# On the Schmidt and analytic ranks for trilinear forms

Karim Adiprasito, David Kazhdan and Tamar Ziegler

###### Abstract.

We discuss relations between different notions of ranks for multilinear forms.
In particular we show that the Schmidt and the analytic ranks for trilinear
forms are essentially proportional.

K. A. is supported by ERC grant 716424 - CASe and ISF Grant 1050/16. T. Z. is
supported by ERC grant ErgComNum 682150 and ISF grant 2112/20.

##  1\. Introduction

In recent years there is growing interest in properties of polynomials which
are independent of the number of variables. To study such properties various
notions were introduced for measuring the complexity of polynomials. In this
paper we compare three notion of rank of polynomials $P$ over a field
${\mathbf{k}}$ \- the Schmidt rank $r_{\mathbf{k}}(P)$, the slice rank
$s_{\mathbf{k}}(P)$ both defined for arbitrary fields ${\mathbf{k}}$, and the
analytic rank $a_{\mathbf{k}}(P)$ defined for finite fields
${\mathbf{k}}=\mathbb{F}_{q}$.

######  Definition 1.1.

Let ${\mathbf{k}}$ be a field, $\mathbf{V}$ be a finite-dimensional k-vector
space, $V=\mathbf{V}({\mathbf{k}})$, let
$\mathcal{S}_{{\mathbf{k}}}=\bigoplus_{d\geq 0}\mathcal{S}^{d}_{{\mathbf{k}}}$
be the graded algebra of polynomial functions on $\mathbf{V}$ and let
$P\in\mathcal{S}^{d}_{\mathbf{k}}$.

  1. (1)

The _Schmidt rank_ $r_{\mathbf{k}}(P)$ is the minimal number $r$ such that $P$
can be written in the form $P=\sum_{i=1}^{r}Q_{i}R_{i}$, where
$Q_{i},R_{i}\in\mathcal{S}$ are polynomials on $V$ of degrees $<d$.

  2. (2)

The _slice rank_ $s_{\mathbf{k}}(P)$ is the minimal number $r$ such that $P$
can be written in the form $P=\sum_{i=1}^{r}Q_{i}R_{i}$, where
$\deg(Q_{i})=1$.

  3. (3)

In the case when ${\mathbf{k}}=\mathbb{F}_{q}$ is a finite field and
$\psi:{\mathbf{k}}\to{\mathbb{C}}^{\star}$ a non-trivial additive character we
write $A_{{\mathbf{k}},\psi}(P):=\frac{\sum_{v\in
V}\psi(P(v))}{q^{\mathrm{dim}(V)}}$. We define the analytic rank
$a_{{\mathbf{k}},\psi}(P):=-\log_{q}(|A_{\mathbb{F}_{q},\psi}(P)|)$

These three notions of complexity of polynomials play an important role in
many problems in number theory, additive combinatorics, and algebraic
geometry. The Schmidt rank, also called the $h$-invariant, was first
introduced by Schmidt in his paper on integer points in varieties defined over
the rationals. In his paper Schmidt showed that over the complex field the
Schmidt rank of a polynomials is proportional to the codimension of the
singular locus of the associated variety. The same notion of complexity was
reintroduced in the work of Ananyan and Hochster [1] as the strength of a
polynomial and was used in the proof of the Stillman conjecture. The notion of
slice rank played an important role in the arguments for the cap set problem
in combinatorics [3, 6, 15]. Finally the notion of analytic rank for
polynomials over finite fields was introduced in Gower and Wolf [10] as a tool
for studying the CS-complexity of systems of linear equations.

######  Remark 1.2.

It is clear that $r_{\mathbf{k}}(P)\leq s_{\mathbf{k}}(P)$ and
$r_{\mathbf{k}}(P)=s_{\mathbf{k}}(P)$ if $d\leq 3.$

In this paper we consider only the case of multilinear polynomials. Namely
$V=V_{1}\times\dots\times V_{d}$ is a product of ${\mathbf{k}}$-vector spaces
and $P:V_{1}\times\dots\times V_{d}\to{\mathbf{k}}$ is a multilinear
polynomial.

We introduce two additional definitions.

######  Definition 1.3.

  1. (1)

In the case when $V=V_{1}\times\dots\times V_{d}$ is a product of
${\mathbf{k}}$-vector spaces and $P:V_{1}\times\dots\times
V_{d}\to{\mathbf{k}}$ is a multilinear polynomial we define a
${\mathbf{k}}$-subvariety
${\mathbf{Z}}_{P}\subset\mathbf{V}_{2}\times\dots\times\mathbf{V}_{d}$ by
${\mathbf{Z}}_{P}:=\\{(v_{2},\dots,v_{d})\in\mathbf{V}_{2}\times\dots\times\mathbf{V}_{d}|P(v_{1},v_{2},\dots,v_{d})=0,\forall
v_{1}\in V_{1}\\}$.

  2. (2)

For a subspace $L\subset(V_{1}\otimes\ldots\otimes V_{d})^{\vee}$ we define
$r_{\mathbf{k}}(L)$ as the minimum of $\sum_{i=1}^{d}\mathrm{codim}(W_{i})$
over the set of subspaces $W_{i}\subset V_{i}$, $i=1,\ldots,m$, such that
$l_{|W_{1}\otimes\ldots\otimes W_{d}}\equiv 0$ for all $l\in L$.

######  Claim 1.4.

For a multilinear polynomial $P:V_{1}\times\dots\times V_{d}\to{\mathbf{k}}$
the slice rank $s_{{\mathbf{k}}}(P)$ is equal to the minimum of
$\sum_{i=1}^{d}\operatorname{codim}_{V_{i}}(W_{i})$ where $W_{i}\subset
V_{i},1\leq i\leq d$ are subspaces such that $P_{|W_{1}\times\dots\times
W_{d}}\equiv 0$. It follows that if $L$ is one dimensional $L={\mathbf{k}}P$
then $r_{{\mathbf{k}}}(L)=s_{{\mathbf{k}}}(P)$.

######  Lemma 1.5.

In the case when ${\mathbf{k}}=\mathbb{F}_{q}$ we have
$A_{{\mathbf{k}},\psi}(P)=\frac{|{\mathbf{Z}}_{P}(\mathbb{F}_{q})|}{q^{\sum_{i=2}^{d}\mathrm{dim}(V_{i})}}$.

###### Proof.

For $(v_{2},\dots,v_{n})\in V_{2}\times\ldots\times V_{d}$ we define
$\psi_{v_{2},\dots,v_{n}}:V_{1}\to{\mathbb{C}}^{\star}$ by

$\psi_{v_{2},\dots,v_{n}}(v_{1}):=\psi(P(v_{1},v_{2},\dots,v_{n})).$

It is clear that $\psi_{v_{2},\dots,v_{n}}$ is an additive character of
$V_{1}$ which trivial if and only if
$(v_{2},\dots,v_{n})\in{\mathbf{Z}}_{P}(\mathbb{F}_{q})$. Therefore
$\sum_{v_{1}\in V_{1}}\psi_{v_{2},\dots,v_{n}}(v_{1})=0$ if
$v_{1}\not\in{\mathbf{Z}}_{P}(\mathbb{F}_{q})$ and $\sum_{v_{1}\in
V_{1}}\psi_{v_{2},\dots,v_{n}}(v_{1})=q^{\mathrm{dim}(V_{1})}$ if
$v_{1}\in{\mathbf{Z}}_{P}(\mathbb{F}_{q})$. ∎

######  Corollary 1.6.

The analytic rank of a multilinear polynomial $P$ does not depend on the
choice of the non-trivial additive character $\psi$. We will denote it by
$a_{\mathbf{k}}(P)$.

Fro a field ${\mathbf{k}}$ we denote by $\bar{\mathbf{k}}$ the algebraic
closure. Obviously we have $r_{\bar{\mathbf{k}}}(P)\leq r_{\mathbf{k}}(P)$. We
conjecture that the reverse inequality is essentially holds as well.

######  Conjecture 1.7 (d).

For any $d\geq 2$ there exists $\kappa_{d}>0$ such that
$r_{\mathbf{k}}(P)\leq\kappa_{d}r_{\bar{\mathbf{k}}}(P)$ for any field
${\mathbf{k}}$ and a multilinear ${\mathbf{k}}$-polynomial $P$ of degree $d$,
where $\bar{\mathbf{k}}$ is the algebraic closure of ${\mathbf{k}}$.

######  Remark 1.8.

It is easy to see that Conjecture 1.7 holds for $d=2$ with $\kappa_{2}=1$.

In [4] Derksen introduced the notion of the $G$-stable rank of a multilinear
polynomial (denoted $r_{{\mathbf{k}}}^{G}(P)$) defined in terms of Geometric
Invariant Theory and proved the following

  1. (1)

$r_{{\mathbf{k}}}^{G}(P)=r_{\bar{\mathbf{k}}}^{G}(P)$ and

  2. (2)

$\frac{2s_{\mathbf{k}}(P)}{d}\leq r_{{\mathbf{k}}}^{G}(P)\leq
s_{\mathbf{k}}(P)$

This immediately implies the inequality
$s_{\mathbf{k}}(P)\leq\frac{3}{2}s_{\bar{\mathbf{k}}}(P)$ for trilinear
polynomials $P$. Since $s_{\mathbf{k}}(P)=r_{\mathbf{k}}(P)$ for trilinear
polynomials $P$ we obtain the following result.

######  Theorem 1.9 (Dersken).

For trilinear polynomials we have
$r_{\mathbf{k}}(P)\leq\frac{3}{2}r_{\bar{\mathbf{k}}}(P)$.

Quantitative estimates for the relation between analytic rank and Schmidt rank
provide a means for obtaining quantitative bounds for important problems in
additive combinatorics number theory and algebraic geometry.

We conjecture the following quantitative relation between the Schmidt rank and
analytic rank.

######  Conjecture 1.10 (d).

For any $d\geq 2$ there exists $c_{d}>0$ such that $r_{\mathbb{F}_{q}}(P)\leq
c_{d}a_{\mathbb{F}_{q}}(P)$ for any multilinear polynomial of degree $d$.

######  Remark 1.11.

  1. (1)

The inequality $a_{\mathbb{F}_{q}}(P)\leq r_{\mathbb{F}_{q}}(P)$ is known. See
[8, 11].

  2. (2)

It is easy to see that Conjecture 1.10 holds for $d=2$ with $c_{2}=1$.

  3. (3)

In [13] it was shown that there exist constants $c_{d},e_{d}$ such that for
any multilinear polynomial of degree $d$ we have $r_{\mathbb{F}_{q}}(P)\leq
c_{d}(a_{\mathbb{F}_{q}}(P))^{e_{d}}$ (earlier work [2] gave ineffective
bounds). Conjecture 1.10 states that one can take $e_{d}=1$.

In his paper [14], Schmidt proved the following result

######  Theorem 1.12 (Schmidt).

For any $d\geq 2$ there exists $D_{d}$ such that for any complex multilinear
polynomial $P:V_{1}\times\dots\times V_{d}\to{\mathbb{C}}$ we have
$r_{\mathbb{C}}(P)\leq D_{d}g$ where
$g:=\mathrm{codim}_{\mathbf{V}_{2}\times\dots\times\mathbf{V}_{d}}{\mathbf{Z}}_{P}$.

The main goal of this paper is the proof of the following result.

######  Theorem 1.13.

  1. (1)

Assuming the validity Theorem 1.12 for multilinear polynomials of degree $d$
over algebraically closed fields of finite characteristic we show that
Conjecture 1.7(d) implies the validity of Conjecture 1.10(d) for
${\mathbf{k}}=\mathbb{F}_{q},q>d$ with
$c_{d}=\frac{D_{d}\kappa_{d}}{1-\log_{q}(d)}$.

  2. (2)

Theorem 1.12 holds for multilinear polynomials of degree $3$ over
algebraically closed fields of characteristic $\geq 3$ with $D_{3}=2$.

The following statement follows now from Theorem 1.9(2).

######  Corollary 1.14.

Conjecture 1.10(3) holds with $c_{3}=3$.

The proof of Theorem 1.13 is based on an extension of the rough bound from
[5], and as such the method of proof provided for Conjecture 1.10(3) is
completely different that the approach taken in [2], [13].

The conjectured bound in 1.10 can be viewed as a special case of the
conjectured bounded for the inverse theorem for the Gowers norms over finite
fields. Let $f:V\to\mathbb{C}$. The Gowers uniformity norms of $f$, introduced
in the study of arithmetic progressions in subsets of the intergers, are
defined as follows

$\|f\|^{2^{d}}_{U_{d}}=\frac{1}{|V|^{d+1}}\sum_{x,,h_{1},\ldots,h_{d}\in
V}\Delta_{h_{d}}\ldots\Delta_{h_{1}}f(x)$

where $\Delta_{h}f(x)=f(x+h)\overline{f(x)}$.

######  Conjecture 1.15.

Let ${\mathbf{k}}=\mathbb{F}_{p}$, and let $d>p$. There exists a constant
$F=F(d,k)$ such that the following holds: for any $\delta>0$, any
${\mathbf{k}}$-vector space $V$, any $f:V\to\mathbb{C}$ satisfying
$\|f\|_{U_{d}}\geq\delta$, there exists a degree $d-1$ polynomial $P$ such
that $|\frac{1}{|V|}\sum_{x\in V}f(x)\psi(P(x))|\geq\delta^{F}$.

When $f=\psi(Q)$ where $Q$ is a polynomial of degree $d$, then
$\Delta_{h_{d}}\ldots\Delta_{h_{1}}f(x)=\psi(\tilde{Q}(h_{1},\ldots,h_{d}))$,
where $\tilde{Q}$ is the multilinear symmetric form associated with $Q$. Via
Fourier analysis the conjectured bound in 1.10 implies the validity of
Conjecture 1.15 in this case.

Finally we present a conjecture relating the rank of a subspace over a field
${\mathbf{k}}$ to its rank over the algebraic closure, and prove it in a
special case.

######  Conjecture 1.16.

There exists a constant $E_{d}$ such that $r_{\mathbf{k}}(L)\leq
E_{d}r_{\bar{\mathbf{k}}}(L)$ for any field ${\mathbf{k}}$ and a linear
subspace $L\subset(V_{1}\otimes\ldots\otimes V_{d})^{\vee}$.

######  Theorem 1.17.

Conjecture 1.16 holds for $d=2$, with $E_{d}=2$.

##  2\. Proof of Theorem 1.17

In this section we prove Theorem 1.17. We start the proof of with the
following result.

######  Proposition 2.1.

For any field ${\mathbf{k}}$ and a linear subspace
$L\subset\mathrm{Hom}_{\mathbf{k}}(W,V)$ there exists ${\mathbf{k}}$-subspaces
$W^{\prime}\subset W$ and $V^{\prime}\subset V$ such that
$\mathrm{dim}(W/W^{\prime})=\mathrm{dim}(V^{\prime})=\tilde{r}_{{\mathbf{k}}}(L)$
and $l(W^{\prime})\subset V^{\prime}$ for all $l\in L$ where
$\tilde{r}(L):=\max_{B\in L}r(B)$.

###### Proof.

We start with the follows result of Kronecker (see [9]).

######  Definition 2.2.

  1. (1)

Let $V,W$ be ${\mathbf{k}}$-vector spaces. A pair $A,B\in\mathrm{Hom}(V,W)$ is
irreducible is there is no non-trivial decomposition $V=V_{1}\oplus V_{2}$,
$W=W_{1}\oplus W_{2}$ such $A(V_{i}),B(V_{i})\subset W_{i}$, for $i=1,2$.

  2. (2)

For any $n\geq 0$ we denote by
$A_{n},B_{n}:{\mathbf{k}}^{n}\to{\mathbf{k}}^{n+1}$ maps such that
$A_{n}(e_{i})=f_{i}$, $B_{n}(e_{i})=f_{i+1}$ for $1\leq i\leq n$, and denote
by $A^{\prime}_{n},B^{\prime}_{n}:{\mathbf{k}}^{n+1}\to{\mathbf{k}}^{n}$ maps
such that $A^{\prime}_{n}(e_{i})=f_{i}$ for $1\leq i\leq n$,
$A_{n}(e_{n+1})=0$ and $B^{\prime}_{n}(e_{1})=0,B^{\prime}_{n}(e_{i+1})=f_{i}$
for $2\leq i\leq n+1$.

######  Claim 2.3.

Let $A,B\in\mathrm{Hom}(V,W)$ be an irreducible pair of linear maps between
are finite dimensional ${\mathbf{k}}$-vector spaces. Then there exist
automorphisms $T\in\mathrm{Aut}(V),S\in\mathrm{Aut}(W)$ such that either

  1. (1)

$A$ is an isomorphism or

  2. (2)

$B$ is an isomorphism or

  3. (3)

there exist automorphisms $T\in\mathrm{Aut}(V)$, $S\in\mathrm{Aut}(W)$ such
that $(SAT,SBT)=(A_{n},B_{n}),n=\dim(V)$ such that $(SAT,SBT)=(A_{n},B_{n})$
or

  4. (4)

there exist automorphisms $T\in\mathrm{Aut}(V)$, $S\in\mathrm{Aut}(W)$ such
that $(SAT,SBT)=(A^{\prime}_{n},B^{\prime}_{n}),n=\dim(W)$.

######  Lemma 2.4.

If $|{\mathbf{k}}|=\infty$ and $A,B\in\mathrm{Hom}_{\mathbf{k}}(V,W)$ linear
maps such that $r(A)\geq r(A+tB)$ for all $t\in{\mathbf{k}}$. Then
$B(\mathrm{Ker}(A))\subset\mathrm{Im}(A)$.

######  Remark 2.5.

The assumption that $|{\mathbf{k}}|=\infty$ is necessary. Indeed in the case
when ${\mathbf{k}}=\mathbb{F}_{q}$ we can take
$V=W={\mathbf{k}}[\mathbb{F}_{q}]$ with the basis
$\\{e_{x}\\}_{x\in\mathbb{F}_{q}}$, $A(e_{x})=xe_{x}$ and $B=Id$. Then
$r(A+tB)=r(A)=q-1$ for all $t\in{\mathbf{k}}$ but
$B(\mathrm{Ker}(A))\not\subset\mathrm{Im}(A)$.

###### Proof.

Consider first the case when a pair $(A,B)$ is irreducible. If
$\mathrm{Im}(A)=W$ or $\mathrm{Ker}(A)=\\{0\\}$ we obviously have the
inclusion $B(\mathrm{Ker}(A))\subset\mathrm{Im}(A)$. In cases $(1)$ and $(4)$
the map $A$ is onto and in the case $(3)$ the map $A$ is an imbedding. In the
case $(2)$ we have $\dim(V)=\dim(W)$ and either $A$ is onto or $r(A)<r(B)$. We
claim that the assumption that $r(A)<r(B)$ leads to a contradiction.

Let $S\subset{\mathbf{A}}$ be the subset of $s_{0}$ such that
$r(s_{0}A+B)<\max_{s\in\bar{\mathbf{k}}}r(sA+B)$. Then
$S({\mathbf{k}})\subset{\mathbf{k}}$ is finite. So $r(sA+B)\geq r(B)$ outside
a finite set of $s$, and since $|{\mathbf{k}}|=\infty$ and there exists
$s\in{\mathbf{k}}$ such that $r(A+s^{-1}B)\geq r(B)>r(A)$, which is a
contradiction.

Now consider the general case when a pair $(A,B)$ is a finite direct sum of
irreducible pairs $(A_{i},B_{i}),i\in I$. Since, as we have seen in the
previous paragraph, the condition $r(A_{i})\leq r(A_{i}+tB_{i})$ is
automatically true outside a finite set of $t\in{\mathbf{k}}$ and
$|{\mathbf{k}}|=\infty$ the assumption that $r(A)\geq r(A+tB)$ for all
$t\in{\mathbf{k}}$ implies that that $r(A_{i})\geq r(A_{i}+tB_{i})$ outside a
finite set of $t\in{\mathbf{k}},i\in I$. So
$B_{i}(\mathrm{Ker}(A_{i}))\subset\mathrm{Im}(A_{i}),i\in I$. Therefore
$B(\mathrm{Ker}(A))\subset\mathrm{Im}(A)$.

∎

######  Lemma 2.6.

Proposition 2.1 holds in the case when $|{\mathbf{k}}|=\infty$.

###### Proof.

Choose $A\in L$ such that $r(A)=\tilde{r}(L)$ and define
$W^{\prime}:=\mathrm{Ker}(A),V^{\prime}:=\mathrm{Im}(A)$. As follows from
Lemma 2.4 we have $l(W^{\prime})\subset V^{\prime}$ for all $l\in L$. But
$\mathrm{dim}(W/W^{\prime})=\mathrm{dim}(V^{\prime})=r(A)$. ∎

Now consider the case when the field ${\mathbf{k}}$ is finite. Let
$\mathrm{{\mathbf{G}}r}$ be the Grassmanian of subspaces $W^{\prime}\subset W$
of codimension $\tilde{r}(L)$, and let $\mathrm{{\mathbf{G}}r^{\prime}}$ be
the Grassmanian of subspaces $V^{\prime}\subset V$ of dimension $\tilde{r}(L)$
and
${\mathbf{X}}_{\tilde{r}_{K}(L)}\subset\mathrm{{\mathbf{G}}r}\times\mathrm{{\mathbf{G}}r^{\prime}}$
be the subvariety of pairs $(W^{\prime},V^{\prime})$ such that and
$l(W^{\prime})\subset V^{\prime}$ for all $l\in L$. It is clear that the
subvariety
${\mathbf{X}}\subset\mathrm{{\mathbf{G}}r}\times{\mathbf{G}}r^{\prime}$ is
closed. Therefore it is proper.

Let $K={\mathbf{k}}(t)$. Since the field $K$ is infinite it follows from Claim
2.6 that ${\mathbf{X}}(K)\neq\emptyset$. So there exists a rational
${\mathbf{k}}$-morphism $\hat{f}:{\mathbf{P}}^{1}\to{\mathbf{X}}$. Any such
morphism is regular outside a finite ${\mathbf{k}}$\- subset
$S\subset{\mathbf{P}}^{1}$. So we obtain a regular ${\mathbf{k}}$-morphism
$\tilde{f}:{\mathbf{P}}^{1}\setminus S\to{\mathbf{X}}$. Since ${\mathbf{X}}$
is proper, $\tilde{f}$ extends to a regular ${\mathbf{k}}$\- morphism
$f:{\mathbf{P}}^{1}\to{\mathbf{X}}$. Let
$(W^{\prime}_{0},V^{\prime}_{0}):=f(0)\in{\mathbf{X}}({\mathbf{k}})$. By
definition of the variety ${\mathbf{X}}$ we see that $l(W^{\prime}_{0})\subset
V^{\prime}_{0}$ for all $l\in L$. ∎

Now we can finish the proof of Theorem 1.17.

As follows from Lemma 1.17 there exists ${\mathbf{k}}$-subspaces
$V^{\prime}\subset V$, $W^{\prime}\subset W$ such that
$l(W^{\prime}_{0})\subset V^{\prime}_{0}$ for all $l\in
L\otimes_{\mathbf{k}}K$ and
$\mathrm{dim}(W/W^{\prime})+\mathrm{dim}(V^{\prime})=2r_{\bar{\mathbf{k}}}(L)$.
So $r_{\mathbf{k}}(L)\leq 2r_{\bar{\mathbf{k}}}(L)$.

##  3\. Rough bound

###  3.1. A lemma on the codimension

Let $K$ be an infinite field, let $A$ be a finitely generated $K$-algebra of
(Krull) dimension $n$ and let $L\subset A$ be linear subspace of $A$. We
denote by $J\subset A$ the ideal generated by $L$ and denote by $A_{L}$ the
quotient algebra $A/J$.

######  Lemma 3.1.

If $\mathrm{dim}A_{L}<n$ then there exists a finite collection of subspaces
$L_{i}\subsetneq L$, $i\in I$ such that the algebra $A_{l}$ has dimension $<n$
for any $l\in L\setminus\bigcup_{i\in I}L_{i}$.

###### Proof.

Let $X_{i},\ i=1,\ldots,k$, be irreducible components of $SpecA$ of dimension
$n$.

For every $i$ we denote by $K_{i}$ the field of rational functions on the
component $X_{i}$, and consider the natural morphism $\nu_{i}:A\to K_{i}$.

Since $\mathrm{dim}A_{L}<n$ the image $\nu_{i}(L)$ is not zero. Hence the
subspace $L_{i}:=L\bigcap\mathrm{Ker}(\nu_{i})\subset L$ is strictly contained
in $L$.

If an element $l\in L$ does not belong to any of the spaces $L_{i}$ then its
image in every field $K_{i}$ is not zero. This implies that the dimension of
the algebra $A_{l}=A/Al$ is less than $n$. ∎

######  Corollary 3.2.

There exist elements $l_{1},\dots,l_{m}\in
L,m:=\mathrm{dim}(A)-\mathrm{dim}(A/L)$ such that
$\mathrm{dim}(A/J)=\mathrm{dim}(A/L)$ where $J\subset A$ is the ideal
generated by $l_{1},\dots,l_{m}$.

###### Proof.

Induction in m. ∎

###  3.2. A proof of the rough bound

In this subsection we present a proof of a generalization of the rough bound
from [5].

For any subset $\Theta\subset\mathbb{F}_{q}[x_{1},\dots,x_{n}]$ we denote by
${\mathbf{X}}_{\Theta}\subset{\mathbf{A}}^{n}$ the subscheme which is the
intersection of zeros of $\theta\in\Theta$.

######  Proposition 3.3.

Let $M\subset\mathbb{F}_{q}[x_{1},\dots,x_{n}]$ be a linear subspace of
polynomials of degrees $d$ such that ${\mathbf{Y}}:={\mathbf{X}}_{M}$ is of
dimension $m$. Then $|{\mathbf{Y}}(\mathbb{F}_{q})|\leq q^{m}d^{c},c:=n-m$.

###### Proof.

Let $F$ be the algebraic closure of $\mathbb{F}_{q}$.

As follows from Lemma 3.1 there exists $P_{i}\in
M\otimes_{\mathbb{F}_{q}}F,1\leq i\leq c$ such that
$\mathrm{dim}({\mathbf{Y}}^{\prime})=m$ where
${\mathbf{Y}}^{\prime}={\mathbf{X}}_{\mathcal{P}}$ ,
$\mathcal{P}:=\\{P_{i}\\}_{1\leq i\leq c}$.

It is clear that ${\mathbf{Y}}(\mathbb{F}_{q})$ is the intersection of
${\mathbf{Y}}$ with hypersurfaces $S_{j},1\leq j\leq n$ defined by the
equations $h_{j}(x_{1},\dots,x_{n})=0$ where
$h_{j}(x_{1},\dots,x_{n})=x_{j}^{q}-x_{j}$. Since
${\mathbf{Y}}\subset{\mathbf{Y}}^{\prime}$ we see that
${\mathbf{Y}}(\mathbb{F}_{q})$ is contained in the intersection of
${\mathbf{Y}}^{\prime}$ with hypersurfaces $S_{j},1\leq j\leq n$.

For $j=1,\ldots,m$ let $H_{j}=\sum_{i=1}^{n}a_{j}^{i}h_{i}$, $a_{j}^{i}\in
F^{\prime}$ be a linear combination of the $h_{j}$ such that coefficients
$a_{j}^{i}$ are algebraically independent over $F$ where $F^{\prime}/F$ is a
transcendental extension. We denote by
${\mathbf{Z}}_{1},\ldots,{\mathbf{Z}}_{m}\subset{\mathbf{A}}^{n}$ be the
corresponding hypersurfaces and define
${\mathbf{B}}_{j}:={\mathbf{Y}}^{\prime}\cap(\bigcap_{i=1}^{j}{\mathbf{Z}}_{i})$.

######  Claim 3.4.

Each component ${\mathbf{C}}$ of ${\mathbf{B}}_{j}$ is of dimension $m-j$.

###### Proof.

The proof is by induction in $j$. The statement obviously is true for $j=0$.

Any component ${\mathbf{C}}$ of ${\mathbf{B}}_{j+1}$ is a component of an
intersection ${\mathbf{C}}^{\prime}\cap{\mathbf{Z}}_{j+1}$ for some component
${\mathbf{C}}^{\prime}$ of ${\mathbf{B}}_{j}$. By induction
$\mathrm{dim}({\mathbf{C}}^{\prime})=m-j$. So not all the functions $h_{j}$
vanish on ${\mathbf{C}}^{\prime}$. Hence by the genericity of the choice of
linear combinations $\\{H_{j}\\}$ we see that $H_{j+1}$ does not vanish on
${\mathbf{C}}^{\prime}$ and therefore
${\mathbf{C}}^{\prime}\cap{\mathbf{Z}}_{j+1}$ is of pure dimension $m-j-1$. ∎

As follows from Claim 3.4 the intersection
${\mathbf{Y}}^{\prime}\cap{\mathbf{Z}}_{1}\cap\dots\cap{\mathbf{Z}}_{m}$ has
dimension $0$. Therefore the Bézout’s theorem implies that
$|{\mathbf{Y}}^{\prime}\cap{\mathbf{Z}}_{1}\cap\dots\mathbf{Z}_{n-c}|\leq
q^{m}d^{c}$. Since
${\mathbf{Y}}(\mathbb{F}_{q})={\mathbf{Y}}\cap{\mathbf{Z}}_{1}\cap\dots\cap{\mathbf{Z}}_{n}\subset{\mathbf{X}}\cap{\mathbf{Y}}_{1}\cap\dots\cap{\mathbf{Y}}_{n-c}$
we see that $|{\mathbf{Y}}(\mathbb{F}_{q})|\leq q^{m}d^{c}$. ∎

##  4\. Proof of Theorem 1.13(1)

###### Proof.

Let ${\mathbf{k}}=\mathbb{F}_{q},P:V_{1}\times\dots\times
V_{d}\to{\mathbf{k}}$ be a multilinear polynomial and
$g=\mathrm{codim}_{\mathbf{V}_{2}\times\dots\times\mathbf{V}_{d}}{\mathbf{Z}}_{P}$
and $\bar{\mathbf{k}}$ be the algebraic closure of $\mathbb{F}_{q}$.

Since ( see Lemma 1.5)
$a_{\mathbf{k}}(P)=\sum_{i=2}^{d}\dim(V_{i})-\log_{q}(|{\mathbf{Z}}_{P}({\mathbf{k}})|)$
it follows from Proposition 3.3 that $|{\mathbf{Z}}_{P}({\mathbf{k}})|\leq
d^{g}q^{n-g}$ where $n=\sum_{i=2}^{d}\dim(V_{i})$. So
$\log_{q}(|{\mathbf{Z}}_{P}({\mathbf{k}})|)\leq g\log_{q}(d)+n-g$. We see that
$a_{\mathbf{k}}(P)\geq g(1-\log_{q}(d))$.

Assuming the validity of Theorem 1.12 for multilinear polynomials of degree
$d$ over $\bar{\mathbf{k}}$ we see that
$g\geq\frac{r_{{\mathbf{k}}}(P)}{\kappa_{d}D_{d}}$ and therefore
$a(P)\geq\frac{r_{{\mathbf{k}}}(P)}{\kappa_{d}D_{d}}(1-\log_{q}(d))$. Theorem
1.13(1) is proven. ∎

In the next section we prove the second part of Theorem 1.13.

##  5\. The adaptation of the Schmidt’s result for fields of finite
characteristic in the case when $d=3$.

We use notations from Definitions 1.1,1.3. In [14] W. Schmidt proved the
following result.

######  Theorem 5.1 (Schmidt).

For any $d\geq 2$ there exists $D_{d}$ such that for any complex multilinear
polynomial $P:V_{1}\times\dots\times V_{d}\to{\mathbb{C}}$ we have
$r_{\mathbb{C}}(P)\leq D_{d}g$ where
$g:=\mathrm{codim}_{\mathbf{V}_{2}\times\dots\times\mathbf{V}_{d}}{\mathbf{Z}}_{P}$

His proof extends to any algebraically closed field, and we sketch it here for
the three-dimensional case.

In this section we fix an algebraically closed field ${\mathbf{k}}$ and write
$r(P)$ instead of $r_{\mathbf{k}}(P)$. Let ${\mathbf{X}}$ be an algebraic
${\mathbf{k}}$-variety. Since our field ${\mathbf{k}}$ is algebraically closed
any constructible subset $Y\subset X$ defines uniquely a subset
${\mathbf{Y}}\subset{\mathbf{X}}$ such that $Y={\mathbf{Y}}({\mathbf{k}})$. We
say that a constructible subset $Y\subset X$ is open if the subset
${\mathbf{Y}}\subset{\mathbf{X}}$ is open and we define
$\dim(Y):=\dim({\mathbf{Y}})$.

For a trilinear polynomial $P:U\times V\times W\to{\mathbf{k}}$ we denote by
$Z_{P}\subset V\times W$ the constructible subset of points $(v,w)$ such that
$P(u,v,w)=0$ for all $u\in U$.

The main goal of this section is to prove the following theorem:

######  Theorem 5.2.

For any trilinear polynomial $P:U\times V\times W\to{\mathbf{k}}$ we have
$r(P)\leq 2g$ where $g:=\mathrm{codim}_{V\times W}Z_{P}$.

Before proving Theorem 5.2 we remind the reader of some results from Algebra.

###  5.1.  Linear Algebra

######  Definition 5.3.

For a bilinear form $B:U\times V\to{\mathbf{k}}$ we write

  1. (1)

$S_{U}(B):=\\{u\in U|B(u,v)=0,\forall v\in V\\}$.

  2. (2)

$S_{V}(B):=\\{v\in V|B(u,v)=0,\forall u\in U\\}$.

######  Remark 5.4.

$r(B)=\operatorname{codim}_{U}S_{U}(B)=\operatorname{codim}_{V}S_{V}(B)$.

Let $X$ be a smooth curve over ${\mathbf{k}}$ , let $t_{0}\in X$ and let
$B(t):U\times V\to{\mathbf{k}},t\in X$ be a family of bilinear forms. We write
$B:=B(t_{0})$, $S_{U}:=S_{U}(B)$ , $S_{V}:=S_{V}(B)$, and $C:=(\partial
B(t)/\partial t)_{t=t_{0}}$.

######  Claim 5.5.

If $r(B(t))\leq r(B),t\in X$ then $C_{|S_{U}\times S_{V}}\equiv 0$.

###### Proof.

We show that the assumption that $C_{|S_{U}\times S_{V}}\not\equiv 0$ leads to
a contradiction. To simplify notations we assume that
${\mathbf{X}}={\mathbf{A}}$ and $t_{0}=0$.

To start with we choose bases $e_{i},f_{j}$, where $1\leq i\leq\dim(U),1\leq
j\leq\dim(V)$ of $U$ and $V$ such that

  1. (1)

$B(e_{i},f_{j})=\delta_{i,j},1\leq i,j\leq r(B)$.

  2. (2)

$B(e_{i},f_{j})=0$ if either $i>r(B)$ or $j>r(B)$.

Then $S_{U}$ is the span of $\\{e_{i}\\}$ , where $r(B)+1\leq i\leq\dim(U)$
and $S_{V}$ is the span of $\\{f_{j}\\}$, where $r(B)+1\leq j\leq\dim(V)$.
Since $C_{|S_{U}\times S_{V}}\not\equiv 0$ there exist a pair $(i_{0},j_{0})$
such that $i_{0},j_{0}>r(B)$ and $C(e_{i_{0}},f_{j_{0}})\neq 0$.

Let $U^{\prime}\subset U$, $\dim(U^{\prime})=r(B)+1$ be the span of
$\\{e_{i}\\}_{1\leq i\leq r(B)}$ and of $e_{i_{0}}$ and $V^{\prime}\subset V$,
$\dim(V^{\prime})=r(B)+1$ be the span of $\\{f_{j}\\},_{1\leq j\leq r(B)}$ and
of $f_{j_{0}}$.

We denote by $\Delta(t)$ be the determinant of the bilinear form of
$B(t)_{U^{\prime}\times V^{\prime}}$ with respect to chosen bases of
$U^{\prime}$ and $V^{\prime}$. The condition $r(B(t))\leq
r(B),t\in{\mathbf{k}}$ implies that $\Delta(t)\equiv 0$. On the other hand
$\Delta(t)\equiv tC(e_{i_{0}},f_{j_{0}})\mod(t^{2})$.

This contradiction proves Lemma 5.5.

∎

###  5.2. Transcendence degree

######  Definition 5.6.

  1. (1)

For a field extension $K/{\mathbf{k}}$ we define the transcendence degree
$tr(K/{\mathbf{k}})$ to be the minimal $n\geq 0$ such that there is no
imbeddings of ${\mathbf{k}}(u_{0},u_{1},\dots,u_{n})\hookrightarrow K$.

  2. (2)

For a point $w\in K^{N}$ we denote by ${\mathbf{k}}(w)\subset K$ the subfield
generated by ${\mathbf{k}}$ and the coordinates of $w$ and write
$tr_{\mathbf{k}}(w):=tr({\mathbf{k}}(w)/{\mathbf{k}})$.

  3. (3)

Let ${\mathbf{X}}\subset{\mathbf{A}}^{N}$ be an irreducible
${\mathbf{k}}$-subvariety. A point $w\in{\mathbf{X}}(K)$ is generic if it is
not contained in any proper ${\mathbf{k}}$-subvariety of $X$.

######  Claim 5.7.

  1. (1)

Let ${\mathbf{X}}\subset{\mathbf{A}}^{M}$ be an irreducible ${\mathbf{k}}$
variety and $w\in{\mathbf{X}}(K)$ be generic point. Then
$tr({\mathbf{k}}(w)/{\mathbf{k}})=\dim(X)$.

  2. (2)

Let $p:{\mathbf{A}}^{M}\to{\mathbf{A}}^{N}$ be a ${\mathbf{k}}$-linear map,
${\mathbf{Z}}\subset{\mathbf{A}}^{M}$ an irreducible variety, and
${\mathbf{Y}}=p({\mathbf{Z}})$. Let $v\in{\mathbf{Z}}(K)$ be generic point and
$w:=p(v)\in{\mathbf{Y}}(K)$. Then $w\in{\mathbf{Y}}(K)$ is a generic point,
${\mathbf{k}}(w)\subset{\mathbf{k}}(v)$ and
$tr({\mathbf{k}}(v)/{\mathbf{k}}(w))=\dim({\mathbf{Z}})-\dim({\mathbf{Y}})$.

###  5.3. Proof of Theorem 5.2

###### Proof.

Let $P:U\times V\times W\to{\mathbf{k}}$ be as in Theorem 5.2 and $Z\subset
Z_{P}$ be an irreducible component of the maximal dimension,
$g:=\operatorname{codim}_{V\times W}Z$. We have to show that $r(P)\leq 2g$.

Let $z_{0}=(v_{0},w_{0})$ be a generic point of $Z$, and let $Y\subset W$ be
the projection of $Z$ on $W$. For $w\in W$ we denote by $P_{w}$ the bilinear
forms on $V$ given by $P_{w}(u,v):=P(u,v,w)$.

Let $S_{U}=\\{u\in U|P_{w_{0}}(u,v)=0,\forall v\in V\\}$, and let
$S_{V}=\\{v\in V|P_{w_{0}}(u,v)=0,\forall u\in U\\}$. Since $w_{0}$ is a
generic point of $Y$, we see that $r(P_{y})\leq r(P_{w_{0}})$ for all $y\in
Y$. Since $w_{0}\in Y$ is a generic point, $Y$ is smooth at $w_{0}$ and we can
define $S_{W}:=T_{Y}(w_{0})$ the tangent space at $w_{0}$..

It follows from Calim 1.4 that the following statement implies the validity of
Theorem 5.2.

######  Proposition 5.8.

  1. (1)

$P_{|S_{U}\times S_{V}\times S_{W}}\equiv 0$.

  2. (2)

$\operatorname{codim}_{U}(S_{U})+\operatorname{codim}_{W}(S_{W})+\operatorname{codim}_{V}(S_{V})\leq
2g$.

###### Proof.

To show that $P_{|S_{U}\times S_{V}\times S_{W}}\equiv 0$ we have to show that
for any $C\in S_{W}$ the restriction of the bilinear form $P_{C}$ on
$S_{U}\times S_{V}$ is equal to $0$. So we fix $C\in S_{W}$. Since $P_{w}$ for
$w\in W$ is a linear function on $W$ we have $P_{C}=(\partial P_{w}/\partial
C)_{w=w_{0}}$. So we have to show that $(\partial P_{w}/\partial
C)_{w=w_{0}}|_{S_{U}\times S_{V}}\equiv 0$.

Choose a smooth curve on $Y$ passing through $w_{0}$ and tangent to $C$. In
other words choose a map $\phi:X\to Y$ of a smooth curve $X$ to $Y$ and a
point $t_{0}\in X$ such such that $\phi(t_{0})=w_{0}$ and
$C=(\partial\phi(t)/\partial t)_{t=t_{0}}$. Since $r(P_{y})\leq r(P_{w_{0})}$
for $y\in Y$ the family $B(t):=P_{\phi(t)}$ of bilinear forms on $U\times V$
satisfies the assumption of Claim 5.5. Therefore $(\partial P_{w}/\partial
C)_{w=w_{0}}|_{S_{U}\times S_{V}}\equiv 0$, and thus $P_{C}|_{S_{U}\times
S_{V}}\equiv 0$.

Since
$\operatorname{codim}_{U}(S_{U})=\operatorname{codim}_{V}(S_{V})=r(P_{w_{0}})$
it follows from Claim 5.7 that
$\operatorname{codim}_{U}(S_{U})+\operatorname{codim}_{W}(S_{W})=\operatorname{codim}_{V}(S_{V})+\operatorname{codim}_{W}(S_{W})\leq
g$. So
$\operatorname{codim}_{U}(S_{U})+\operatorname{codim}_{W}(S_{W})+\operatorname{codim}_{V}(S_{V})\leq
2g$. ∎

∎

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