###### Abstract

This paper gives the first quantitative bounds for the inverse theorem for the
Gowers $U^{4}$-norm over $\mathbb{F}_{p}^{n}$ when $p=2,3$. We build upon
earlier work of Gowers and Milićević who solved the corresponding problem for
$p\geq 5$. Our proof has two main steps: _symmetrization_ and _integration_ of
low-characteristic trilinear forms. We are able to solve the integration
problem for all $k$-linear forms, but the symmetrization problem we are only
able to solve for trilinear forms. We pose several open problems about
symmetrization of low-characteristic $k$-linear forms whose resolution,
combined with recent work of Gowers and Milićević, would give quantitative
bounds for the inverse theorem for the Gowers $U^{k+1}$-norm over
$\mathbb{F}_{p}^{n}$ for all $k,p$.

title = Quantitative Bounds for the $U^{4}$-Inverse Theorem over Low
Characteristic Finite Fields, author = Jonathan Tidor, plaintextauthor =
Jonathan Tidor, plaintexttitle = Quantitative Bounds for the U4-Inverse
Theorem over Low Characteristic Finite Fields, runningtitle = Quantitative
$U^{4}$-Inverse Theorem over Low Characteristic Finite Fields, year=2022,
number=14, received=12 October 2021, published=28 October 2022,
doi=10.19086/da.38591,

[classification=text]

##  1 Introduction

A central problem in additive combinatorics is to understand the inverse
theory of the Gowers uniformity norms. Given a 1-bounded function $f\colon
G\to\mathbb{C}$ with large Gowers uniformity norm where $G$ is a finite
abelian group, the goal is to show that $f$ must correlate with some
structured object.

This inverse problem has primarily received attention in the cases that $G$ is
a cyclic group of prime order or a vector space over a finite field. For
$G=\mathbb{Z}/N\mathbb{Z}$ the inverse problem was solved by Green, Tao, and
Ziegler [9, 10] and the structured objects require the theory of nilsequences
to describe. For vector spaces over finite fields the situation is known to be
somewhat simpler. For $p\geq k$, a 1-bounded function
$f\colon\mathbb{F}_{p}^{n}\to\mathbb{C}$ has large $U^{k+1}$-norm if and only
if it correlates with a (classical) polynomial phase function, i.e., $e^{2\pi
iP(x)/p}$ for a polynomial $P\colon\mathbb{F}_{p}^{n}\to\mathbb{F}_{p}$ of
degree at most $k$, as shown by Bergelson, Tao, and Ziegler [2, 17] (see also
[3] for a discussion of the $p=k$ case). For $p<k$, one needs the theory of
non-classical polynomials to describe the structured objects [18].

The original proofs of these inverse theorems give extremely bad or even
ineffective quantitative bounds. Recently Manners proved quantitative bounds
for the $U^{k+1}$-inverse theorem over $\mathbb{Z}/N\mathbb{Z}$ [14] and
Gowers and Milićević proved quantitative bounds for the $U^{k+1}$-inverse
theorem over $\mathbb{F}_{p}^{n}$ when $p>k$ [5, 6].

The goal of this paper is to build upon the work of Gowers and Milićević to
give quantitative bounds for the $U^{4}$-inverse theorem over
$\mathbb{F}_{p}^{n}$ in the low-characteristic regime $p=2,3$. Our proof has
two main steps: _symmetrization_ and _integration_ of low-characteristic
trilinear forms. We are able to solve the integration problem for all
$k$-linear forms, but the symmetrization problem we are only able to solve for
trilinear forms.

###  1.1 Statement of main result

In this paper we always use $V$ to denote a finite-dimensional
$\mathbb{F}_{p}$-vector space and we write $\omega=e^{2\pi i/p}$. Given a
function $f\colon V\to\mathbb{C}$ and a shift $h\in V$, we write
$\partial_{h}f(x)=f(x+h)\overline{f(x)}$ for the multiplicative derivative.

######  Definition 1.1.

Given a function $f\colon V\to\mathbb{C}$ and $d\geq 2$, the Gowers uniformity
norm $\|f\|_{U^{d}}$ is defined by

$\|f\|_{U^{d}}^{2^{d}}=\mathbb{E}_{x,h_{1},\ldots,h_{d}\in
V}(\partial_{h_{1}}\cdots\partial_{h_{d}}f)(x).$

This definition has many useful properties, including being a well-defined
norm for all $d\geq 2$. Another useful property is the inductive formula
$\|f\|_{U^{d}}^{2^{d}}=\mathbb{E}_{h}\|\partial_{h}f\|_{U^{d-1}}^{2^{d-1}}$.
See [18, Lemma B.1] for more properties and references about the Gowers
uniformity norms.

Consider a function $f\colon V\to G$ where $G$ is an abelian group. Given a
shift $h\in V$, we write $\Delta_{h}f(x)=f(x+h)-f(x)$ for the additive
derivative. It follows from the definition of the Gowers uniformity norm that
a 1-bounded function $f\colon V\to\mathbb{C}$ satisfies $\|f\|_{U^{d}}\leq 1$
with equality if and only if $f=e^{2\pi iP}$ where $P\colon
V\to\mathbb{R}/\mathbb{Z}$ satisfies
$\Delta_{h_{1}}\Delta_{h_{2}}\cdots\Delta_{h_{d}}P(x)=0$.

######  Definition 1.2.

A non-classical polynomial of degree at most $k$ is a map $P\colon
V\to\mathbb{R}/\mathbb{Z}$ that satisfies

$(\Delta_{h_{1}}\cdots\Delta_{h_{k+1}}P)(x)=0$

for all $h_{1},\ldots,h_{k+1},x\in V$. We write
$\operatorname{Poly}_{\leqslant k}(V\to\mathbb{R}/\mathbb{Z})$ for the set of
non-classical polynomials of degree at most $k$.

A classical polynomial is a map $P\colon V\to\mathbb{F}_{p}$ satisfying the
same definition. Thus we use $\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{F}_{p})$ to denote the set of classical polynomials of degree
at most $k$. In this paper we commonly abuse notation to identify
$\mathbb{F}_{p}$ with $\\{0,1/p,\ldots,(p-1)/p\\}\subset\mathbb{R}/\mathbb{Z}$
when convenient. In this way we consider $\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{F}_{p})$ to be a subset of $\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{R}/\mathbb{Z})$.

######  Theorem 1.3.

Fix a prime $p$ and $\delta>0$. There exists an $\epsilon>0$ satisfying

$1/\epsilon=\exp\left(\exp\left(\exp\left(O_{p}\left(\log(1/\delta)^{O(1)}\right)\right)\right)\right)$

such that the following holds. Let $V$ be a finite-dimensional
$\mathbb{F}_{p}$-vector space. Given a function $f\colon V\to\mathbb{C}$
satisfying $\|f\|_{\infty}\leq 1$ and $\|f\|_{U^{4}}>\delta$, there exists a
non-classical cubic polynomial $P\in\operatorname{Poly}_{\leqslant
3}(V\to\mathbb{R}/\mathbb{Z})$ such that

$\left\lvert\mathbb{E}_{x\in V}f(x)e^{-2\pi iP(x)}\right\rvert\geq\epsilon.$

Furthermore, if $p\geq 3$, the polynomial can be taken to be classical.

This theorem was proved with ineffective bounds by Tao and Ziegler [18,
Theorem 1.10] and with effective bounds for $p\geq 5$ by Gowers and Milićević
[6, Theorem 7]. The bounds we prove are of the same shape as Gowers and
Milićević’s result.

###  1.2 Proof strategy

We start by summarizing the Gowers-Milićević proof of the $U^{4}$-inverse
theorem for $p\geq 5$ and then explain where the difficulties arise when
$p<5$.

We are given a function $f\colon\mathbb{F}_{p}^{n}\to\mathbb{C}$ which is
1-bounded and satisfies $\|f\|_{U^{4}}>\delta$. By expanding the definition of
the $U^{4}$-norm and applying the $U^{2}$-inverse theorem, this implies

$\delta^{16}<\|f\|_{U^{4}}^{16}=\mathbb{E}_{a,b}\|\partial_{a}\partial_{b}f\|_{U^{2}}^{4}\leq\mathbb{E}_{a,b}\|\widehat{\partial_{a}\partial_{b}f}\|_{\infty}^{2}.$

By an application of Markov’s inequality, this implies that there is a fairly
dense set $A\subset V^{2}$ and a function $\phi\colon A\to V$ such that
$|\widehat{\partial_{a}\partial_{b}f}(\phi(a,b))|$ is large for each $(a,b)\in
A$.

One can show that the function $\phi$ has some weak bilinearity properties.
The main bulk of the work done by Gowers and Milićević is to show that this
local bilinearity can be upgraded to some global structure. Namely, they prove
that $\phi$ agrees with a biaffine map $\Phi\colon V^{2}\to V$ on a fairly
dense set. This immediately implies that there is a triaffine form $T\colon
V^{3}\to\mathbb{F}_{p}$ that satisfies

$\left\lvert\mathbb{E}_{a,b,c,x}\partial_{a}\partial_{b}\partial_{c}f(x)\omega^{T(a,b,c)}\right\rvert\geq\eta.$
(1.1)

This part of the proof has no assumptions on the characteristic $p$.

To finish the proof, we would like to find a cubic polynomial $P$ such that
$\|f\omega^{P}\|_{U^{3}}$ is large. From this point we would be able to
conclude by a single application of the $U^{3}$-inverse theorem. We expand

$\|f\omega^{P}\|_{U^{3}}^{8}=\mathbb{E}_{a,b,c,x}\partial_{a}\partial_{b}\partial_{c}f(x)\omega^{\Delta_{a}\Delta_{b}\Delta_{c}P(x)}.$

Thus the object $\Delta_{a}\Delta_{b}\Delta_{c}P(x)$ naturally appears. We
call this object the total derivative of $P$. The total derivative of $P$
clearly does not depend on $x$ so we will just write
$\Delta_{a}\Delta_{b}\Delta_{c}P$ from now on. Furthermore,
$\Delta_{a}\Delta_{b}\Delta_{c}P$ is symmetric and trilinear in $a,b,c$.

Thus our next step is to massage Eq. 1.1 to turn the triaffine form $T$ into a
symmetric trilinear form. For simplicity of exposition, let us assume that $T$
is a trilinear. (The parts of $T$ that are not trilinear can often be removed
by an appropriate application of the Cauchy-Schwarz inequality.) We will first
explain the argument in the $p\geq 5$ case.

Symmetrization: We start with an argument based on an idea of Green and Tao.
By several applications of the Cauchy-Schwarz inequality, Eq. 1.1 implies that
$T(a,b,c)$ is “close” to $T(b,a,c)$, more precisely, we can prove that the map
$(a,b,c)\mapsto T(a,b,c)-T(b,a,c)$ has bounded rank, and similarly for all
other permutations of $a,b,c$.111See Section 2 for the definition of tensor
rank that we use in this paper.

To conclude the symmetrization step, we define

$S(a,b,c)=\frac{1}{6}\left(T(a,b,c)+T(a,c,b)+T(b,a,c)+T(b,c,a)+T(c,a,b)+T(c,b,a)\right).$
(1.2)

Clearly $S$ is a symmetric trilinear form. Furthermore, by the Green-Tao
argument, we know that $\operatorname{rank}(T-S)\ll_{\eta}1$.

Integration: We now integrate the symmetric trilinear form $S$ to a cubic
polynomial $P$. Namely, define

$P(x)=\frac{1}{6}S(x,x,x).$ (1.3)

One can easily check that $\Delta_{a}\Delta_{b}\Delta_{c}P=S(a,b,c)$.

Starting from Eq. 1.1, with some involved but not particularly interesting
additional inequalities, we can conclude that
$\|f\omega^{P}\|_{U^{3}}\gg_{\eta}1$. Applying the $U^{3}$-inverse theorem
lets us complete the proof.

###  1.3 Low characteristic

Now we explain the difficulties that appear in characteristic $p<5$.
Fortunately the bulk of the proof, up to Eq. 1.1, works in arbitrary
characteristic. The Green-Tao argument also goes through, showing that $T$ is
close to each of its permutations. However, the remainder of the
symmetrization step and the integration step, which were essentially trivial
in high characteristic, fail badly when $p<5$. In particular, Eq. 1.2 and Eq.
1.3 both involve dividing by 6.

For $p\geq 5$ we have the convenient characterization that $S$ satisfies
$S(a,b,c)=\Delta_{a}\Delta_{b}\Delta_{c}P$ for some cubic polynomial $P$ if
and only if $S$ is a symmetric trilinear form. However, this is no longer the
case in low characteristic.

Following Tao and Ziegler, we define a classical symmetric trilinear form to
be a trilinear form $S\colon V^{3}\to\mathbb{F}_{p}$ that satisfies
$S(a,b,c)=\Delta_{a}\Delta_{b}\Delta_{c}P$ for some classical cubic polynomial
$P$. As part of their proof of the inverse theorem Tao and Ziegler
characterized classical symmetric $k$-linear forms for all $k$. In the same
vein, we define a non-classical symmetric trilinear form to be $S\colon
V^{3}\to\mathbb{F}_{p}$ that satisfies
$S(a,b,c)=\Delta_{a}\Delta_{b}\Delta_{c}P$ for some non-classical cubic
polynomial $P$. One problem we solve in this paper is the characterization of
non-classical symmetric $k$-linear forms for all $k$.

As a special case, we will show that if $S$ is a symmetric trilinear form and
also has some further symmetries which will be defined in Section 3, then we
can integrate $S$ to a non-classical cubic polynomial $P$ (i.e.,
$\Delta_{a}\Delta_{b}\Delta_{c}P=S(a,b,c)$). We are able to characterize non-
classical symmetric $k$-linear forms for all $k$, which solves the integration
problem necessary for the $U^{k+1}$-inverse theorem for all $k$.

For the symmetrization problem, we have a trilinear form $T$ which we know is
close in rank to all of its permutations. We wish to find, for $p=2$, an non-
classical symmetric trilinear form $S$ such that $\operatorname{rank}(T-S)\ll
1$ and, for $p=3$, a classical symmetric trilinear $S$ such that
$\operatorname{rank}(T-S)\ll 1$.

Even the problem of finding a symmetric $k$-linear $S$ that is close to $T$ is
quite hard and we do not know how to solve it for general $k$. When $k=3$, the
hypothesis that $T$ is close to all of its permutations implies that there is
a subspace $U\leq V$ of bounded codimension so that $T|_{U}$ is symmetric.
This lets us easily find a symmetric trilinear $S^{\prime}$ such that
$\operatorname{rank}(T-S^{\prime})\ll 1$. For $p=3$, turning $S^{\prime}$ into
a classical symmetric trilinear form is easy. For $p=2$, we need to turn
$S^{\prime}$ into a non-classical symmetric trilinear form which takes some
more work, but we are able to do so.

Structure of paper. We give the necessary preliminaries in Section 2. We solve
the integration problem for all $k$ in Section 3 and the symmetrization
problem for $k=3$ in Section 4. We combine these tools to prove our main
result in Section 5. In Section 6 we give some conjectures.

##  2 Preliminaries

We gave a local definition of non-classical polynomials in Section 1. There is
an equivalent global definition known. (Also see [18, Lemma 1.7] for some more
basic facts about non-classical polynomials.)

######  Lemma 2.1 ([18, Lemma 1.7(iii)]).

$P\colon\mathbb{F}_{p}^{n}\to\mathbb{R}/\mathbb{Z}$ is a non-classical
polynomial of degree at most $k$ if and only if it can be expressed in the
form

$P(x_{1},\ldots,x_{n})=\alpha+\sum_{\genfrac{}{}{0.0pt}{}{0\leq
i_{1},\ldots,i_{n}<p,\,j\geq 0:}{0<i_{1}+\cdots+i_{n}\leq
k-j(p-1)}}\frac{c_{i_{1},\ldots,i_{n},j}|x_{1}|^{i_{1}}\cdots|x_{n}|^{i_{n}}}{p^{j+1}}\pmod{1},$

for some $\alpha\in\mathbb{R}/\mathbb{Z}$ and coefficients
$c_{i_{1},\ldots,i_{n},j}\in\\{0,\ldots,p-1\\}$ where $|\cdot|$ is the
standard map $\mathbb{F}_{p}\to\\{0,\ldots,p-1\\}$. Furthermore, this
representation is unique.

For a tuple $(h_{1},\ldots,h_{k})\in V^{k}$ and a subset $I\subseteq[k]$, we
use the notation $h_{I}$ to denote the tuple $(h_{i})_{i\in I}\in V^{I}$.

######  Definition 2.2.

Given a $k$-linear form $T\colon V^{k}\to\mathbb{F}_{p}$ we use two different
notions for the rank of $T$.

The analytic rank of $T$, denoted $\operatorname{arank}(T)$, is defined by the
equation

$p^{-\operatorname{arank}(T)}=\mathbb{E}_{h_{1},\ldots,h_{k}\in
V}\omega^{T(h_{1},\ldots,h_{k})}.$

The partition rank of $T$, denote $\operatorname{prank}(T)$, is the length of
the shortest decomposition of $T$ as a sum of partition rank 1 forms. We
define $T\colon V^{k}\to\mathbb{F}_{p}$ to have partition rank 1 if we can
write $T(h_{1},\ldots,h_{k})=R(h_{I})S(h_{[k]\setminus I})$ where $R\colon
V^{I}\to\mathbb{F}_{p}$ and $S\colon V^{[k]\setminus I}\to\mathbb{F}_{p}$ are
multilinear forms and $\emptyset\neq I\subsetneq[k]$.

These definitions of rank enjoy many useful properties. One which will be
important to us is the subadditivity of rank. Obviously
$\operatorname{prank}(S+T)\leq\operatorname{prank}(S)+\operatorname{prank}(T)$.
For analytic rank, we also have
$\operatorname{arank}(S+T)\leq\operatorname{arank}(S)+\operatorname{arank}(T)$,
but the proof is more subtle [12, Theorem 1.5].

For $k=2$, the partition rank and analytic rank coincide and both are equal to
the usual notion of matrix rank. For general $k$ we have the following bounds.

######  Theorem 2.3.

There are constants $C_{p,k}$ and $D_{k}$ such that for a $k$-linear form
$T\colon V^{k}\to\mathbb{F}_{p}$ we have the bounds

$\operatorname{arank}(T)\leq\operatorname{prank}(T)\leq
C_{p,k}\operatorname{arank}(T)^{D_{k}}.$

The lower bound is due to Lovett [12] and the upper bound was proved
independently by Milićević [15] and Janzer [11]. Note that a recent result of
Cohen and Moshkovitz and independently Adiprasito, Kazhdan, and Ziegler
improves the upper bound to linear when $k=3$ and $p>2$ [4, 1].

One simple property of rank that we use multiple times through the argument is
the following.

######  Lemma 2.4.

Let $T\colon V^{k}\to\mathbb{F}_{p}$ be a $k$-linear form and let $U\leq V$ be
a subspace such that $T|_{U}\equiv 0$. Then $\operatorname{prank}(T)\leq
k\cdot\operatorname{codim}U$.

###### Proof.

Write $n=\dim V$ and $r=\operatorname{codim}U$. Pick linearly independent
linear forms $\alpha_{1},\ldots,\alpha_{n}\colon V\to\mathbb{F}_{p}$ such that
$\alpha_{i}|_{U}\equiv 0$ for $1\leq i\leq r$. (In other words
$\alpha_{1},\ldots,\alpha_{n}$ is a dual basis for $V$ and
$\alpha_{r+1},\ldots,\alpha_{n}$ is a dual basis for $U$.) The tensor $T$ can
be expressed in these coordinates as

$T(x_{1},\ldots,x_{k})=\sum_{1\leq i_{1},\ldots,i_{k}\leq
n}T_{i_{1},\ldots,i_{k}}\alpha_{i_{1}}(x_{1})\cdots\alpha_{i_{k}}(x_{k})$

for some coefficients $T_{i_{1},\ldots,i_{k}}\in\mathbb{F}_{p}$ satisfying
$T_{i_{1},\ldots,i_{k}}=0$ if $i_{1},\ldots,i_{k}>r$.

Now one can easily group the above expression into the sum of $kr$ terms, each
of partition rank 1. Explicitly,

$T(x_{1},\ldots,x_{k})=\sum_{j=1}^{k}\sum_{\ell=1}^{r}\alpha_{\ell}(x_{j})\beta_{j,\ell}(x_{[k]\setminus
j})$

where

$\beta_{j,\ell}(x_{[k]\setminus
j})=\sum_{\genfrac{}{}{0.0pt}{}{r<i_{1},\ldots,i_{j-1}\leq n}{1\leq
i_{j+1},\ldots,i_{n}\leq
n}}T_{i_{1},\ldots,i_{j-1},\ell,i_{j+1},\ldots,i_{k}}\alpha_{i_{1}}(x_{1}),\ldots,\alpha_{i_{j-1}}(x_{j-1})\alpha_{i_{j+1}}(x_{j+1})\cdots\alpha_{i_{k}}(x_{k}).\qed$

##  3 Non-classical symmetric multilinear forms

In this section we define non-classical symmetric multilinear forms
($\operatorname{nCSM}$ for short) and classical symmetric multilinear forms
($\operatorname{CSM}$ for short) as multilinear forms with additional symmetry
properties. Then we will show that these objects are exactly the total
derivatives of non-classical polynomials and classical polynomials
respectively.

######  Definition 3.1.

A non-classical symmetric multilinear form is a map $T\colon
V^{k}\to\mathbb{F}_{p}$ that is:

  * •

(multilinear) fixing all the variables but one, the map $h_{i}\mapsto
T(h_{1},\ldots,h_{k})$ is a linear map;

  * •

(symmetric) $T(h_{1},\ldots,h_{k})$ is invariant under permutations of the $k$
variables $h_{1},\ldots,h_{k}$;

  * •

(non-classical)
$T(h_{1},\ldots,h_{1},h_{2},\ldots,h_{k-p+1})=T(h_{1},h_{2},\ldots,h_{2},h_{3},\ldots,h_{k-p+1})$
where the variable $h_{1}$ appears $p$ times in the first expression and
$h_{2}$ appears $p$ times in the second expression.

We write $\operatorname{nCSM}^{k}(V)$ for the space of non-classical symmetric
multilinear forms $T\colon V^{k}\to\mathbb{F}_{p}$.

A classical symmetric multilinear form additionally satisfies:

  * •

(classical) $T(h_{1},\ldots,h_{1},h_{2},\ldots,h_{k-p+1})=0$ where the
variable $h_{1}$ appears $p$ times in the left-hand side.

We write $\operatorname{CSM}^{k}(V)$ for the space of classical symmetric
multilinear forms $T\colon V^{k}\to\mathbb{F}_{p}$.

Note that the classical condition is stronger than the non-classical
condition, so $\operatorname{CSM}^{k}(V)\subseteq\operatorname{nCSM}^{k}(V)$.
For $k\leq p$ the non-classical condition is vacuous while for $k<p$ the
classical condition is vacuous. In their proof of the inverse theorem Tao and
Ziegler introduced classical symmetric multilinear forms and (implicitly) non-
classical symmetric multilinear forms [18].

######  Definition 3.2.

For $P\in\operatorname{Poly}_{\leqslant k}(V\to\mathbb{R}/\mathbb{Z})$, define
the total derivative of $P$ to be the map $d^{k}P\colon
V^{k}\to\mathbb{F}_{p}$ defined by

$d^{k}P(h_{1},\ldots,h_{k})=(\Delta_{h_{1}}\cdots\Delta_{h_{k}}P)(0).$

The right-hand side of this equation lies in
$\\{0,1/p,\ldots,(p-1)/p\\}\subset\mathbb{R}/\mathbb{Z}$ which we identify
with $\mathbb{F}_{p}$; furthermore
$d^{k}P(h_{1},\ldots,h_{k})=(\Delta_{h_{1}}\cdots\Delta_{h_{k}}P)(x)$ for all
$h_{1},\ldots,h_{k},x\in V$.222One way to see these facts is to notice that
$\Delta_{h_{2}}\cdots\Delta_{h_{k}}P\in\operatorname{Poly}_{\leqslant
1}(V\to\mathbb{R}/\mathbb{Z})$. By Lemma 2.1 this expression is of the form
$\alpha+c_{1}|x_{1}|/p+\cdots+c_{n}|x_{n}|/p\pmod{1}$. Applying
$\Delta_{h_{1}}$, we see that
$\Delta_{h_{1}}\cdots\Delta_{h_{k}}P(x)=(h_{1,1}c_{1}+\cdots+h_{1,n}c_{n})/p$
is independent of $x$ and lies in the set $\\{0,1/p,\ldots,(p-1)/p\\}$.

Tao and Ziegler prove that the total derivatives of classical polynomials are
exactly CSMs.

######  Proposition 3.3 ([18, Lemma 4.5]).

For $k\geq 1$ and a classical polynomial $P\in\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{F}_{p})$, the total derivative $d^{k}P$ lies in
$\operatorname{CSM}^{k}(V)$. Furthermore this map is surjective. In other
words, we have the short exact sequence

$0\to\operatorname{Poly}_{\leqslant
k-1}(V\to\mathbb{F}_{p})\to\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{F}_{p})\xrightarrow{d^{k}}\operatorname{CSM}^{k}(V)\to 0.$

They also show that the total derivative of a non-classical polynomial is an
nCSM.

######  Proposition 3.4 ([18, Eq. (4.1)]).

For $P\in\operatorname{Poly}_{\leqslant k}(V\to\mathbb{R}/\mathbb{Z})$, the
total derivative $d^{k}P$ lies in $\operatorname{nCSM}^{k}(V)$.

Our integration result gives the implication in the opposite implication.

######  Proposition 3.5 (Integrating $\operatorname{nCSM}$s).

For $k\geq 1$ and $T\in\operatorname{nCSM}^{k}(V)$, there exists a non-
classical polynomial $P\in\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{R}/\mathbb{Z})$ such that $d^{k}P=T$. In other words, we have
the short exact sequence

$0\to\operatorname{Poly}_{\leqslant
k-1}(V\to\mathbb{R}/\mathbb{Z})\to\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{R}/\mathbb{Z})\xrightarrow{d^{k}}\operatorname{nCSM}^{k}(V)\to
0.$

###### Proof.

First note that $\ker(d^{k})=\operatorname{Poly}_{\leqslant
k-1}(V\to\mathbb{R}/\mathbb{Z})$ since $\Delta_{h_{1}}\cdots\Delta_{h_{k}}P$
is identically zero for all $h_{1},\ldots,h_{k}$ if and only if $P$ is a non-
classical polynomial of degree at most $k-1$.

We show that $d^{k}$ is surjective in an inexplicit manner. First define
$\overline{\operatorname{Poly}}_{\leqslant k}(V\to\mathbb{R}/\mathbb{Z})$ to
be the quotient of $\operatorname{Poly}_{\leqslant
k}(V\to\mathbb{R}/\mathbb{Z})$ under the equivalence relation $P\sim P+\alpha$
for $\alpha\in\mathbb{R}/\mathbb{Z}$. Note that for $k\geq 1$, the total
derivative $d^{k}$ acts on this quotient space. We will prove that

$\frac{|\overline{\operatorname{Poly}}_{\leqslant
k}(V\to\mathbb{R}/\mathbb{Z})|}{|\overline{\operatorname{Poly}}_{\leqslant
k-1}(V\to\mathbb{R}/\mathbb{Z})|}=|\operatorname{nCSM}^{k}(V)|.$ (3.1)

This suffices to complete the proof.

Choose an isomorphism $V\cong\mathbb{F}_{p}^{n}$. By Lemma 2.1, we see that
$|\overline{\operatorname{Poly}}_{\leqslant
k}(V\to\mathbb{R}/\mathbb{Z})|=p^{C^{\prime}_{k}}$ where $C^{\prime}_{k}$ is
defined to be the number of tuples $(i_{1},\ldots,i_{n},j)$ where
$i_{1},\ldots,i_{n}\in\\{0,\ldots,p-1\\}$ and $j\geq 0$ satisfy
$0<i_{1}+\cdots+i_{n}\leq k-j(p-1)$. Then the left-hand side of Eq. 3.1 is
$p^{C^{\prime}_{k}-C^{\prime}_{k-1}}$. Define
$C_{k}=C^{\prime}_{k}-C^{\prime}_{k-1}$. We can easily see that $C_{k}$ is the
number of tuples $(i_{1},\ldots,i_{n},j)$ where
$i_{1},\ldots,i_{n}\in\\{0,\ldots,p-1\\}$ and $0\leq j\leq(k-1)/(p-1)$ satisfy
$i_{1}+\cdots+i_{n}=k-j(p-1)$.

Now we compute $|\operatorname{nCSM}^{k}(V)|$. Clearly any multilinear form
$T\colon V^{k}\to\mathbb{F}_{p}$ can be expressed uniquely as

$T(x_{1},\ldots,x_{k})=\sum_{j_{1},\ldots,j_{k}\in[n]}c_{j_{1},\ldots,j_{k}}x_{1,j_{1}}\cdots
x_{k,j_{k}}$

where $c_{j_{1},\ldots,j_{k}}\in\mathbb{F}_{p}$. For a tuple
$\bm{j}=(j_{1},\ldots,j_{k})$, define $i(\bm{j})=(i_{1},\ldots,i_{n})$ such
that $i_{\ell}$ is the number of $j_{1},\ldots,j_{k}$ which are equal to
$\ell$. With this notation, we can say that a multilinear form $T$ is
symmetric if and only if $c_{\bm{j}_{1}}=c_{\bm{j}_{2}}$ whenever
$i(\bm{j}_{1})=i(\bm{j}_{2})$.

Finally define $i^{\prime}(\bm{j})=(i^{\prime}_{1},\ldots,i^{\prime}_{n})$
where $i^{\prime}_{\ell}=0$ if $i_{\ell}=0$ and otherwise
$i^{\prime}_{\ell}\in\\{1,\ldots,p-1\\}$ satisfies $i^{\prime}_{\ell}\equiv
i_{\ell}\pmod{p-1}$. We can see that a multilinear form $T$ is an
$\operatorname{nCSM}$ if and only if $c_{\bm{j}_{1}}=c_{\bm{j}_{2}}$ whenever
$i^{\prime}(\bm{j}_{1})=i^{\prime}(\bm{j}_{2})$. Thus
$|\operatorname{nCSM}^{k}(V)|=p^{D_{k}}$ where $D_{k}$ is the number of tuples
$\bm{i}^{\prime}\in\\{0,\ldots,p-1\\}^{n}$ such that there exists
$\bm{j}\in[n]^{k}$ such that $\bm{i}^{\prime}=i^{\prime}(\bm{j})$. Clearly the
only constraint on such a tuple $\bm{i}^{\prime}\in\\{0,\ldots,p-1\\}^{n}$ is
that $s=i^{\prime}_{1}+\cdots+i^{\prime}_{n}$ satisfies $0<s\leq n$ and
$s\equiv n\pmod{p-1}$. Thus we see that $C_{k}=D_{k}$, proving the desired
result. ∎

##  4 Symmetrization in low characteristic

For a $k$-linear form $T\colon V^{k}\to\mathbb{F}_{p}$ and a permutation
$\pi\in\mathfrak{S}_{k}$, write $T_{\pi}\colon V^{k}\to\mathbb{F}_{p}$ for

$T_{\pi}(x_{1},\ldots,x_{k})=T(x_{\pi(1)},\ldots,x_{\pi(k)}).$

We have one tool that lets us show that forms are close to symmetric. The
following result is based on an idea of Green and Tao.

######  Lemma 4.1 (cf. [7, Lemma 2.8]).

Let $A\colon V^{2}\to\mathbb{F}_{p}$ be a bilinear form. Let
$b_{1},b_{2},b_{3}\colon V\to\mathbb{C}$ be arbitrary 1-bounded functions. If

$\left\lvert\mathbb{E}_{u,v\in
V}b_{1}(u)b_{2}(v)b_{3}(u+v)\omega^{A(u,v)}\right\rvert\geq\delta,$

then

$p^{-\operatorname{arank}(A-A_{(12)})}=\mathbb{E}_{u,v}\omega^{A(u,v)-A(v,u)}\geq\delta^{8}.$

###### Proof.

By an application of the Cauchy-Schwarz inequality and the 1-boundedness of
$b_{2}$, we have

$\delta^{2}\leq\mathbb{E}_{u,u^{\prime},v\in
V}b_{1}(u)\overline{b_{1}(u^{\prime})}b_{3}(u+v)\overline{b_{3}(u^{\prime}+v)}\omega^{A(u-u^{\prime},v)}.$

We make the change of variables $u^{\prime\prime}=u+u^{\prime}+v$.
Rearranging, we obtain

$\displaystyle\delta^{2}$
$\displaystyle\leq\mathbb{E}_{u,u^{\prime},u^{\prime\prime}}b_{1}(u)\overline{b_{1}(u^{\prime})}b_{3}(u^{\prime\prime}-u^{\prime})\overline{b_{3}(u^{\prime\prime}-u)}\omega^{A(u-u^{\prime},u^{\prime\prime}-u-u^{\prime})}$
$\displaystyle=\mathbb{E}_{u,u^{\prime},u^{\prime\prime}}\left(b_{1}(u)\overline{b_{3}(u^{\prime\prime}-u)}\omega^{A(u,u^{\prime\prime}-u)}\right)\left(\overline{b_{1}(u^{\prime})}b_{3}(u^{\prime\prime}-u^{\prime})\omega^{A(-u^{\prime},u^{\prime\prime}-u^{\prime})}\right)\omega^{A(u^{\prime},u)-A(u,u^{\prime})}.$

Write $B=A-A_{(12)}$. By first averaging over $u^{\prime\prime}$ and then
choosing $b_{1}^{\prime},b_{2}^{\prime}\colon V\to\mathbb{C}$ appropriate
1-bounded functions, the above inequality becomes

$\delta^{2}\leq\left\lvert\mathbb{E}_{u,u^{\prime}}b_{1}^{\prime}(u)b_{2}^{\prime}(u^{\prime})\omega^{B(u,u^{\prime})}\right\rvert.$

By two more applications of the Cauchy-Schwarz inequality, we conclude

$\delta^{8}\leq\mathbb{E}_{u,v,u^{\prime},v^{\prime}}\omega^{B(u-v,u^{\prime}-v^{\prime})}=\mathbb{E}_{u,u^{\prime}}\omega^{B(u,u^{\prime})}=p^{-\operatorname{arank}(B)}.\qed$

######  Corollary 4.2.

Let $T\colon V^{3}\to\mathbb{F}_{p}$ be a trilinear form and let
$b_{1},\ldots,b_{7}\colon V\to\mathbb{C}$ be 1-bounded functions such that

$\left\lvert\mathbb{E}_{x,y,z\in
V}b_{1}(x)b_{2}(y)b_{3}(z)b_{4}(x+y)b_{5}(x+z)b_{6}(y+z)b_{7}(x+y+z)(-1)^{T(x,y,z)}\right\rvert\geq\delta.$

Then we have $\operatorname{arank}(T-T_{\pi})\leq 16\log_{p}(1/\delta)$ for
all $\pi\in\mathfrak{S}_{3}$.

###### Proof.

Suppose $\pi=(12)$. By the triangle inequality, we see that there exists a
function $\delta\colon V\to[0,1]$ and 1-bounded functions
$b_{1,z},b_{2,z},b_{3,z}\colon V\to\mathbb{C}$ such that

$\left\lvert\mathbb{E}_{x,y}b_{1,z}(x)b_{2,z}(y)b_{3,z}(x+y)\omega^{T(x,y,z)}\right\rvert\geq\delta(z)$

for all $z\in V$ and additionally $\mathbb{E}_{z\in V}\delta(z)\geq\delta$.

By Lemma 4.1 for all $z\in V$,

$\mathbb{E}_{x,y}\omega^{T(x,y,z)-T(y,x,z)}\geq\delta(z)^{8}.$

Averaging over $z$ and applying convexity,

$p^{-\operatorname{arank}(T-T_{(12)})}=\mathbb{E}_{x,y,z}\omega^{T(x,y,z)-T(y,x,z)}\geq\mathbb{E}_{z}\delta(z)^{8}\geq\delta^{8},$

so $\operatorname{arank}(T-T{(12)})\leq 8\log_{p}(1/\delta)$. The same holds
for all transpositions. Since every permutation in $\mathfrak{S}_{3}$ can be
written as the product of at most two transpositions, we conclude by the
subadditivity of analytic rank. ∎

######  Proposition 4.3.

Let $T\colon V^{3}\to\mathbb{F}_{p}$ be a trilinear form. Suppose that $T$ is
close to symmetric in the sense that $\operatorname{prank}(T-T_{\pi})\leq r$
for all permutations $\pi\in\mathfrak{S}_{3}$. Then there exists a subspace
$U\leq V$ with $\operatorname{codim}U\leq 5r$ such that $T|_{U}\colon
U^{3}\to\mathbb{F}_{p}$ is a symmetric trilinear form.

###### Proof.

By definition, there exists a partition rank decomposition of each $T-T_{\pi}$
(for $\pi\in\mathfrak{S}_{3}\setminus\\{\operatorname{id}\\}$) as the sum of
at most $r$ terms, each of which is the product of a linear form and a
bilinear form. Let $U$ be the subspace of $V$ where all of the linear forms
involved in this decomposition vanish. Since there are at most $5r$ linear
forms in this decomposition, we have the desired bound on the codimension of
$U$. Furthermore, since $(T-T_{\pi})|_{U}$ is identically zero for each
$\pi\in\mathfrak{S}_{3}$, we see that $T|_{U}$ is symmetric. ∎

######  Remark.

This result immediately implies that there exists a symmetric trilinear form
$S\colon V^{3}\to\mathbb{F}_{p}$ with $\operatorname{prank}(T-S)\leq 15r$. To
see this, pick an arbitrary decomposition $V=U\oplus W$. Then define $S$ by
$S(u_{1}+w_{1},u_{2}+w_{2},u_{3}+w_{3})=T|_{U}(u_{1},u_{2},u_{3})$ for all
$u_{1},u_{2},u_{3}\in U$ and $w_{1},w_{2},w_{3}\in W$. Since $T|_{U}$ is a
symmetric trilinear form it follows that $S$ is a symmetric trilinear form. By
Lemma 2.4, we have the bound $\operatorname{prank}(T-S)\leq 15r$.

4.3 easily handles the initial symmetrization step. To complete the
symmetrization, we need to find some $R\in\operatorname{nCSM}^{3}(V)$ such
that $\operatorname{prank}(T-R)$ is small. For $p\geq 5$ this is trivial,
since every symmetric trilinear form is an $\operatorname{nCSM}$ and actually
a $\operatorname{CSM}$.

For $p=3$ every symmetric trilinear form is also an $\operatorname{nCSM}$
though not necessarily a $\operatorname{CSM}$. We note however, that for $p=3$
every $\operatorname{nCSM}$ is close to a $\operatorname{CSM}$. This allows us
to prove the $U^{4}$-inverse theorem over $\mathbb{F}_{3}^{n}$ with classical
polynomials.

######  Proposition 4.4.

Let $T\colon V^{3}\to\mathbb{F}_{3}$ be a symmetric trilinear form. Then there
exists a codimension 1 subspace $U\leq V$ such that
$T|_{U}\in\operatorname{CSM}^{3}(U)$.

###### Proof.

The only additional condition that $T$ needs to satisfy to be a
$\operatorname{CSM}$ is $T(x,x,x)=0$. Thus if $U$ is any subspace of $V$ such
that $T(x,x,x)=0$ for all $x\in U$, we would have
$T|_{U}\in\operatorname{CSM}^{3}(U)$.

Consider the map $x\mapsto T(x,x,x)\in\mathbb{F}_{3}$. Because $T$ is a
symmetric trilinear form, this map is linear. Indeed,
$T(x+y,x+y,x+y)=T(x,x,x)+3T(x,x,y)+3T(x,y,y)+T(y,y,y)=T(x,x,x)+T(y,y,y)$.
Defining $U$ to be the codimension 1 subspace on which $T(x,x,x)$ vanishes, we
have the desired conclusion. ∎

Applying 4.3, 4.4, and Lemma 2.4 we conclude the following.

######  Corollary 4.5.

Fix $p\geq 3$. Let $T\colon V^{3}\to\mathbb{F}_{p}$ be a trilinear form that
satisfies $\operatorname{prank}(T-T_{\pi})\leq r$ for all permutations
$\pi\in\mathfrak{S}_{3}$. Then there exists $S\in\operatorname{CSM}^{3}(V)$
such that $\operatorname{prank}(T-S)\leq 15r+3$.

For $p=2$ the situation is more complicated. It is known, due to examples of
Green and Tao [8] and Lovett, Meshulam, and Samorodnitsky [13] that classical
polynomials are not sufficient for the $U^{4}$-inverse theorem over
$\mathbb{F}_{2}$. This means that we will not be able to find a
$\operatorname{CSM}$ that is close to $T$. However the following argument
shows that $T$ is close to an $\operatorname{nCSM}$.

######  Proposition 4.6.

Let $T\colon V^{3}\to\mathbb{F}_{2}$ be a symmetric trilinear form. Suppose
there are 1-bounded functions $b_{1},\ldots,b_{7}\colon V\to\mathbb{C}$ such
that

$\left\lvert\mathbb{E}_{x,y,z\in
V}b_{1}(x)b_{2}(y)b_{3}(z)b_{4}(x+y)b_{5}(x+z)b_{6}(y+z)b_{7}(x+y+z)(-1)^{T(x,y,z)}\right\rvert\geq\delta.$

Then there exists a subspace $U\leq V$ satisfying $\operatorname{codim}U\leq
8\log_{2}(1/\delta)$ such that $T|_{U}\in\operatorname{nCSM}^{3}(U)$.

###### Proof.

Make the change of variables $z=x+w$. Then the hypothesis becomes

$\displaystyle\delta$ $\displaystyle\leq\left\lvert\mathbb{E}_{x,y,w\in
V}b_{1}(x)b_{2}(y)b_{3}(x+w)b_{4}(x+y)b_{5}(w)b_{6}(x+y+w)b_{7}(y+w)(-1)^{T(x,x,y)+T(x,y,w)}\right\rvert$
$\displaystyle=\left\lvert\mathbb{E}_{w}b_{5}(w)\mathbb{E}_{x,y}(b_{1}(x)b_{3}(x+w))(b_{2}(y)b_{7}(y+w))(b_{4}(x+y)b_{6}(x+y+w))(-1)^{T(x,x,y)+T(x,y,w)}\right\rvert.$

By averaging (and the 1-boundedness of $b_{5}$), there exists $w_{0}\in V$
such that

$\delta\leq\left\lvert\mathbb{E}_{x,y}(b_{1}(x)b_{3}(x+w_{0}))(b_{2}(y)b_{7}(y+w_{0}))(b_{4}(x+y)b_{6}(x+y+w_{0}))(-1)^{T(x,x,y)+T(x,y,w_{0})}\right\rvert.$

Consider the map $A\colon V^{2}\to\mathbb{F}_{2}$ defined by
$A(x,y)=T(x,x,y)+T(x,y,w_{0})$. We claim that $A$ is bilinear. Using the
symmetry and trilinearity of $T$,

$\displaystyle A(x+x^{\prime},y)$
$\displaystyle=T(x+x^{\prime},x+x^{\prime},y)+T(x+x^{\prime},y,w_{0})$
$\displaystyle=T(x,x,y)+2T(x,x^{\prime},y)+T(x^{\prime},x^{\prime},y)+T(x,y,w_{0})+T(x^{\prime},y,w_{0})$
$\displaystyle=A(x,y)+A(x^{\prime},y).$

Similarly,

$\displaystyle A(x,y+y^{\prime})$
$\displaystyle=T(x,x,y+y^{\prime})+T(x,y+y^{\prime},w_{0})$
$\displaystyle=T(x,x,y)+T(x,x,y^{\prime})+T(x,y,w_{0})+T(x,y^{\prime},w_{0})$
$\displaystyle=A(x,y)+A(x,y^{\prime}).$

Since $A$ is bilinear, we can apply Lemma 4.1 to conclude that

$\delta^{8}\leq\mathbb{E}_{x,y}(-1)^{A(x,y)-A(y,x)}.$

Note that

$A(x,y)-A(y,x)=T(x,x,y)+T(x,y,w_{0})-T(y,y,x)-T(y,x,w_{0})=T(x,x,y)-T(x,y,y)$

by the symmetry of $T$.

Define $B\colon V^{2}\to\mathbb{F}_{2}$ by $B(x,y)=T(x,x,y)-T(x,y,y)$. Note
that $B$ is bilinear. (This can be seen since $B(x,y)=A(x,y)-A(y,x)$ or by
direct computation.) We have shown that $\operatorname{arank}B\leq
8\log_{2}(1/\delta)$. Let $U$ be the nullspace of $B$.333For the purpose of
the following argument we could take $U$ to either be the left or right
nullspace of $B$, though since $B$ is antisymmetric its left and right
nullspaces agree. This satisfies
$\operatorname{codim}U=\operatorname{prank}B=\operatorname{arank}B\leq
8\log_{2}(1/\delta)$. Furthermore $B(x,y)=0$ for all $x,y\in U$, which implies
that $T|_{U}$ is an $\operatorname{nCSM}$. ∎

This essentially implies the desired symmetrization result. To put the result
in the form we need, we use the following application of the Cauchy-Schwarz
inequality which will also appear again later in the paper.

######  Lemma 4.7.

Let $\phi\colon V^{3}\to\mathbb{F}_{p}$ be a triaffine form. Suppose there are
1-bounded functions $b_{1},\ldots,b_{7}\colon V\to\mathbb{C}$ such that

$\left\lvert\mathbb{E}_{x,y,z\in
V}b_{1}(x)b_{2}(y)b_{3}(z)b_{4}(x+y)b_{5}(x+z)b_{6}(y+z)b_{7}(x+y+z)\omega^{\phi(x,y,z)}\right\rvert\geq\delta.$

Let $T\colon V^{3}\to\mathbb{F}_{p}$ be the trilinear part of $\phi$. Then
there exist 1-bounded functions $b^{\prime}_{1},\ldots,b^{\prime}_{7}\colon
V\to\mathbb{C}$ such that

$\left\lvert\mathbb{E}_{x,y,z\in
V}b^{\prime}_{1}(x)b^{\prime}_{2}(y)b^{\prime}_{3}(z)b^{\prime}_{4}(x+y)b^{\prime}_{5}(x+z)b^{\prime}_{6}(y+z)b^{\prime}_{7}(x+y+z)\omega^{T(x,y,z)}\right\rvert\geq\delta^{8}.$

###### Proof.

This result follows from three applications of the Cauchy-Schwarz inequality,
one in each variable. We have

$\displaystyle\delta^{2}$
$\displaystyle\leq\left\lvert\mathbb{E}_{y,z}b_{2}(y)b_{3}(z)b_{6}(y+z)\mathbb{E}_{x}b_{1}(x)b_{4}(x+y)b_{5}(x+z)b_{7}(x+y+z)\omega^{\phi(x,y,z)}\right\rvert^{2}$
$\displaystyle\leq\left(\mathbb{E}_{y,z}\left\lvert
b_{2}(y)b_{3}(z)b_{6}(y+z)\right\rvert^{2}\right)$
$\displaystyle\qquad\qquad\cdot\left(\mathbb{E}_{x,x^{\prime},y,z}b_{1}(x)\overline{b_{1}(x^{\prime})}b_{4}(x+y)\overline{b_{4}(x^{\prime}+y)}b_{5}(x+z)\overline{b_{5}(x^{\prime}+z)}\right.$
$\displaystyle\qquad\qquad\qquad\cdot\left.b_{7}(x+y+z)\overline{b_{7}(x^{\prime}+y+z)}\omega^{\phi(x,y,z)-\phi(x^{\prime},y,z)}\right).$

The 1-boundedness of $b_{2},b_{3},b_{6}$ lets us remove the first term.

The next application of the Cauchy-Schwarz inequality gives

$\displaystyle\delta^{4}$
$\displaystyle\leq\mathbb{E}_{x,x^{\prime},y,y^{\prime},z}b_{4}(x+y)\overline{b_{4}(x^{\prime}+y)}\overline{b_{4}(x+y^{\prime})}{b_{4}(x^{\prime}+y^{\prime})}b_{7}(x+y+z)\overline{b_{7}(x^{\prime}+y+z)}$
$\displaystyle\qquad\qquad\cdot\overline{b_{7}(x+y^{\prime}+z)}{b_{7}(x^{\prime}+y^{\prime}+z)}\omega^{\phi(x,y,z)-\phi(x^{\prime},y,z)-\phi(x,y^{\prime},z)+\phi(x^{\prime},y^{\prime},z)}.$

The third gives

$\displaystyle\delta^{8}$
$\displaystyle\leq\mathbb{E}_{x,x^{\prime},y,y^{\prime},z,z^{\prime}}b_{7}(x+y+z)\overline{b_{7}(x^{\prime}+y+z)}\overline{b_{7}(x+y^{\prime}+z)}{b_{7}(x^{\prime}+y^{\prime}+z)}$
$\displaystyle\qquad\qquad\cdot\overline{b_{7}(x+y+z^{\prime})}{b_{7}(x^{\prime}+y+z^{\prime})}{b_{7}(x+y^{\prime}+z^{\prime})}\overline{b_{7}(x^{\prime}+y^{\prime}+z^{\prime})}$
$\displaystyle\qquad\qquad\cdot\omega^{\phi(x,y,z)-\phi(x^{\prime},y,z)-\phi(x,y^{\prime},z)+\phi(x^{\prime},y^{\prime},z)-\phi(x,y,z^{\prime})+\phi(x^{\prime},y,z^{\prime})+\phi(x,y^{\prime},z^{\prime})-\phi(x^{\prime},y^{\prime},z^{\prime})}.$

Since $T$ is defined to be the trilinear part of the triaffine form $\phi$,
the exponent in the last term is equal to
$T(x-x^{\prime},y-y^{\prime},z-z^{\prime})$. Applying a change of variables
and averaging, we can find $x_{0},y_{0},z_{0}$ such that

$\displaystyle\delta^{8}$
$\displaystyle\leq\mathbb{E}_{x,y,z}b_{7}(x+y+z+x_{0}+y_{0}+z_{0})\overline{b_{7}(y+z+x_{0}+y_{0}+z_{0})}\overline{b_{7}(x+z+x_{0}+y_{0}+z_{0})}{b_{7}(z+x_{0}+y_{0}+z_{0})}$
$\displaystyle\qquad\qquad\cdot\overline{b_{7}(x+y+x_{0}+y_{0}+z_{0})}{b_{7}(y+x_{0}+y_{0}+z_{0})}{b_{7}(x+x_{0}+y_{0}+z_{0})}\overline{b_{7}(x_{0}+y_{0}+z_{0})}\omega^{T(x,y,z)}.$

This expression is of the desired form, completing the proof. ∎

######  Corollary 4.8.

Let $T\colon V^{3}\to\mathbb{F}_{2}$ be a trilinear form and let
$b_{1},\ldots,b_{7}\colon V\to\mathbb{C}$ be 1-bounded functions such that

$\left\lvert\mathbb{E}_{x,y,z\in
V}b_{1}(x)b_{2}(y)b_{3}(z)b_{4}(x+y)b_{5}(x+z)b_{6}(y+z)b_{7}(x+y+z)(-1)^{T(x,y,z)}\right\rvert\geq\delta.$

Then there exists $S\in\operatorname{nCSM}^{3}(V)$ such that
$\operatorname{prank}(T-S)\leq 432\log_{2}(1/\delta)$.

###### Proof.

By 4.2, we have that $\operatorname{arank}(T-T_{\pi})\leq
16\log_{2}(1/\delta)$ for all $\pi\in\mathfrak{S}_{3}$. Then applying 4.3,
there exists a subspace $U\leq V$ with $\operatorname{codim}U\leq
80\log_{2}(1/\delta)$ such that $T|_{U}\colon U^{3}\to\mathbb{F}_{2}$ is a
symmetric trilinear form.

By averaging, there exist cosets $x_{0}+U,y_{0}+U,z_{0}+U$ such that

$\displaystyle\delta$
$\displaystyle\leq\left\lvert\mathbb{E}_{\begin{subarray}{c}x\in x_{0}+U\\\
y\in y_{0}+U\\\ z\in
z_{0}+U\end{subarray}}b_{1}(x)b_{2}(y)b_{3}(z)b_{4}(x+y)b_{5}(x+z)b_{6}(y+z)b_{7}(x+y+z)(-1)^{T(x,y,z)}\right\rvert$
$\displaystyle=\left\lvert\mathbb{E}_{x,y,z\in
U}b^{\prime}_{1}(x)b^{\prime}_{2}(y)b^{\prime}_{3}(z)b^{\prime}_{4}(x+y)b^{\prime}_{5}(x+z)b^{\prime}_{6}(y+z)b^{\prime}_{7}(x+y+z)(-1)^{T(x_{0}+x,y_{0}+y,z_{0}+z)}\right\rvert.$

Now by Lemma 4.7, this implies that

$\delta^{8}\leq\left\lvert\mathbb{E}_{x,y,z\in
U}b^{\prime\prime}_{1}(x)b^{\prime\prime}_{2}(y)b^{\prime\prime}_{3}(z)b^{\prime\prime}_{4}(x+y)b^{\prime\prime}_{5}(x+z)b^{\prime\prime}_{6}(y+z)b^{\prime\prime}_{7}(x+y+z)(-1)^{T(x,y,z)}\right\rvert.$

We can now apply 4.6 to find a subspace $W\leq U$ such that
$\operatorname{codim}_{V}W\leq 144\log_{2}(1/\delta)$ and such that
$T|_{W}\in\operatorname{nCSM}^{3}(W)$. To conclude, all that remains is to
extend $T|_{W}$ to an $\operatorname{nCSM}$ on all of $V$. To do this we first
pick an arbitrary decomposition $V=W\oplus W^{\prime}$. For
$w_{1},w_{2},w_{3}\in W$ and $w_{1}^{\prime},w_{2}^{\prime},w_{3}^{\prime}\in
W^{\prime}$ define
$S(w_{1}+w^{\prime}_{1},w_{2}+w^{\prime}_{2},w_{3}+w^{\prime}_{3})=T|_{W}(w_{1},w_{2},w_{3})$.
Clearly $S\in\operatorname{nCSM}^{3}(V)$ and $\operatorname{prank}(T-S)\leq
3\operatorname{codim}_{V}W\leq 432\log_{2}(1/\delta)$ by Lemma 2.4. ∎

##  5 Proof of main theorem

We now have the tools to prove the main theorem. We first record the result of
Gowers and Milićević that we require. This result can be found at the
beginning of Section 10 of [5] and its proof takes up the bulk of that paper.

######  Theorem 5.1 ([5]).

Fix a prime $p$ and $\delta>0$. There exists

$\epsilon=\left(\exp\left(\exp\left(\exp\left(O_{p}\left(\log(1/\delta)^{O(1)}\right)\right)\right)\right)\right)^{-1}$

such that the following holds. Let $V$ be a finite-dimensional
$\mathbb{F}_{p}$-vector space. Given a function $f\colon V\to\mathbb{C}$
satisfying $\|f\|_{\infty}\leq 1$ and $\|f\|_{U^{4}}>\delta$, there exists a
triaffine form $\phi\colon V^{3}\to\mathbb{F}_{p}$ such that

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}\in
V}(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}f)(x)\omega^{\phi(h_{1},h_{2},h_{3})}\right\rvert\geq\epsilon.$
(5.1)

###### Proof of 1.3.

Applying 5.1, we find a triaffine form $\phi\colon V^{3}\to\mathbb{F}_{p}$
such that

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}\in
V}(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}f)(x)\omega^{\phi(h_{1},h_{2},h_{3})}\right\rvert\geq\epsilon.$

Let $T$ be the trilinear part of $\phi$. First average over $x$ and expand out
the derivative to conclude that there exist 1-bounded functions
$b_{1},\ldots,b_{7}\colon V\to\mathbb{C}$ such that

$\left\lvert\mathbb{E}_{x,y,z\in
V}b_{1}(x)b_{2}(y)b_{3}(z)b_{4}(x+y)b_{5}(x+z)b_{6}(y+z)b_{7}(x+y+z)\omega^{\phi(x,y,z)}\right\rvert\geq\epsilon.$

Then applying Lemma 4.7 and then 4.2, we see that
$\operatorname{arank}(T-T_{\pi})\leq 128\log_{p}(1/\epsilon)$ for all
$\pi\in\mathfrak{S}_{3}$. By 2.3, we conclude that
$\operatorname{prank}(T-T_{\pi})\leq O_{p}(\log_{p}(1/\epsilon)^{O(1)})$ for
all $\pi\in\mathfrak{S}_{3}$.

Now for $p\geq 3$, we can apply 4.5 to find $S\in\operatorname{CSM}^{3}(V)$
such that $\operatorname{prank}(T-S)\leq O_{p}(\log_{p}(1/\epsilon)^{O(1)})$.
Then by 3.3, there exists a classical cubic polynomial
$P\in\operatorname{Poly}_{\leqslant 3}(V\to\mathbb{F}_{p})$ such that
$d^{3}P=S$. For $p=2$, we apply 4.8 to find $S\in\operatorname{nCSM}^{3}(V)$
such that $\operatorname{prank}(T-S)\leq O_{p}(\log_{p}(1/\epsilon)^{O(1)})$.
Then by 3.5, there exists a non-classical cubic polynomial
$P\in\operatorname{Poly}_{\leqslant 3}(V\to\mathbb{R}/\mathbb{Z})$ such that
$d^{3}P=S$.

In either case we define
$r=\max\\{\operatorname{prank}(T-S),\log_{p}(1/\epsilon)\\}$ satisfying $r\leq
O_{p}(\log_{p}(1/\epsilon)^{O(1)})$. We have a decomposition
$T-S=\sum_{i=1}^{r}\gamma_{i}$ where each $\gamma_{i}\colon
V^{3}\to\mathbb{F}_{p}$ is the product of a linear form in one of the
variables with a bilinear form in the other two variables. Write $\Gamma\colon
V^{3}\to\mathbb{F}_{p}^{2r}$ for the list of $2r$ linear and bilinear forms
that factor the $\gamma_{i}$’s. For example, if
$\gamma_{1}(x,y,z)=\alpha(x)\beta(y,z)$, then we define
$\Gamma_{1}(x,y,z)=\alpha(x)$ and $\Gamma_{2}(x,y,z)=\beta(y,z)$.

Define $g=f\omega^{P}$. Note that
$(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)=(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}f)(x)\omega^{S(h_{1},h_{2},h_{3})}$.
The inequality we started with can now be written as

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}}b(h_{1},h_{2},h_{3})\omega^{\sum_{i=1}^{r}\gamma_{i}(h_{1},h_{2},h_{3})}(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)\right\rvert\geq\epsilon$

where $b(h_{1},h_{2},h_{3})$ is a product of linear and bilinear phase
functions. We first remove the middle term.

By averaging, there exists $c\in\mathbb{F}_{p}^{2r}$ such that

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}}b(h_{1},h_{2},h_{3})\mathbbm{1}(\Gamma(h_{1},h_{2},h_{3})=c)(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)\right\rvert\geq\epsilon
p^{-2r}.$ (5.2)

Now by Fourier analysis, we can write

$\mathbbm{1}(\Gamma(h_{1},h_{2},h_{3})=c)=\mathbb{E}_{\xi\in\mathbb{F}_{p}^{2r}}\omega^{(\Gamma(h_{1},h_{2},h_{3})-c)\cdot\xi}.$

Plugging this expression into Eq. 5.2, by averaging again, there exists
$\xi_{0}\in\mathbb{F}_{p}^{2r}$ such that

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}}b(h_{1},h_{2},h_{3})\omega^{(\Gamma(h_{1},h_{2},h_{3})-c)\cdot\xi_{0}}(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)\right\rvert\geq\epsilon
p^{-2r}.$

Since $\Gamma$ is a list of linear and bilinear forms, the middle term,
$\omega^{(\Gamma(h_{1},h_{2},h_{3})-c)\cdot\xi_{0}}$, is a product of linear
and bilinear phase functions. Thus we conclude that

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}}b^{\prime}(h_{1},h_{2},h_{3})(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)\right\rvert\geq\epsilon
p^{-2r},$

where $b^{\prime}$ is a product of linear and bilinear phase functions.

Write

$b^{\prime}(h_{1},h_{2},h_{3})=\omega^{\beta_{1}(h_{2},h_{3})+\beta_{2}(h_{1},h_{3})+\beta_{3}(h_{1},h_{2})+\alpha_{1}(h_{1})+\alpha_{2}(h_{2})+\alpha_{3}(h_{3})}.$

By averaging, there exist $x,h_{1}\in V$ such that

$\left\lvert\mathbb{E}_{h_{2},h_{3}}b^{\prime}(h_{1},h_{2},h_{3})(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)\right\rvert\geq\epsilon
p^{-2r}.$

Expanding this out, we see that this inequality is of the correct form to
apply Lemma 4.1 which then implies that
$\operatorname{arank}(\beta_{1}-(\beta_{1})_{(12)})\leq
8(2r+\log_{p}(1/\epsilon))\leq 24r$. The same is true of
$\beta_{2},\beta_{3}$.

The next step is to find
$\beta^{\prime}_{1},\beta^{\prime}_{2},\beta^{\prime}_{3}\in\operatorname{nCSM}^{2}(V)$
such that $\operatorname{prank}(\beta_{i}-\beta^{\prime}_{i})\leq 48r$. First
note that for bilinear forms we have
$\operatorname{arank}(\beta_{i})=\operatorname{prank}(\beta_{i})$. Thus there
is a subspace $U\leq V$ of codimension at most $24r$ such that
$\beta_{i}-(\beta_{i})_{(12)}$ vanishes on $U\times U$. This is the only
condition we need to imply that $\beta_{i}|_{U}$ is an nCSM. Extending
$\beta_{i}|_{U}$ to $\beta_{i}^{\prime}\in\operatorname{nCSM}^{2}(V)$
arbitrarily, we have $\operatorname{prank}(\beta_{i}-\beta_{i}^{\prime})\leq
48r$ by Lemma 2.4. Finally, by 3.3, there exist non-classical quadratic
polynomials $Q_{i}\in\operatorname{Poly}_{\leqslant
2}(V\to\mathbb{R}/\mathbb{Z})$ such that $d^{2}Q_{i}=\beta^{\prime}_{i}$.

We repeat the same averaging and Fourier analysis argument from above to
conclude that

$\left\lvert\mathbb{E}_{x,h_{1},h_{2},h_{3}}b^{\prime\prime}(h_{1},h_{2},h_{3})\omega^{\beta^{\prime}_{1}(h_{2},h_{3})+\beta^{\prime}_{2}(h_{1},h_{3})+\beta^{\prime}_{3}(h_{1},h_{2})}(\partial_{h_{1}}\partial_{h_{2}}\partial_{h_{3}}g)(x)\right\rvert\geq\epsilon
p^{-290r},$

where $b^{\prime\prime}(h_{1},h_{2},h_{3})$ is a product just of linear phase
functions, say
$b^{\prime\prime}(h_{1},h_{2},h_{3})=\omega^{L_{1}(h_{1})+L_{2}(h_{2})+L_{3}(h_{3})}$.

Now define

$\displaystyle g_{000}$ $\displaystyle=f\omega^{P},$ $\displaystyle g_{001}$
$\displaystyle=f\omega^{P+Q_{1}},$ $\displaystyle g_{010}$
$\displaystyle=f\omega^{P+Q_{2}},$ $\displaystyle g_{011}$
$\displaystyle=f\omega^{P+Q_{1}+Q_{2}+L_{3}},$ $\displaystyle g_{100}$
$\displaystyle=f\omega^{P+Q_{3}},$ $\displaystyle g_{101}$
$\displaystyle=f\omega^{P+Q_{1}+Q_{3}+L_{2}},$ $\displaystyle g_{110}$
$\displaystyle=f\omega^{P+Q_{2}+Q_{3}+L_{1}},$ $\displaystyle g_{111}$
$\displaystyle=f\omega^{P+Q_{1}+Q_{2}+Q_{3}+L_{1}+L_{2}+L_{3}}.$

One can verify that the inequality above can be rewritten as

$\begin{split}\left|\mathbb{E}_{x,h_{1},h_{2},h_{3}}\overline{g_{000}(x)}\right.&{g_{001}(x+h_{1})g_{010}(x+h_{2})}\overline{g_{011}(x+h_{1}+h_{2})}{g_{100}(x+h_{3})}\\\
&\left.\cdot\overline{g_{101}(x+h_{1}+h_{3})g_{110}(x+h_{2}+h_{3})}{g_{111}(x+h_{1}+h_{2}+h_{3})}\right|\geq\epsilon
p^{-290r}.\end{split}$

The left-hand side is in the correct form to apply the Gowers-Cauchy-Schwarz
inequality [18, Lemma B.1.(iv)], which implies that
$\|f\omega^{P}\|_{U^{3}}\geq\epsilon p^{-290r}\geq
p^{-O_{p}(\log_{p}(1/\epsilon)^{O(1)})}$. Finishing with a single application
of the $U^{3}$-inverse theorem, we prove the desired result. ∎

##  6 Concluding remarks

The symmetrization results in this paper are very specific to trilinear forms.
We pose the following conjecture, which seems surprisingly difficult and may
be of independent interest.

######  Conjecture 6.1.

Let $T\colon V^{k}\to\mathbb{F}_{p}$ be a $k$-linear form. Suppose that $T$ is
close to symmetric in the sense that $\operatorname{prank}(T-T_{\pi})\leq r$
for all permutations $\pi\in\mathfrak{S}_{k}$. Then there exists a symmetric
$k$-linear form $S\colon V^{k}\to\mathbb{F}_{p}$ such that
$\operatorname{prank}(S-T)\leq r^{\prime}$ where $r^{\prime}$ is a polynomial
function of $r$.

Note that this conjecture is only interesting if $p\leq k$, since for $p>k$,
the symmetric form $S=\frac{1}{k!}\sum_{\pi}T_{\pi}$ can easily be seen to
satisfy $\operatorname{prank}(S-T)\leq k!r$.

Combined with the recent work of Gowers and Milićević [6], 6.1 is the missing
ingredient which would suffice to give quantitative bounds on the
$U^{p+1}$-inverse theorem. For the $U^{k+1}$-inverse theorem with $k>p$ there
is additional symmetrization necessary to produce a non-classical symmetric
multilinear form from a symmetric form.

######  Conjecture 6.2.

Let $T\colon V^{k}\to\mathbb{F}_{p}$ be a $k$-linear form. Suppose that $T$
satisfies

$\left\lvert\mathbb{E}_{x_{1},\ldots,x_{k}}\prod_{I\subseteq[k]}b_{I}\left(\sum_{i\in
I}x_{i}\right)\omega^{T(x_{1},\ldots,x_{k})}\right\rvert\geq\epsilon$

for some 1-bounded functions $b_{I}\colon V\to\mathbb{C}$. Then there exists
$S\in\operatorname{nCSM}^{k}(V)$ such that $\operatorname{prank}(S-T)\leq
r^{\prime}$ where $r^{\prime}$ is a polynomial function of $\log(1/\epsilon)$.

Combined with the work of Gowers and Milićević and this paper’s integration
result, 3.5, this result would give quantitative bounds on the
$U^{k+1}$-inverse theorem over $\mathbb{F}_{p}^{n}$ for all $k,p$.

######  Remark.

After the first version of this paper appeared on arXiv, Milićević gave a
counterexample disproving 6.1 in the case of $k=4$ and $p=2$ [16]. It remains
an interesting question to determine for which pairs $(k,p)$ 6.1 holds as well
as to determine the validity of 6.2.

## Acknowledgments

The author thanks Aaron Berger for many helpful discussions.

## References

  * [1] Karim Adiprasito, David Kazhdan, and Tamar Ziegler, _On the Schmidt and analytic ranks for trilinear forms_ , arXiv:2102.03659.
  * [2] Vitaly Bergelson, Terence Tao, and Tamar Ziegler, _An inverse theorem for the uniformity seminorms associated with the action of $\mathbb{F}^{\infty}_{p}$_, Geom. Funct. Anal. 19 (2010), 1539–1596.
  * [3] Aaron Berger, Ashwin Sah, Mehtaab Sawhney, and Jonathan Tidor, _Non-classical polynomials and the inverse theorem_ , Math. Proc. Cambridge Philos. Soc. (2021), 1–13.
  * [4] Alex Cohen and Guy Moshkovitz, _Structure vs. randomness for bilinear maps_ , Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC 2021), ACM, New York, 2021, pp. 800–808.
  * [5] W. T. Gowers and L. Milićević, _A quantitative inverse theorem for the $U^{4}$ norm over finite fields_, arXiv:1712.00241.
  * [6] W. T. Gowers and L. Milićević, _An inverse theorem for Freiman multi-homomorphisms_ , arXiv:2002.11667.
  * [7] Ben Green, _Montréal notes on quadratic Fourier analysis_ , Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 69–102.
  * [8] Ben Green and Terence Tao, _The distribution of polynomials over finite fields, with applications to the Gowers norms_ , Contrib. Discrete Math. 4 (2009), 1–36.
  * [9] Ben Green, Terence Tao, and Tamar Ziegler, _An inverse theorem for the Gowers $U^{4}$-norm_, Glasg. Math. J. 53 (2011), 1–50.
  * [10] Ben Green, Terence Tao, and Tamar Ziegler, _An inverse theorem for the Gowers $U^{s+1}[N]$-norm_, Ann. of Math. (2) 176 (2012), 1231–1372.
  * [11] Oliver Janzer, _Polynomial bound for the partition rank vs the analytic rank of tensors_ , Discrete Anal. (2020), Paper No. 7, 18.
  * [12] Shachar Lovett, _The analytic rank of tensors and its applications_ , Discrete Anal. (2019), Paper No. 7, 10.
  * [13] Shachar Lovett, Roy Meshulam, and Alex Samorodnitsky, _Inverse conjecture for the Gowers norm is false_ , Theory Comput. 7 (2011), 131–145.
  * [14] Freddie Manners, _Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups_, arXiv:1811.00718.
  * [15] Luka Milićević, _Polynomial bound for partition rank in terms of analytic rank_ , Geom. Funct. Anal. 29 (2019), 1503–1530.
  * [16] Luka Milićević, _Approximately symmetric forms far from being exactly symmetric_ , arXiv:2112.14755.
  * [17] Terence Tao and Tamar Ziegler, _The inverse conjecture for the Gowers norm over finite fields via the correspondence principle_ , Anal. PDE 3 (2010), 1–20.
  * [18] Terence Tao and Tamar Ziegler, _The inverse conjecture for the Gowers norm over finite fields in low characteristic_ , Ann. Comb. 16 (2012), 121–188.

[jtidor] Jonathan Tidor
Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
jtidormitedu