be an infinite integral domain equipped with a pseudo-norm
$\phi$ such that $(\mathbf{A},\phi)$ has linear growth. Let ${\mathbf{k}}$ be
the fraction field of $\mathbf{A},$
$P:{\mathbf{k}}^{s_{1}}\times\ldots\times{\mathbf{k}}^{s_{d}}\to{\mathbf{k}}$
a multilinear form and $D$ a positive constant. Write
$s=s_{1}+\ldots+s_{d-1}.$ Then there exists $R_{0}(s,d,D)$ independent of $P$
such that if we have

$|Z_{P}({\mathbf{k}})\cap(B_{R}(\mathbf{A}))^{s}|\geq D|B_{R}(A)|^{s-t}$

for some $R\geq R_{0}$ then $\textnormal{prk}_{\mathbf{k}}(P)\leq 2^{d-1}\cdot
t.$

###### Proof.

Combine lemma 5.6 and theorem 5.2. ∎

Corollary 5.7 is the result that we will actually use in our proof of
proposition 5.3, the other lemmas were preliminary.

###### Proof of proposition 5.3.

Let $s=s_{1}+\ldots+s_{d-1}.$ First we claim that our hypothesis implies
$Z_{\mathcal{P}}({\mathbf{k}}^{s})={\mathbf{k}}^{s}.$ Indeed, suppose this
isn’t the case. Then there is some ${\mathbf{x}}\in{\mathbf{k}}^{s}\setminus
Z_{\mathcal{P}}({\mathbf{k}}^{s}),$ meaning $P_{i}({\mathbf{x}},\cdot)$ are
linearly independent and $P_{i}({\mathbf{x}},\cdot)=c_{i}$ has a solution in
${\mathbf{k}}^{s_{d}}$. The proof of the proposition will be similar for both
types of fields. Number fields: Let $\mathbf{A}$ be the ring of integers of
${\mathbf{k}}.$ Recalling that $\mathbf{A}\cong{\mathbb{Z}}^{m}$ as
${\mathbb{Z}}$-modules, we define a pseudo-norm on $\mathbf{A}$ with linear
growth by

$\phi(n_{1},\ldots,n_{m})=\max_{i}|n_{i}|.$

Assume without loss of generality that the collection $\mathcal{P}$ has
coefficients in $\mathbf{A}.$ Now, suppose ${\mathbf{x}}\in
Z_{\mathcal{P}}({\mathbf{k}})\cap B_{R}(\mathbf{A})^{s}.$ Then the matrix
$M=(m_{i,j}),$ where
$m_{i,j}=P_{j}({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},e_{i}),$ is of rank
$\leq n-1$ and has $\phi(m_{i,j})\ll R^{d-1}$ for all $i,j.$ By a standard
linear algebra argument, we obtain a small vector in the kernel of $M.$

######  Claim 5.8.

Suppose $(\mathbf{A},\phi)$ is an integral domain equipped with a pseudo-norm.
Suppose $M\in M_{m\times n}(\mathbf{A})$ has $\textnormal{rk}(M)\leq n-1$ and
$\phi(m_{i,j})\leq T$ for all $i,j.$ Then there exists $0\neq
a\in\mathbf{A}^{n}$ such that $Ma=0$ and $\phi(a_{i})\ll_{n}T^{n-1}$ for all
$i.$

###### Proof.

Suppose without loss of generality that the first $j\leq n-1$ rows of $M$ are
a ${\mathbf{k}}$-basis for $\textnormal{row}(M).$ We obtain a new matrix by
deleting the other rows and then adding $n-j$ rows of standard basis vectors
to obtain an invertible $n\times n$ matrix which we call $M^{\prime}.$ Let
$D=\det M^{\prime}.$ Then Cramer’s rule gives a solution $0\neq
a\in\mathbf{A}^{n}$ to the equation $M^{\prime}a=(0,\ldots,0,D)$ which
satisfies $\phi(a_{i})\ll_{n}T^{n-1}$ for all $i.$ ∎

Therefore, there is a nontrivial solution $a\in\mathbf{A}^{n}$ to the equation
$Ma=0$ which satisfies $\phi(a_{i})\ll R^{(d-1)(n-1)}$ for all $i.$ The number
of possibilities for such an $a$ is $\ll R^{mn(d-1)(n-1)}.$ Hence there must
be some nontrivial linear combination $P=a_{1}P_{1}+\ldots+a_{n}P_{n}$ with

$Z_{P}({\mathbf{k}})\cap(B_{R}(\mathbf{A}))^{s}\gg R^{m[s-(d-1)n^{2}]}.$

If $R$ is sufficiently large, we can apply corollary 5.7 to obtain

$\textnormal{prk}_{{\mathbf{k}}}(P)\leq 2^{d-1}(d-1)n^{2}.$

Finite separable extensions of $\mathbb{F}_{q}(t)$: Let $\mathbf{A}$ be the
integral closure of $\mathbb{F}_{q}[t]$ in ${\mathbf{k}}.$ Then
$\mathbf{A}\cong\mathbb{F}_{q}[t]^{m}$ as a $\mathbb{F}_{q}[t]$-module, with
the same proof as for rings of integers in number fields. Define a pseudo-norm
on $\mathbb{F}_{q}[t]$ with linear growth by $\phi(f)=q^{\deg(f)}$ and extend
it to $\mathbf{A}$ by $\phi(f_{1},\ldots,f_{m})=\max_{i}\phi(f_{i}).$ Assume
without loss of generality that $\mathcal{P}$ has coefficients in
$\mathbf{A}.$ From here the proof proceeds in a similar manner: If $x\in
Z_{\mathcal{P}}({\mathbf{k}})\cap B_{q^{R}}(\mathbf{A})^{s}$ then the matrix
$M=(m_{i,j}),$ where
$m_{i,j}=P_{j}({\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{d-1},e_{i}),$ is of rank
$\leq n-1$ and has $\phi(m_{i,j})\ll q^{R(d-1)}$ for all $i,j.$ By claim 5.8
there is a nontrivial solution $a\in\mathbf{A}^{n}$ to the equation $Ma=0$
which satisfies $\phi(a_{i})\ll q^{R(d-1)(n-1)}$ for all $i.$ The number of
possibilities for such an $a$ is $\ll q^{mnR(d-1)(n-1)}.$ Hence there must be
some nontrivial linear combination $P=a_{1}P_{1}+\ldots+a_{n}P_{n}$ with

$Z_{P}({\mathbf{k}})\cap(B_{q^{R}}(\mathbf{A}))^{s}\gg q^{mR[s-(d-1)n^{2}]}.$

If $R$ is sufficiently large, we can apply corollary 5.7 to obtain

$\textnormal{prk}_{{\mathbf{k}}}(P)\leq 2^{d-1}(d-1)n^{2}.$

∎

Now we can finish the proof of theorem 3.5, and hence of theorems 3.4 and 1.5.

###### Proof.

We will show that theorem 3.5 holds for number fields and finite separable
extensions of $\mathbb{F}_{q}(t)$ with constants $C=2^{d-1}(d-1),D=2.$ Suppose
$P_{1},\ldots,P_{n}$ satisfy
$\textnormal{prk}_{\mathbf{k}}(P_{1},\ldots,P_{n})>2^{d-1}(d-1)(nt^{d})^{2}.$
Recall that we proved that this implies
$\textnormal{prk}_{\mathbf{k}}(P^{l}_{j_{1},\ldots,j_{d}})>2^{d-1}(d-1)(nt^{d})^{2}.$
By proposition 5.3 the system of equations

$P^{l}_{j_{1},\ldots,j_{d}}(A_{1},\ldots,A_{d})=r^{l}_{j_{1},\ldots,j_{d}}$

from the end of section $3$ has a ${\mathbf{k}}$-solution as desired. Recall
that for the constants $\tilde{A},\tilde{B}$ of theorem 3.4 we can take

$\tilde{A}=C=2^{d-1}(d-1),\ \tilde{B}=dD=2d,$

and for the constants $A,B$ of theorem 1.5 we can take

$A=\tilde{A}\binom{d}{\lfloor
d/2\rfloor}^{\tilde{B}}=2^{d-1}(d-1)\binom{d}{\lfloor d/2\rfloor}^{2d},\
B=\tilde{B}=2d.$

∎

## References

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# Asymmetric Polynomial Loss For Multi-Label Classification

###### Abstract

Various tasks are reformulated as multi-label classification problems, in
which the binary cross-entropy (BCE) loss is frequently utilized for
optimizing well-designed models. However, the vanilla BCE loss cannot be
tailored for diverse tasks, resulting in a suboptimal performance for
different models. Besides, the imbalance between redundant negative samples
and rare positive samples could degrade the model performance. In this paper,
we propose an effective Asymmetric Polynomial Loss (APL) to mitigate the above
issues. Specifically, we first perform Taylor expansion on BCE loss. Then we
ameliorate the coefficients of polynomial functions. We further employ the
asymmetric focusing mechanism to decouple the gradient contribution from the
negative and positive samples. Moreover, we validate that the polynomial
coefficients can recalibrate the asymmetric focusing hyperparameters.
Experiments on relation extraction, text classification, and image
classification show that our APL loss can consistently improve performance
without extra training burden.111We release our code at
https://github.com/LUMIA-Group/APL.

Index Terms—  Multi-Label Classification, Taylor Expansion, Asymmetric
Focusing, Binary Cross-Entropy Loss

##  1 Introduction

Many tasks, including text classification [1] and image classification [2, 3],
can be formulated into multi-label classification problems, and BCE loss is
often used as the training objective. Specifically, the multi-label
classification problem is reduced to a series of independent binary
classification subproblems, and in each of them the negative log-likelihood
loss is optimized.

However, since recent researches show that BCE loss could be a suboptimal
solution for different tasks [4, 5, 6, 7, 8, 9], we aim to design a simply
tuned loss function with a larger design space that can be tailored for
various tasks. Specifically, we focus on two aspects. Firstly, for each
independent binary classification subproblem, the negative log-likelihood loss
is suboptimal in terms of the polynomial coefficients of the logarithm’s
Taylor expansion [10]. This result is also consistent with the optimization of
each independent label in the multi-label classification problem. Secondly,
the imbalance between the number of rare positive labels and redundant
negative labels is an obstacle to multi-label classification [11], and
consequently, BCE loss is a suboptimal solution for learning the features of
positive samples [6]. Because the massive negative samples contribute more
gradient weights than the fewer positve samples, neural networks may focus on
learning features from negative samples and pay less attention to positive
samples. This will lead to a low confidence in the prediction of positive
samples. The models would under-emphasize the gradient of positive samples and
get suboptimal solutions, which could degrade the model performance.

In this paper, we propose an Asymmetric Polynomial Loss (APL) to counteract
these deficiencies and improve the performance of multi-label classification.
As stated before, the coefficients of the leading polynomials are suboptimal,
and we propose to adjust them for different tasks and datasets. While a
prominent question is the number of classes may be rather large, it is
impracticable to adjust the coefficient for each class separately. Therefore,
we propose to collectively tune the coefficients of leading polynomial bases
for all classes, which makes the parameter tuning feasible.

To mitigate the positive-negative imbalance issue, we further exploit the
asymmetric focusing mechanism [6] proposed in the ASL loss to enhance the
gradient contribution from rare positive samples, which aims to improve the
feature learning. Specifically, we set different focusing parameters to the
Taylor expansion of the BCE loss. Different focusing parameters enable our
loss to decouple the gradient contribution, and we could emphasize the
positive samples. To further reduce the interference of easy negative samples,
we discard them by excluding them from the calculation. Furthermore, we have
validated theoretically and experimentally that the polynomial coefficients of
Taylor expansion maintain the ability to recalibrate the parameters of the
asymmetric focusing mechanism.

In summary, our contributions include:

(1) We propose an asymmetric polynomial loss APL from the perspective of
Taylor expansion, which could be tailored for various tasks. (2) We analyze
the APL properties and elaborate on parameter correlations between polynomial
coefficients and asymmetric focusing. (3) We conduct experiments on four
datasets, and APL achieves superior performance, which includes state-of-the-
art results on MS-COCO and NUS-WIDE.

##  2 Methodology

###  2.1 Taylor Expansion for BCE

The BCE loss could be decomposed into $C$ independent binary classification
subproblems $L_{\rm BCE}=\sum_{i=0}^{C}(y_{i}L_{+}+(1-y_{i})L_{-})/C$,
$y_{i}\in\\{1,0\\}$, where $L_{+}=-\log(p_{i})$ is for the positive class and
$L_{-}=-\log(1-p_{i})$ is for the negative class. $p_{i}$ is the prediction
probability after the sigmoid function.

Since we aim to optimize the subproblems of BCE and adjust the polynomial
coefficients following [10], we firstly apply Taylor series expansion [12] on
$L_{\rm BCE}$. For positive classes where $y_{i}=1$, we set the expansion
point to be $1$, and for negative classes where $y_{i}=0$, we set the
expansion point to be $0$. Thus, Taylor series for BCE loss is $L_{\rm
t-BCE}=\sum_{i=0}^{C}[y_{i}\sum_{m=1}^{\infty}\alpha_{i,m}(1-p_{i})^{m}+(1-y_{i})\sum_{n=1}^{\infty}\beta_{i,n}p_{i}^{n}]/C,$
where $\alpha_{i,m}=\frac{1}{m}$, $\beta_{i,n}=\frac{1}{n}$,
$m,n\in\mathbb{N}_{+}$ are polynomial coefficients. We would amend the leading
polynomial coefficients such as $\alpha_{i,1},\alpha_{i,2},\beta_{i,1}$
because they contribute a higher proportion of gradients in the later training
phase.

###  2.2 Asymmetric Polynomial Loss

To mitigate the positive-negative imbalance problem, we scale the loss of
positive and negative classes to prevent massive negative samples from
overwhelming the training process. Specifically, to differentiate the positive
and negative classes, we propose to leverage the asymmetric focusing mechanism
that employs distinct scaling factors $\gamma^{+/-}$ to $L_{+/-}$. In this
way, we can adjust the order of starting polynomials of $L_{+}$ and $L_{-}$,
which enables us to manually change the gradient proportions of the positive
and negative classes.

To further reduce the influence of easy negative classes on model
optimization, we discard these negative classes with small prediction
probabilities. In detail, we prohibit them with predicted probability $p_{i}$
lower than the threshold $p_{th}$ from participating in the calculation of
loss, i.e.,
$L_{-}=\sum_{n=1}^{\infty}\beta_{i,n}{\text{max}(p_{i}-p_{th},0)}^{n+\gamma^{-}},$
where $p_{th}>0$ is a hard threshold.

Based on the facts that individually tuning parameters is prohibitive and
amending the leading polynomial coefficients leads to maximal gain, we propose
a simple implementation that collectively adjusts coefficients of leading
polynomial bases $(1-p_{i})^{1\&2}$ for all positive/negative classes.
Aggregating the above techniques, our proposed Asymmetric Polynomial Loss
(APL) is formalized as follows,

$L_{\rm
APL}=\sum_{i=0}^{C}[y_{i}\sum_{m=1}^{\infty}\alpha_{i,m}(1-p_{i})^{m+\gamma^{+}}+\\\
\indent\indent\ \ \
(1-y_{i})\sum_{n=1}^{\infty}\beta_{i,n}{\text{max}(p_{i}-p_{th},0)}^{n+\gamma^{-}}]/C\\\
=\sum_{i=0}^{C}\\{y_{i}(1-p_{i})^{\gamma^{+}}[-\log
p_{i}+(\alpha_{1}-1)(1-p_{i})+(\alpha_{2}-\frac{1}{2})(1-p_{i})^{2}]+(1-y_{i})p_{res}^{\gamma^{-}}[-\log(1-p_{i})+(\beta_{1}-1)p_{res}]\\}/C,$

where $\alpha_{1}=\alpha_{i,1},\alpha_{2}=\alpha_{i,2},\beta_{1}=\beta_{i,1},$
$i=1,2,...,C$ are coefficients, and $p_{res}$ denotes
$\text{max}(p_{i}-p_{th},0)$.

Fig. 1: Polynomial coefficients of $L_{+}$ loss for different loss functions
in the base of $(1-p_{i})^{k},k\in\mathbb{N}+$.

Our APL loss maintains good flexibility in the choice of parameters for the
polynomial basis. As presented in Figure 1, compared with BCE loss with fixed
coefficients and ASL loss that can only be shifted horizontally, our APL loss
allows for simultaneous vertical and horizontal adjustments, which greatly
increases the design space for miscellaneous tasks. So $L_{-}$ loss shares the
property.

Fig. 2: Comparison of $L_{-}$ curves for different loss functions.
$\gamma_{-}=2$ and $p_{th}=0.2$ for both ASL and APL for comparison.

By drawing $L_{-}$ loss curves in Figure 2, we can directly tell that easy
negative classes whose prediction probabilities are lower than the threshold
$p_{th}$ have no effect on training, and adjusting the leading coefficient
$\beta_{1}$ is beneficial to further optimize the loss ratio of positive and
negative classes. Besides, adjusting the leading coefficient $\beta_{1}$ can
theoretically alleviate the mislabelling problem, which will be further
discussed in Section 2.3.

###  2.3 Gradient Analysis

We separately analyze gradients for positive and negative classes described in
$L_{\rm APL}$ for clarity. For positive classes, without loss of generality,
we set $\gamma^{+}=0$ and the gradient with respect to $p_{i}$ is
$-\frac{\partial L^{+}}{\partial
p_{i}}=\alpha_{1}+2\alpha_{2}(1-p_{i})+\sum_{j=2}^{\infty}(1-p_{i})^{j}.$ We
observe that $\alpha_{1}$ provides a constant gradient regardless of $p_{i}$,
and $\alpha_{2}$ maintains a linear correlation with $p_{i}$. But the rest
terms are strongly suppressed as $p_{i}$ approaches $1$. Similar property
holds for $\gamma^{+}>0$. Therefore, adjusting $\alpha_{1}$ and $\alpha_{2}$
is reasonable from the gradient perspective.

For negative classes, the gradient with respect to logits $l_{i}$ is
$\frac{\partial L^{-}}{\partial l_{i}}=\frac{\partial L^{-}}{\partial
p_{i}}\frac{\partial p_{i}}{\partial
l_{i}}=p_{i}(1-p_{i})p_{res}^{\gamma^{-}}[\frac{1}{1-p_{res}}-\frac{\gamma^{-}\log(1-p_{res})}{p_{res}}+(\beta_{1}-1)(\gamma^{-}+1)].$
We can find that adjusting $\beta_{1}$ can theoretically alleviate the
mislabelling problem. Specifically, following previous works [6], very hard
negative samples are suspected to be mislabelled, i.e., predicted probability
$p_{i}>p^{*}$ where $p^{*}$ is the solution to $\frac{\partial}{\partial
p_{i}}(\frac{\partial L^{-}}{\partial l_{i}})=0$. However, simply changing
$\gamma^{-}$ has little effect on shifting the solution $p^{*}$ when keeping
$\beta_{1}$ as the default value $1$. But as shown in Figure 3, slightly
changing $\beta_{1}$ can greatly alter $p^{*}$ and the amplitude of the
gradient curves. This finding might provide some insights to mitigate the
mislabelling problem. Related experimental results on the AAPD [13] dataset
are reported in Section 3.1.

Fig. 3: We compare the gradients of $L_{-}$ losses of different loss
functions. We set $\gamma_{-}=1.8$ and $p_{th}=0.01$ for both ASL and APL
losses for comparison.

###  2.4 Parameter Interaction Analysis

We validate that the polynomial coefficients (i.e., $\alpha_{1,2}$) can
recalibrate the asymmetric focusing hyperparameters (i.e., $\gamma^{+}$). For
an imbalanced dataset, people usually keep the weight of rare positive classes
unchanged and down-weight the negative classes, i.e., set $\gamma^{+}=0$ and
fine-tune $\gamma^{-}>0$. Assume that we have already found the best
$\gamma^{-}$, but the target $\gamma^{+}=1$ is better. Then theoretically, we
will find that $\alpha_{1}=0$ when we alter it, which means the first-order
polynomial $\alpha_{1}(1-p)$ is an inappropriate term. By now, we can choose
to either set $\gamma^{+}=1$ to raise the order for all polynomials or
continue to alter $\alpha_{2}$ for better performance. The difference between
the two choices is
$\sum_{i=1}^{\infty}\frac{1}{i}(1-p)^{i+1}-\sum_{i=2}^{\infty}\frac{1}{i}(1-p)^{i}=\frac{1}{2}(1-p)^{2}+\frac{1}{6}(1-p)^{3}+\frac{1}{12}(1-p)^{4}+\cdots.$
We can see that (i) the second choice to adjust $\alpha_{2}$ maintains smaller
coefficients for polynomial bases; (ii) the difference between the same
polynomial base decreases rapidly with the increase of polynomial order; (iii)
high-order terms bring less gradients. Thus, we need to continue to adjust
$\alpha_{2}$ at most, and the original smaller coefficient could reach a
better solution. Besides, to obtain maximal gain with minimum parameter-
tuning, we mildly suggest to compare the results of $\gamma^{+}=0$ and
$\gamma^{+}=1$ after finding the optimal $\gamma^{-}$. Related experimental
results on the MS-COCO dataset [14] are reported in Section 3.3.

##  3 Experiments

###  3.1 Text Classification: AAPD Dataset

Dataset and Metrics. We use the arXiv academic paper dataset AAPD [13], which
contains 54 labels. Following the previous work [15], we adopt both the
precision at top K ($P@k$) and the normalized discounted cumulated gains at
top K ($nDCG@K$) as our evaluation metrics.

Main Results. Since our APL-based solution does not involve model architecture
modifications, our APL can be combined with different structures to improve
the performance. Here, we employ LSAN [15] model that performs best in the
open-source codes as our baseline. For a fair comparison, to reduce the bias
caused by devices, we present both the original LSAN results and our re-
implemented results. Our experiments are conducted on NVIDIA A100 GPU with
CUDA version being 11.2 and PyTorch version being 1.9.0.

We firstly re-implement the LSAN model with BCE loss and report the results in
Table 1. We further implement the ASL loss for comparison. In detail, we set
$\gamma^{+}=0$, $p_{th}=0.05$ and tune $\gamma^{-}$ by grid search from $1$ to
$5$. We obtain $85.31\%$, $85.73\%$, $86.15\%$, $85.83\%$ and $85.73\%$ P@1
metric scores respectively, and we therefore choose $\gamma^{-}=3$. By setting
$\gamma^{-}=3$, we also change $\gamma^{+}$ from $1$ to $4$, and obtain
$85.52\%,85.42\%,84.27\%$ and $83.65\%$ P@1 metric scores accordingly, which
means $\gamma^{+}=0$ and $\gamma^{-}=3$ are the best choices for ASL loss. We
consider this outcome as the ASL results reported in Table 1.

Method | P@1 | P@3 | G@3 | G@5
---|---|---|---|---
LSAN with BCE | 85.28 | 61.12 | 80.84 | 84.78
LSAN* | 85.10 | 60.80 | 80.55 | 84.52
ASL Loss | 86.15 | 61.01 | 80.90 | 84.50
APL Loss | 86.56 | 61.35 | 81.34 | 84.86

Table 1: Results (%) on AAPD test set. G@K is abbreviated for $nDCG@K,K=3,5$.
* denotes re-implemented results. _Improvement_ is compared with the results
of LSAN*.

We then adjust the leading polynomial coefficient $\alpha_{1}$ of our APL with
the above hyperparameters. In detail, the result when $\alpha_{1}=1$ is the
benchmark since this is the result of the ASL loss. We increase the value of
$\alpha_{1}$, we can observe that the metric values are rising consistently,
and we report the results when $\alpha_{1}=2.5$ in Table 1. Results show that
adjusting the coefficient of the first-order polynomial is effective.

Model | Dev | Test
---|---|---
| Ign $F_{1}$ | $F_{1}$ | Ign $F_{1}$ | $F_{1}$
MRN with ASL | 56.62 | 58.69 | 56.19 | 58.46
MRN with APL | 56.96 | 58.89 | 56.43 | 58.60

Table 2: Results ($\%$) on the dev and test sets of DocRED.

Another finding is that based on the ASL setting, i.e., $\alpha_{1}=1$ and we
set $\beta_{1}=1.4$, we obtain the results<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>Compared with ASL results in Table 1,
we get $-0.21\%$ lower scores on precision at Top $1$ and achieve performance
improvement of $+0.8\%,+0.72\%$ and $+0.53\%$ on the latter three metrics. The
reason might be that the number of paper subjects could be limited when
labeling, and some positive labels could be mislabelled as negative labels.
Then, raising $\beta_{1}$ mitigates this issue as it would shift $p^{*}$ to
the left as described in Section 2.3. However, it would affect the
optimization of some ordinary hard negative samples, resulting in the
reduction of $P@1$.

###  3.2 Relation Extraction: DocRED Dataset

Dataset and Metrics. We adopt the large-scale human-annotated document-level
relation extraction dataset DocRED [16], and we adopt the Ign F1 and F1 as the
metrics.

Main Results. We choose the recent open-sourced MRN model [17] as our
baseline, which leverages the ASL loss and GloVe [18] word embedding. In
detail, they adopt $\gamma^{+}=1,\gamma^{-}=4$ and $p_{th}=0.05$ for ASL loss.
Following the same setting, we adjust $\alpha_{1}$ for our APL loss.
Considering that $\alpha_{1}=1$ can be regarded as the ASL baseline, we set
$\alpha_{1}=1.4$ and report the test set results as shown in Table 2. Although
the multi-label problem is not significant on the DocRED dataset (7%), our APL
loss still achieves better results on both development set and test set.

We also conduct experiments using the pre-trained language model BERT [19].
Our APL loss achieves $59.78\%$ Ign F1 and $61.86\%$ F1 scores on the test
set, which are higher than MRN+BERT with $59.52\%$ Ign F1 and $61.74\%$ F1
score. This shows that our APL loss can be effective using different word
embeddings.

###  3.3 Image Classification: MS-COCO and NUS-WIDE

Dataset and Metrics. We conduct experiments on MS-COCO [14] and NUS-WIDE [20]
multi-label image classification datasets. Following the recent work
ML_Decoder [21], we present the mean average precision (mAP) score.

Main Results. We choose the ML_Decoder model using ASL loss as the baseline
and the results are reported in Table 3. For the MS-COCO dataset that contains
$122,218$ images with $80$ labels, we firstly report the original ASL [6] and
ML_Decoder results, where they both adopt $\gamma^{+}=0$, $\gamma^{-}=4$,
$p_{th}=0.05$ ,and use TResNet-L [22] as the backbone.

Method | MS-COCO | NUS-WIDE
---|---|---
ASL Loss | 86.6 | -
ML_Decoder with ASL | 90.0 | 31.1
ML_Decoder* | 89.81 | 30.89
APL Loss | 90.08 | 31.27

Table 3: mAP scores on MS-COCO and NUS-WIDE test sets. * denotes re-
implemented results.

We firstly test different $\alpha_{1}$ for APL loss based on the above
settings, and we notice that $\alpha_{1}=0$ achieves the best mAP scores,
which indicates that $\gamma^{+}=0$ is not the best parameter for this model.
As discussed in Section 2.4, we conduct the following experiments:

(i) We further adjust $\alpha_{2}$ by fixing $\alpha_{1}=0$ and find that
increasing $\alpha_{2}$ can further consistently enhance the performance. We
report this result in Table 3 by setting $\alpha_{1}=0$ and $\alpha_{2}=3.0$.
Particularly, tuning $\alpha_{2}$ brings less improvement than tuning
$\alpha_{1}$, which also shows that it is right to give priority to adjusting
the leading polynomial coefficient.

(ii) We only raise $\gamma^{+}=1$, and the performance is improved from
$89.81\%$ to $89.99\%$. We further tune $\alpha_{1}\in\\{2.0,2.5,3.0,3.5\\}$
and obtain $90.00\%$, $90.02\%$, $90.02\%$, $90.02\%$ mAP scores. Raising
$\alpha_{1}$ from $2.5$ to $3.5$ brings no additional improvement. We believe
the reason might be that the coefficients of the same-order polynomial bases
are large than those of the original Taylor expansion, resulting in a lower
proportion of the gradient of the quadratic polynomial. Therefore, the effect
of adjusting $\alpha_{1}$ is smaller when $\gamma^{+}>0$.

For the NUS-WIDE dataset which is the multi-label zero-shot learning scenario
with $925$ seen labels from Flicker user tags and $81$ unseen human-annotated
labels, the original ML_Decoder uses the TResNet-L as the backbone and
leverages the ASL loss with $\gamma^{+}=0,\gamma^{-}=4,p_{th}=0.05$ for
optimization in Table 3. We re-implement the ML_Decoder model and we adjust
$\alpha_{1}\in\\{1.2,1.3,1.4,1.5\\}$ and obtain $31.10\%,31.27\%,31.12\%$ and
$31.09\%$ mAP scores, respectively. We present the result using
$\alpha_{1}=1.3$ in Table 3. Results show that our APL loss could further
enhance the model performance on the multi-label zero-shot scenario.

##  4 Conclusion

In this paper, we propose an asymmetric polynomial loss (APL) for multi-label
classification. Through conducting Taylor expansion on the BCE loss, our APL
loss maintains high flexibility to tailor the polynomial coefficients for
various tasks and models. Moreover, our APL loss enables us to alleviate the
positive-negative imbalance problem by leveraging the asymmetric focusing
mechanism. The polynomial coefficients and asymmetric focusing parameters are
highly correlated both for positive and negative classes. Extensive
experiments verity the effectiveness of our APL loss.

##  5 Acknowledgement

This work was sponsored by the National Natural Science Foundation of China
(NSFC) grant (No. 62106143), and Shanghai Pujiang Program (No. 21PJ1405700).

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