# When any three solutions are independent

James Freitag  James Freitag
University of Illinois Chicago
Department of Mathematics, Statistics, and Computer Science
851 S. Morgan Street
Chicago, IL, 60607-7045
USA<EMAIL_ADDRESS>and  Rahim Moosa  Rahim Moosa
University of Waterloo
Department of Pure Mathematics
200 University Avenue West
Waterloo, Ontario N2L 3G1
Canada<EMAIL_ADDRESS>
###### Abstract.

Given an algebraic differential equation of order $>1$, it is shown that if
there is any nontrivial algebraic relation amongst any number of distinct
nonalgebraic solutions, along with their derivatives, then there is already
such a relation between three solutions. This is deduced as an application of
the following model-theoretic result: Suppose $p$ is a stationary nonalgebraic
type in the theory of differentially closed fields of characteristic zero; if
any three distinct realisations of $p$ are independent then $p$ is minimal. An
algebro-geometric formulation in terms of $D$-varieties is given. The same
methods yield also an analogous statement about families of compact Kähler
manifolds.

###### Key words and phrases:

Geometric stability theory, differentially closed fields, algebraic
differential equations, transcendence, $D$-varieties, compact Kähler manifolds

###### 2020 Mathematics Subject Classification:

03C45, 12H05, 11J81, 32J27

The first author was partially supported by NSF grant DMS-1700095 and NSF
CAREER award 1945251. The second author was partially supported by an NSERC
Discovery Grant.

###### Contents

  1. 1 Introduction
  2. 2 Generic transitivity and degree of nonminimality
  3. 3 Distinct solutions being independent
  4. 4 A formulation for $D$-varieties
  5. 5 The case of compact complex manifolds

##  1\. Introduction

Understanding algebraic relations between solutions of a differential equation
is, perhaps along with understanding solutions of a particular form, the
central problem in algebraic differential equations. Characterizing algebraic
relations between solutions is an important output of various approaches to
differential Galois theory [14, 29, 4], while work using the model theory of
differential fields frequently applies techniques from stability theory to
obtain conclusions in this vein [24, 25, 10]. A number of recent transcendence
results for analytic functions (equivalently, problems of _bi-algebraic
geometry_ [12]) have played an important role in diophantine geometry and have
natural interpretations in terms of characterizing algebraic relations between
solutions of differential equations and their derivatives [26, 6, 3].

To state our main theorem let us fix a differential field $(k,\delta)$ of
characteristic zero, as well as an order $n$ algebraic differential equation

(1) $P(y,\delta y,\dots,\delta^{n}y)=0,$

where $P\in k[X_{0},\dots,X_{n}]$ is irreducible. For each $m\geq 1$, consider
the following condition:

  * $(C_{m})$

For any $m$ distinct solutions $a_{1},\dots,a_{m}\notin
k^{\operatorname{alg}}$ of (1), the sequence

$(\delta^{(i)}a_{j}:i=0,\dots,n-1,j=1,\dots,m)$

is algebraically independent over $k$.

In [5], as a byproduct of working on a related problem, we showed that as long
as $n>1$, one always has $(C_{n+2})\implies(C_{m})$ for all $m$. We can now do
much better:

######  Theorem A.

If $n>1$ then $(C_{3})\implies(C_{m})$ for all $m$.

The theorem is sharp in both of the natural possible ways: $(C_{2})$ is in
general not enough and when $n=1$ then $(C_{3})$ is not enough. See Examples
3.7 and 3.8 below. However, if $n=1$ then we do have $(C_{4})\implies(C_{m})$
for all $m$, as we will explain.

Let us consider some specific examples that have been studied recently.

######  Example 1.

Consider a Painlevé equation from the families $P_{II}$ through $P_{V}$ with
generic coefficients. For instance, one might take
$y^{\prime\prime}=2y^{3}+ty+\pi$ over the differential field
$(\mathbb{C}(t),\frac{d}{dt})$. Nagloo and Pillay [25] show that $(C_{m})$
holds of all $m$.

######  Example 2.

In [10], Jaoui studied the case when $P(y,\delta y,\delta^{2}y)=0$ is a
general order two and degree at least 3 algebraic differential equation over a
constant field of parameters. He showed that in that case
$(C_{2})\implies(C_{m})$ for all $m$. Indeed, it is his phrasing of the
problem in the Introduction to [10] that lead us to consider the question in
these terms.

######  Example 3.

Consider the following equation satisfied by the $j$-function

$S(y)+\frac{y^{2}-1968y+2654208}{2y^{2}(y-1728)^{2}}(y^{\prime})^{2}=0,$

where $S$ is the Schwarzian derivative:
$S(y)=\left(\frac{y^{\prime\prime}}{y^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{y^{\prime\prime}}{y^{\prime}}\right)^{2}$.
It follows from results of [6] that $(C_{2})$ fails of this equation. In fact,
if $y_{1},y_{2}$ are nonalgebraic solutions then
$(y_{1},y_{1}^{\prime},y_{1}^{\prime\prime},y_{2},y_{2}^{\prime},y_{2}^{\prime\prime})$
are algebraically dependent if and only if a modular polynomial vanishes at
$(y_{1},y_{2})$.

A generalisation of this equation and setting is considered in [3], where the
$j$-function is replaced by the uniformisation function of the quotient of the
upper half plane by an arbitrary genus zero Fuchsian group $\Gamma$ of the
first kind. For such equations, it is shown that $(C_{2})$ fails if and only
if $(C_{m})$ fails for some $m$, if and only if $\Gamma$ is a proper subgroup
of its commensurator. This analysis of the algebraic relations amongst the
solutions and their derivatives, in [3], is key to proving the Ax-Lindemann-
Weierstrass theorem for uniformising functions of Fuchsian groups and
answering an old open question of Painlevé (1895). Ax-Lindemann-Weierstrass
results for the uniformizing functions of Shimura varieties are an essential
component of the resolution of a number of recent diophantine conjectures and
can naturally be phrased in terms of algebraic relations between solutions of
differential equations (see [12] and the references therein).

Another way of talking about algebraic differential equations is in terms of
algebraic vector fields; namely algebraic varieties $V$ equipped with a
polynomial section $s$ to the tangent bundle. To study the algebraic relations
between the solutions of an equation amounts to looking at subvarieties of
cartesian powers of $V$ that are invariant under the vector field. (Here we
put on $V^{m}$ the canonical vector field induced by $s$, obtained by
identifying $T(V^{m})$ with the $m$-fold fibred product of $TV$ with itself.)
We can formulate our main theorem as follows:

######  Theorem B.

Suppose $(V,s)$ is an algebraic vector field over a field $k$ of
characteristic zero, and $\dim V>1$. If the third cartesian power $V^{3}$
admits no invariant subvarieties over $k$ projecting dominantly onto each co-
ordinate, other than the diagonals, then the same holds of $V^{m}$ for all
$m\geq 0$.

######  Remark.

We have restricted our attention here to algebraic vector fields only to keep
the exposition concise. In fact, the theorem holds more generally of
“$D$-varieties” in the sense of Buium [2]. That is, our results work
identically in the non-autonomous situation where the derivation is not
assumed to be trivial on the base field $k$. The tangent bundlle must then be
replaced with the prolongation, a natural torsor of the tangent bundle that
takes into account the twisting induced by $\delta$. See $\S$4 below for a
discussion, and Corollary 4.2 for the general statement of Theorem B in the
non-autonomous setting.

There is an apparent discrepancy between Theorems A and B. The latter is
really about relations among generic solutions to the differential equation –
this is the meaning of only considering invariant subvarieties that project
dominantly on each co-ordinate – while Theorem A seems to be stronger, making
conclusions about all nonalgebraic solutions. In fact, however, $(C_{1})$
already rules out the possibility of lower order equations, so that all
nonalgebraic solutions are generic.

These theorems are applications of a more fundamental theorem about the model
theory of differentially closed fields ($\operatorname{DCF}_{0}$), which we
now state. We work model-theoretically, assuming familiarity with geometric
stability theory as well as the particular case of $\operatorname{DCF}_{0}$.
We suggest [27] as a general reference for the former, and [17] as an
introduction to the latter.

The following is the main clause of Theorem 3.6 below:

######  Theorem C.

Suppose $p$ is a stationary nonalgebraic type in $\operatorname{DCF}_{0}$ such
that any three distinct realisations of $p$ are independent. Then $p$ is
minimal.

Once we have a minimal generic type we can apply the Zilber trichotomy in
differentially closed fields and deduce Theorems A and B quite readily. The
proof of minimality itself goes via reducing to the case when $p$ is internal
to the constants and with a binding group that acts $3$-transitively. As the
binding group is an algebraic group action, a theorem of Knop [13] implying
that the only $3$-transitive algebraic group action is that of
$\operatorname{PGL}_{2}$ on $\mathbb{P}$ now applies, and settles the matter.

Other than standard facts from geometric stability theory and the model theory
of differentially closed fields, as well as the theorem of Knop just
mentioned, the paper is self-contained. In particular, the results of [5] are
not used here, though the approach taken there does inform what we do.

The proof of Theorem C works in any totally transcendental theory where one
has the following strong form of the Zilber dichotomy: every non locally
modular minimal type over $A$ is nonorthogonal to an $A$-definable pure
algebraically closed field. So, for example, we could just as easily have
worked with several commuting derivations in partial differentially closed
fields ($\operatorname{DCF}_{0,m}$). Or in the theory $\operatorname{CCM}$ of
compact complex manifolds. Indeed, concerning the latter, a short final
section is dedicated to articulating the result as a theorem in bimeromorphic
geometry that may be of independent interest, see Theorem 5.1 below.

Throughout, we work in a sufficiently saturated model
$\mathcal{U}\models\operatorname{DCF}_{0}$ with field of constants
$\mathcal{C}$.

##  2\. Generic transitivity and degree of nonminimality

Recall from [1] that a definable group action $(G,S)$ of finite Morley rank is
_generically transitive_ if the action of $G$ on $S$ admits an orbit
$\mathcal{O}$ such that

$\operatorname{RM}(S\setminus\mathcal{O})<\operatorname{RM}(S).$

Assuming that $\operatorname{dM}(S)=1$, this is equivalent to asking that any
two generic elements of $S$ over $A$ are conjugate.

Actually, we are interested in the multiple-transitivity version:

######  Definition 2.1.

A finite rank definable group action $(G,S)$ is generically $k$-transitive if
the diagonal action $(G,S^{k})$ is generically transitive.

Generic $k$-transitivity, and the Borovik-Cherlin conjecture about such group
actions (articulated in [1]), played an important role in our earlier work in
[5] on bounding the size of a witness to the nonminimality of a type in
$\operatorname{DCF}_{0}$. Recall:

######  Definition 2.2.

The degree of nonminimality of $p\in S(A)$ is the least $d$ such that there
are distinct realisations $a_{1},\dots,a_{d}$ of $p$ and a nonalgebraic
forking extension of $p$ to $Aa_{1}\cdots a_{d}$.

The main result of [5] was that $\operatorname{nmdeg}(p)\leq U(p)+1$. One step
in that proof was the following observation, which we repeat here for the sake
of completeness.

######  Proposition 2.3.

Suppose $p$ is a stationary type with $U(p)>1$ that is $\mathcal{C}$-internal
and weakly $\mathcal{C}$-orthogonal. Let $d:=\operatorname{nmdeg}(p)$. Then
the binding group of $p$ acts transitively and generically $d$-transitively on
$p(\mathcal{U})$.

###### Proof.

Denote the binding group by $G:=\operatorname{Aut}_{A}(p/\mathcal{C})$. It
acts definably and faithfully on $S:=p(\mathcal{U})$. Weak orthogonality
implies that the action is transitive and $p$ is isolated. So $(G,S)$ is a
definable homogeneous space. To show that the action is generically
$d$-transitive is precisely to show that $G$ acts transitively on $p^{(d)}$.
As $G$ is also the binding group of the $\mathcal{C}$-internal type $p^{(d)}$,
it suffices to show that $p^{(d)}$ is weakly orthogonal to $\mathcal{C}$.

Let $r>1$ be least such that $p^{(r)}$ is not weakly orthogonal to
$\mathcal{C}$. (This exists by $\mathcal{C}$-internality.) Fix
$\overline{a}=(a_{1},\dots,a_{r-1})\models p^{(r-1)}$ and let $q$ be the
nonforking extension of $p$ to $A\overline{a}$. Then $q$ is not weakly
orthogonal to $\mathcal{C}$. It follows (using elimimination of imaginaries
for the structure induced on $\mathcal{C}$; namely that of a pure
algebraically closed field) that there is $a\models q$ and a constant
$c\in\mathcal{C}$ such that
$c\in\operatorname{dcl}(A\overline{a}a)\setminus\operatorname{acl}(A\overline{a})$.
Write $c=f(a)$ where $f$ is a partial $A\overline{a}$-definable function from
$S$ to $\mathcal{C}$. The fibre $f^{-1}(c)$ is infinite as $U(q)>1$, and so we
may assume that all the fibres are infinite. Let
$c_{0}\in\operatorname{Im}(f)\cap\mathbb{Q}$, which exists as
$\operatorname{Im}(f)$ is an infinite and hence cofinite subset of
$\mathcal{C}$. Let $a_{0}$ be generic in $f^{-1}(c_{0})$ over $A\overline{a}$.
Then $U(a_{0}/A\overline{a})>0$ as $f^{-1}(c_{0})$ is infinite. On the other
hand, $U(a_{0}/A\overline{a})<U(p)$, as otherwise $a_{0}\models q$, which is
impossible as $f(a)\notin\operatorname{acl}(A\overline{a})$ while
$f(a_{0})\in\mathbb{Q}=\operatorname{dcl}(\emptyset)$. Hence
$\operatorname{tp}(a_{0}/A\overline{a})$ is a nonalgebraic forking extension
of $p$ witnessing that $d\leq r-1$. By minimal choice of $r$, it follows that
$p^{(d)}$ is weakly $\mathcal{C}$-orthogonal, as desired. ∎

##  3\. Distinct solutions being independent

We are interested in the following condition on a stationary type $p\in S(A)$.

######  Definition 3.1 (The condition $D_{m}$).

Given $m>1$, let us say that $p\in S(A)$ satisfies $D_{m}$ if every $m$-tuple
of distinct realisations of $p$ is independent over $A$.

We begin with a few straightforward consequences of $D_{2}$.

######  Lemma 3.2.

Every type satisfying $D_{2}$ is of finite rank.

###### Proof.

As some co-ordinate of any realisation of an infinite rank type must be
$\delta$-transcendental, it suffices to observe that the $1$-type of a
$\delta$-transcendental element does not satisfy $D_{2}$. Indeed, if $a$ is
$\delta$-transcendental over a differential field $k$ then so is $\delta(a)$,
which is distinct for $a$, and $a\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mathchar 12854\relax$\kern
8.00134pt\hss}\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mathchar 12854\relax$\kern
3.92064pt\hss}\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mathchar 12854\relax$\kern
2.00034pt\hss}\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower
0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{k}\delta a$. ∎

######  Lemma 3.3.

No type satisfying $D_{2}$ can be a nontrivial finite cover. That is, if
$p=\operatorname{tp}(a/A)$ is a nonalgebraic stationary type satisfying
$D_{2}$, and $a$ is interalgebraic with $b$ over $A$, then
$a\in\operatorname{dcl}(Ab)$.

###### Proof.

If $a\notin\operatorname{dcl}(Ab)$ then we can find $a^{\prime}\neq a$ such
that $\operatorname{tp}(a^{\prime}/Ab)=\operatorname{tp}(a/Ab)$. By $D_{2}$ we
must have $a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{A}a$. But
$a\in\operatorname{acl}(Ab)=\operatorname{acl}(Aa^{\prime})$, which forces
$a\in\operatorname{acl}(A)$ and contradicts nonalgebraicity of $p$. ∎

######  Lemma 3.4.

If $p=\operatorname{tp}(a/A)$ is a nonalgebraic stationary type satisfying
$D_{2}$ then it satisfies the following form of exchange: if
$b\in\operatorname{acl}(Aa)\setminus\operatorname{acl}(A)$ then
$a\in\operatorname{dcl}(Ab)$.

###### Proof.

Note that the conclusion is a strengthening of the property called admitting
no proper fibrations in [22]. In any case, by Lemma 3.3, it suffices to show
that $a\in\operatorname{acl}(Ab)$. Suppose $a\notin\operatorname{acl}(Ab)$,
and seek a contradiction. It follows that if $a^{\prime}$ realises the
nonforking extension of $\operatorname{tp}(a/Ab)$ to $Aa$ then $a\neq
a^{\prime}$. On the other hand, using the Lascar inequality,
$U(a/Aa^{\prime})=U(a/Ab)<U(a/A)$ as
$b\in\operatorname{acl}(Aa)\setminus\operatorname{acl}(A)$. So $a$ and
$a^{\prime}$ are dependent over $A$, contradicting $D_{2}$. ∎

######  Proposition 3.5.

Every stationary type of $U$-rank $>1$ satisfying $D_{2}$ is internal and
weakly orthogonal to the constants.

###### Proof.

Suppose $p=\operatorname{tp}(a/A)$ is stationary, nonminimal, and satisfies
$D_{2}$. As $p$ is of finite rank it is nonorthogonal to some minimal type
$r\in S(B)$ for some $B\supseteq A$. We first show that $r$ is not locally
modular. Indeed, in [22, Proposition 2.3], it is shown that, because $p$ has
no proper fibrations (Lemma 3.4), if $r$ is locally modular than in fact $p$
is interalgebraic with some $q^{(k)}$ where $q\in S(A)$ is a locally modular
minimal type and $k\geq 1$. Now, $k=1$ is impossible by the assumption that
$U(p)>1$. But $k>1$ is also impossible: taking $(b_{1},\dots,b_{k})\models
q^{(k)}$, we obtain

$b_{1}\in\operatorname{acl}(Ab_{1}\cdots
b_{k})\setminus\operatorname{acl}(A)=\operatorname{acl}(Aa)\setminus\operatorname{acl}(A)$

while

$\operatorname{acl}(Aa)=\operatorname{acl}(Ab_{1}\cdots
b_{k})\not\subseteq\operatorname{acl}(Ab_{1})$

contradicting Lemma 3.4.

We have shown that $p$ is nonorthogonal to a non locally modular minimal type.
In $\operatorname{DCF}_{0}$ this means that $p$ is nonorthogonal to the
constants $\mathcal{C}$. It follows by [27, Corollary 7.4.6] that there is
$c\in\operatorname{dcl}(Aa)\setminus\operatorname{acl}(A)$ such that
$q:=\operatorname{tp}(c/A)$ is $\mathcal{C}$-internal. By Lemma 3.4,
$a\in\operatorname{dcl}(Ac)$. That is, $p$ and $q$ are interdefinable, and
hence $p$ is also $\mathcal{C}$-internal.

If $p$ were not weakly orthogonal to $\mathcal{C}$ then there would be
$c\in\mathcal{C}$ such that
$c\in\operatorname{dcl}(Aa)\setminus\operatorname{acl}(A)$. Lemma 3.4 would
then imply that $a\in\operatorname{dcl}(Ac)$, contradicting nonminimality of
$p$ (and also $D_{2}$). ∎

Here is our main theorem.

######  Theorem 3.6.

Suppose $p$ is a stationary nonalgebraic type satisfying $D_{3}$. Then $p$ is
minimal. Moreover, there are in this case exactly two possibilities:

  * (a)

Either $p$ is trivial and satisfies $D_{m}$ for all $m>1$, or,

  * (b)

$p$ is nonorthogonal to the constants and $D_{4}$ fails.

###### Proof.

We assume $p$ is not minimal and seek a contradiction. By Proposition 3.5 we
have that $p$ is $\mathcal{C}$-internal and weakly $\mathcal{C}$-orthogonal.
Denote the binding group by $G:=\operatorname{Aut}_{A}(p/\mathcal{C})$ and set
$S:=p(\mathcal{U})$. Proposition 2.3 tells us that $(G,S)$ is a generically
$d$-transitive definable homogeneous space, where $d$ is the degree of
nonminimality of $p$.

Let $a_{1},\dots,a_{d}$ be distinct realisations of $p$ and $q$ a nonalgebraic
forking extension of $p$ to $Aa_{1}\cdots a_{d}$. Let $a_{d+1}\models q$. Then
$a_{1},\dots,a_{d+1}$ witnesses the failure of $D_{d+1}$. Since $D_{3}$ holds,
we must have $d\geq 3$. In particular, $G$ acts generically $3$-transitively
on $S$. But $D_{3}$ tells us that every tuple of three distinct elements is
generic. Hence $(G,S)$ is outright $3$-transitive.

As $(G,S)$ is the binding group action of a $\mathcal{C}$-internal type, it is
definably isomorphic (over possibly additional parameters) with a definable
homogeneous space in the induced structure on $\mathcal{C}$. That induced
structure being a pure algebraically closed field, and using the Weil group-
chunk theorem, we have that $(G,S)$ is definably isomorphic to an algebraic
group action $(\overline{G},\overline{S})$ in the constants. So
$(\overline{G},\overline{S})$ is $3$-transitive. But the only $3$-transitive
algebraic group action is $\operatorname{PGL}_{2}$ acting naturally on
$\mathbb{P}$. Indeed, Knop [13, Satz 2] classifies the (faithful)
$2$-transitive actions of algebraic groups as being of only two kinds:

  * •

The standard action of $\operatorname{PGL}_{n+1}$ on $\mathbb{P}^{n}.$

  * •

Certain subgroups of the affine group of transformations on a vector space.

In both cases, the action preserves colinearity of three points, and hence the
only possibility for $3$-transitivity is if we are in the first case and
$n=1$. So $(\overline{G},\overline{S})$ is isomorphic to
$(\operatorname{PGL}_{2}(\mathcal{C}),\mathbb{P}(\mathcal{C}))$. This
contradicts nonminimality of $p$.

We have proven that $p$ is minimal. The “moreover” clause now follows by
applying the Zilber trichotomy as it is manifested in
$\operatorname{DCF}_{0}$. That trichotomy says that exactly one of the
following must hold:

  * (i)

$p$ is trivial;

  * (ii)

$p$ is nonorthogonal to the generic type of the Manin kernel $\mathcal{G}$ of
a simple abelian variety over $\operatorname{acl}(A)$ that does not descend to
the constants; or,

  * (iii)

$p$ is nonorthogonal to the constants.

This is originally an unpublished theorem of Hrushovski and Sokolovic [9].
Published references and explanations can be found in various places, see for
example the discussion around Fact 4.1 of [16].

We first point out that $D_{3}$ rules out case (ii). Indeed, assuming we are
in case (ii), and setting $q$ to be the nonforking extension of $p$ to $Aa$,
it would follow that $q$ is interalgebraic with the generic type $r$ of
$\mathcal{G}$ over $Aa$. This is because both $q$ and $r$ would be modular
minimal types, and for such types weak orthogonality implies orthogonality
(see [27, Corollary 2.5.5]). As $p$ satisfies $D_{3}$, $q$ satisfies $D_{2}$,
and hence Lemma 3.4 implies that $r$ is a finite cover of $q$, say $\pi:r\to
q$. Given $g\models r$, since $\mathcal{G}$ has finite $n$-torsion, $ng$ also
realises $r$ for all $n$, and hence we can find $n>1$ such that
$\pi(g)\neq\pi(ng)$. But these two elements are distinct realisations of $q$
that are dependent over $Aa$. It follows that $\\{a,\pi(g),\pi(ng)\\}$
witnesses the failure of $p$ to satisfy $D_{3}$.

We must, therefore, be in case (i) or (iii); either $p$ is trivial or it is
nonorthogonal to the constants. In the trivial case, $D_{2}$ already implies
$D_{m}$ for all $m$, as desired. So it remains to consider the case of a
minimal type nonorthogonal to the constants. We will show, and this will
suffice, that in this case $D_{4}$ fails.

Indeed, from nonorthogonality to the constants, and using $D_{2}$, exactly as
in the proof of Proposition 3.5, we conclude that $p$ is actually
$\mathcal{C}$-internal and either interdefinable with the generic type of the
constants or weakly $\mathcal{C}$-orthogonal. In the former case $D_{2}$ would
clearly fail, so $p$ is weakly $\mathcal{C}$-orthogonal. It follows that
$S:=p(\mathcal{U})$ is a strongly minimal homogeneous space for the binding
group action. These are completely classified (see, for example, [27, Fact
1.6.25]), and it follows from that classification that if
$\overline{a}=(a_{1},a_{2},a_{3})\models p^{(3)}$ then the nonforking
extension of $p$ to $A\overline{a}$ is not weakly orthogonal to $\mathcal{C}$.
This gives rise to an $A\overline{a}$-definable function $f:X\to\mathcal{C}$
where $X\subseteq S\setminus\\{a_{1},a_{2},a_{3}\\}$ is cofinite, $f(X)$ is
cofinite, and the fibres of $f$ are finite. Let $c\in f(X)\cap\mathbb{Q}$ and
choose any $a_{4}\in f^{-1}(c)$. Then
$a_{4}\in\operatorname{acl}(A\overline{a})$, and hence
$(a_{1},a_{2},a_{3},a_{4})$ witnesses the failure of $D_{4}$. ∎

The following example shows that Theorem 3.6 is sharp in the sense that
$D_{2}$ would not suffice for minimality.

######  Example 3.7.

In [5, $\S 4.2$] we gave an example, for each $n\geq 3$, of a
$\mathcal{C}$-internal type $p$ of $U$-rank $n-1$ whose binding group action
is that of $\operatorname{PGL}_{n}$ on $\mathbb{P}^{n-1}$. In particular, the
binding group action is $2$-transitive, and hence $D_{2}$ holds for these
nonminimal types. The construction is based on one from [11] when $n=2$.
Essentially, $p$ is the generic type of the projectivisation of the space of
solutions to $\delta X=BX$ where $B$ is an $n\times n$ matrix whose entries
form a differentially transcendental set, and $X$ is an $n$-column of
variables.

Let us point out that both cases of the “moreover” clause of Theorem 3.6
actually occur. That there are minimal trivial types satisfying $D_{m}$ for
all $m>1$ is well known, with constructions appearing in [8, 18, 28]. The
following shows that case (b) also occurs: there are minimal types
nonorthogonal to the constants satisfying $D_{3}$.

######  Example 3.8.

Consider the order $1$ Riccati equation $\delta y=y^{2}+c$ with
$c\in\mathbb{C}(t)$ a transcendental rational function. The generic type $p$
of this equation is minimal and $\mathcal{C}$-internal. Nagloo [23] has shown
that it does satisfy $D_{3}$. (Another argument is by [5, $\S 4.3$] where we
show that the binding group acts $3$-transitively on $p$.)

###  3.1. Deducing Theorem A

Let us spell out how Theorem 3.6 implies Theorem A of the Introduction. This
is a standard translation from model-theoretic to differential-algebraic
language. We are working over a differential field $(k,\delta)$ of
characteristic zero, with an order $n>1$ algebraic differential equation

$P(y,\delta y,\dots,\delta^{n}y)=0$

where $P\in k[X_{0},\dots,X_{n}]$ is irreducible. Denote by
$Y\subseteq\mathcal{U}$ the Kolchin closed set defined by this equation. For
$m>1$, the condition $(C_{m})$ on the equation translates precisely into
saying that any distinct $m$ elements of $Y\setminus k^{\operatorname{alg}}$
are independent (in the stability-theoretic sense). Suppose that $(C_{3})$
holds. In particular, $(C_{1})$ holds, and this already implies that $Y$ has
no infinite Kolchin closed subsets over $k$ of order less than $n$. Together
with irreducibility of $P$, this implies that $Y$ is Kolchin irreducible and
that all points of $Y\setminus k^{\operatorname{alg}}$ realise the Kolchin
generic type $p\in S(k)$. So condition $(C_{3})$ implies that $p$ satisfies
$D_{3}$, and hence is minimal by Theorem 3.6. That $n>1$ now implies that $p$
cannot be nonorthogonal to the constants. We thus conclude that we are in case
(a) of Theorem 3.6, and so $D_{m}$ holds of $p$ for all $m$. This in turn
implies that $(C_{m})$ holds of the equation, for all $m$.∎

##  4\. A formulation for $D$-varieties

There is a well studied correspondence between types in
$\operatorname{DCF}_{0}$ and so called “$D$-varieties”, and it is worth
reformulating Theorem 3.6 in these terms. The idea goes back to Buium [2], who
considered algebraic varieties over a differential field $(k,\delta)$ equipped
with an extension of the derivation $\delta$ to the structure sheaf of the
variety. Later, an equivalent formulation, paralleling more closely the
language of algebraic vector fields, came into use. We point the reader to
[15, $\S 2$] for a more detailed survey with references. We restrict ourselves
here to a very concise explanation.

Fix a differential field $(k,\delta)$ of characteristic zero. If
$V\subseteq\mathbb{A}^{n}$ is an irreducible affine algebraic variety over
$k$, then the prolongation of $V$ is the algebraic variety $\tau
V\subseteq\mathbb{A}^{2n}$ over $k$ whose defining equations are

$\displaystyle P(x_{1},\dots,x_{n})$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle P^{\delta}(x_{1},\dots,x_{n})+\sum_{i=1}^{n}\frac{\partial
P}{\partial x_{i}}(x_{1},\dots,x_{n})\cdot y_{i}$ $\displaystyle=$
$\displaystyle 0$

for each $P$ vanishing on $V$. Here $P^{\delta}$ denotes the polynomial
obtained by applying $\delta$ to the coefficients of $P$. The projection onto
the first $n$ coordinates gives us a surjective morphism $\pi:\tau V\to V$.

######  Definition 4.1.

A $D$-variety over $k$ is a pair $(V,s)$ where $V$ is an irreducible algebraic
variety over $k$ and $s:V\to\tau V$ is a regular section to the prolongation
defined over $k$. A $D$-subvariety of $(V,s)$ is then a $D$-variety $(W,t)$
where $W$ is a closed subvariety of $V$ over $k$ and $t=s|_{W}$.

Note that if $\delta=0$ on $k$, then the above equations for $\tau V$ reduce
to the familiar equations for the tangent bundle $TV$. A $D$-variety in that
case is nothing other than an algebraic vector field, and a $D$-subvariety is
an invariant subvariety of the vector field. In general, $\tau V$ will be a
torsor for $TV$; for each $a\in V$ the fibre $\tau_{a}V$ is an affine
translate of the tangent space $T_{a}V$.

Associated to a $D$-variety $(V,s)$ over $k$ is a certain complete type over
$k$ in $\operatorname{DCF}_{0}$, namely the generic type of $(V,s)$ over $k$.
This type, $p(x)$, is axiomatised by asserting that $x$ is Zariski-generic in
$V$ over $k$, and that the following system of order $1$ algebraic
differential equations holds: $s(x)=(x,\delta(x))$. In fact, up to
interdefinability, all finite rank types in $\operatorname{DCF}_{0}$ arise in
this way; as the generic types of $D$-varieties. Results about finite rank
types in $\operatorname{DCF}_{0}$, such as Theorem 3.6 above, can therefore be
translated into algebro-geometric statements about $D$-varieties. For example,
the condition we called $D_{m}$ in the previous section, namely that of having
every distinct $m$-tuple of realisations being independent, when applied to
the generic type of $(V,s)$ translates to the absence of any proper
$D$-subvarieties of $(V^{m},s^{m})$ projecting dominantly onto each co-
ordinate, other than the diagonals. In particular, we obtain

######  Corollary 4.2.

Suppose $(V,s)$ is a $D$-variety over $(k,\delta)$ with $\dim V>1$. If the
third cartesian power $(V^{3},s^{3})$ admits no proper $D$-subvarieties over
$k$ projecting dominantly onto each co-ordinate, other than the diagonals,
then the same holds of $(V^{m},s^{m})$ for all $m$.

###### Proof.

Let $p\in S(k)$ be the generic type of $(V,s)$ over $k$. The assumption on
$(V^{3},s^{3})$ tells us that $p$ satisfies $D_{3}$. Hence, by Theorem 3.6,
$p$ is minimal. Now, the fact that $\dim V>1$ rules out the possibility of $p$
being nonorthogonal to the constants. So we are in case (a) of the Theorem,
and $p$ satisfies $D_{m}$ for all $m$. This implies the desired conclusion
about $D$-subvarieties of $(V^{m},s^{m})$ over $k$. ∎

Specialising to the case when $\delta=0$ on $k$ yeilds Theorem B of the
Introduction. When $\dim V=1$, then one has to make the assumption on
$(V^{4},s^{4})$ in order to rule out case (b) of Theorem 3.6 and thus conclude
it for all $m$.

##  5\. The case of compact complex manifolds

As mentioned in the introduction, the main part of Theorem 3.6 holds also in
$\operatorname{CCM}$, the theory of compact complex manifolds. This translates
into the following statement in bimeromorphic geometry that may be of
independent interest:

######  Theorem 5.1.

Suppose $f:X\to Y$ is a fibre space of compact Kähler manifolds111More
generally our results apply to compact complex anlaytic varieties in Fujiki’s
class $\mathscr{C}$. In fact, all that is needed is that they be essentially
saturated in the sense of [20].. Suppose that the three-fold fibred product
$X\times_{Y}X\times_{Y}X$ contains no proper complex analytic subvarieties
that projects onto each co-ordinate, other than the diagonals. Then the
‘general’ fibre of $f$ is simple.

Let us define the terminology used here. That $f$ is a fibre space means that
off a proper complex analytic subset of $Y$ the fibres are irreducible. A
property holds of the ‘general’ fibre of $f$ if it holds of all fibres off a
countable union of proper complex analytic subsets of $Y$. Finally, we mean
simple in the sense of [7]; namely, not covered by an analytic family of
proper positive-dimensional analytic subsets.

We will not give a proper proof of this theorem, as it is really the same as
(part of) Theorem 3.6. But let us give some explanations. First of all, we
work in $\operatorname{CCM}$ which is the theory of the multisorted structure
$\mathcal{A}$ where there is a sort for each compact complex analytic variety
and a predicate for every complex analytic subset of a cartesian product of
sorts. The role of the constants is played by the projective line which is a
sort on which the induced structure is bi-interpretable with that of a pure
algebraically closed field. The truth of the Zilber dichotomy in
$\operatorname{CCM}$ tells us that every non locally modular minimal type is
nonorthogonal to the projective line. We suggest [21] for an introduction to,
and survey of, the subject.

Theorem 5.1 is proved by considering the generic type $p$ of the generic fibre
of $f$, observing that the assumption on the three-fold fibred product
expresses precisely that $p$ satisfies $D_{3}$, and then deducing, exactly as
in the proof of Theorem 3.6, that $p$ is minimal. But for a compact complex
analytic variety to be simple is equivalent to its generic type in
$\operatorname{CCM}$ being minimal.

But there is a subtlety: the structure $\mathcal{A}$ is not saturated and so
“generic” in the above paragraph has to be understood, a priori, in the sense
of a nonstandard elementary extension of $\mathcal{A}$. See [19, $\S 2$] for
details about the Zariski geometry in nonstandard models. The reason for
restricting to the Kähler case is then precisely so that we can, following
[20], find a suitable countable language in which the standard model is
saturated. We can thus conclude simplicity of the ‘general’ fibre in
$\mathcal{A}$ from simplicity of the generic fibre in an elementary extension.

Note that we only concluded minimality, and did not state an analogue of the
“moreover” clause in Theorem 3.6. The reason for this is that there remains
some ambiguity about the characterisation of nontrivial locally modular types
in $\operatorname{CCM}$. It is known that they are nonorthogonal to simple
complex tori, but it seems as yet unclear whether one has the same control
over parameters that one does in $\operatorname{DCF}_{0}$. This is related to
Question 5.1 of [16].

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# Envision of mmWave Wireless Communication with Artificial Intelligence

Quanda Zhang
School of Telecommunication
Daqing Normal College
<EMAIL_ADDRESS>
Hudi Wang
School of Telecommunication
Daqing Normal College
<EMAIL_ADDRESS>

###### Abstract

The future wireless communication system faces the bottleneck of the shortage
of traditional spectrum resources and the explosive growth of the demand for
wireless services. Millimeter-wave communication with spectral resources has
become an effective choice for the next generation of wireless broadband
cellular communication. However, the transmission path loss is large and
oxygen and water molecules absorb Characteristics such as seriousness have
brought great challenges to millimeter wave communication, and it is necessary
to seek a technical approach different from low-frequency wireless
communication. In the analysis of millimeter wave transmission characteristics
After the analysis, the research progress of millimeter wave communication
technology and the RF front-end is comprehensively analyzed, and the
technology of millimeter wave communication is thoroughly analyzed with
technical challenges and proposed corresponding research directions. Finally,
the advances of artificial intelligence and machine learning are also applied
into millimeter wave communication, so we also cover these parts in this
paper.

##  1 Introduction

With the rapid development of electronic information technology, especially
wireless With the development of communication technology and the
popularization of 4G wireless communication technology, in just 10 years, the
amount of IP data processed by wireless networks has exceeded more than 100x:
from 2010 to less than 3 EB (Exabyte, Exabyte bytes) to 190 exabytes in 2018,
and even on track for 2020 More than 500 exabytes. Video data processing is by
far the main factor behind the growth of data processing in wireless networks.
Adaptive Personal Communication Technology. Emerging Mobile Internet and
Internet of Things Technologies are also advancing by leaps and bounds. In the
future, the number of network devices and smart terminals continues to grow,
the demand for communication services is getting higher and higher, resulting
in the demand for network traffic exponential growth.

In order to meet the growth of communication traffic thousands of times in the
next few years, development needs, and research on wireless communication
technology in industry and academia People are in full swing on key
technologies for 5G mobile communications research [1]. The key technology of
5G mobile communication is embodied primarily in wireless transmission
technology and network technology [2, 3]; the former mainly includes large-
scale line array, ultradense networking, new multiple access and full-spectrum
access, etc. [4]. The latter mainly include self-organizing network (SON),
soft-defined network network (SDN), network function virtualization (NFV),
etc. [5, 6, 7, 8, 9, 10]. The main development goal of mobile communication
technology is to communicate with other wireless mobile devices. Communication
technology is closely linked, providing for the rapid development of the
mobile Internet. Provide ubiquitous basic business capabilities, mainly from
three aspects Carry out research and exploration: first, through the
introduction of new wireless transmission technology [11] The resource
utilization rate is increased by more than 10 times on the basis of 4G;
second, by introducing a new architecture (such as ultra-dense networking,
control noodle making and data surface separation, etc.) and deeper
intelligent capabilities Improve the throughput of the whole system by about
25 times; the final further mining new frequency resources (millimeter wave,
visible light, laser communication, etc.), so that the mobile communication
frequency resource is expanded by about 4 times [12, 13, 14, 15, 16].

Millimeter-wave (mmWave) wireless communication is a key technology that
enables the high data rates required for 5G communication. These high
frequencies, which range from 30 to 300 GHz, allow for the transmission of
large amounts of data over short distances, making them ideal for high-speed,
high-capacity applications such as 5G.

One of the main advantages of mmWave technology is its ability to support
massive amounts of data transmission. This is because the high frequencies
used in mmWave communication allow for a wider bandwidth, which means that
more data can be transmitted in a given amount of time. This increased
capacity is critical for 5G, which is expected to support a wide range of
applications, from enhanced mobile broadband to the Internet of Things (IoT)
and beyond.

Another important benefit of mmWave technology is its ability to support
ultra-low latency communication. Latency is the time it takes for a signal to
travel from one point to another, and it is a crucial factor in the
performance of any wireless communication system. In mmWave systems, the short
distances over which the signals travel and the high frequencies used allow
for ultra-low latency communication, making it possible to support real-time
applications such as augmented and virtual reality.

In addition to its high data rates and ultra-low latency, mmWave technology
also offers several other benefits for 5G communication. For example, mmWave
signals are highly directional, which means that they can be focused on
specific areas, reducing interference and improving signal strength. This is
especially useful in urban environments, where there are many potential
sources of interference. Additionally, mmWave technology can support multiple-
input multiple-output (MIMO) communication, which uses multiple antennas to
transmit and receive signals simultaneously, further increasing capacity and
reliability.

Overall, mmWave technology is an essential component of 5G communication,
offering high data rates, ultra-low latency, and other key benefits that make
it possible to support a wide range of applications and services. As 5G
continues to develop and expand, mmWave technology will play a critical role
in enabling the high-speed, high-capacity communication that is required to
support the growing demand for data.

##  2 Challenges

According to Shannon’s Theorem, it is known that expanding the channel
capacity is the most direct way. The most effective way is to increase the
system bandwidth. Currently used for mobile communication, the bandwidth of
the microwave band is only 600 MHz, and 5G mobile in 2020 requires about 1 GHz
of bandwidth, so one needs to dig digging for new spectrum resources to meet
the needs of mobile Internet traffic in the next 10 years, 1,000 times growth
in volume. Using the millimeter wave frequency band Potentially large
bandwidth (30 300 GHz) to provide higher data rates However, the millimeter
wave [17, 18, 19, 20, 21, 22, 23] has fairly rich spectrum resources, which
can be utilized for other communication purposes. For example, it can be used
for realizing high-speed wireless transmission output, but many technical
challenges are encountered in the process of implementing mmWave cellular
network communication many technical challenges:

  * •

Shadowing and Rapid Channel Fluctuations: Shadowing effects can cause an
outage or poor mmWave propagation channel quality. Meanwhile, the security of
the communication channels is also very important, so there have been works,
which use encryption techniques [24, 25] to protect the security of it. The
linear relationship of the rate means that in the mmWave channel phase dry
times are very small, e.g. for a moving speed of 60 km/h In terms of degrees,
the Doppler spread at 60 GHz is about 3 kHz, that is, millimeter wave channels
vary in milliseconds characteristics, so how to quickly track mmWave channel
changes. It is an important problem in millimeter wave communication.

  * •

Multiuser coordination: research and application of millimeter wave
communication Focus on point-to-point links (such as cellular backhaul) or
limit LANs and personal area networks that limit the number of users and
prohibit the MAC protocol for multi-user transmission. For spatial
multiplexing with spectral efficiency, a new mechanism needs to be studied to
coordinate the mmWave spatial multiplexing transmission mechanism.

  * •

Power Consumption: Power consumption is linear with the sample rate, while it
is exponentially related to the number of sampling bits [26]. For low power
consumption and low-cost equipment, on a large bandwidth and large scale,
achieving high-resolution quantization on antennas is not practical;
especially in terms of high-definition video transmission, millimeter wave
communication is facing more severe technical challenges.

  * •

Range and Directional Communication: According to Friis’ law of propagation,
the loss of an omnidirectional path in free space is proportional to the
square of the frequency. Therefore, millimeter wave. The communication system
requires the use of large-scale array antenna communication technology
technologies such as beamforming and directional transmission to compensate
for millimeter wave propagation path loss.

In detail, we extend some of the above challenges and present more details and
solutions to them.

One of the main challenges of millimeter wave (mmWave) wireless communication
is its limited range. Because mmWave signals are highly directional, they have
a limited range and are susceptible to interference from obstacles such as
buildings and trees. This can make it difficult to establish and maintain a
stable mmWave connection, particularly in urban environments where there are
many potential sources of interference.

To overcome this challenge, several solutions have been proposed, including
the use of beamforming technology. Beamforming uses multiple antennas to focus
the mmWave signal in a specific direction, increasing its range, and reducing
interference. Additionally, adaptive beamforming algorithms can be used to
automatically adjust the beamforming pattern in response to changes in the
environment, further improving the performance of the mmWave system.

Another challenge of mmWave technology is its sensitivity to weather
conditions. Because mmWave signals are attenuated more easily than lower-
frequency signals, they can be significantly affected by rain, snow, and other
weather conditions. This can result in reduced signal strength and reduced
capacity, making it difficult to maintain a stable mmWave connection in
adverse weather conditions.

To address this challenge, several solutions have been proposed, including the
use of advanced modulation schemes and error correction codes. By using more
advanced modulation schemes and error correction codes, it is possible to
increase the robustness of the mmWave signal, allowing it to maintain its
performance in adverse weather conditions. Additionally, the use of multiple-
input multiple-output (MIMO) communication can also help to improve the
robustness of the mmWave signal by using multiple antennas to transmit and
receive signals simultaneously.

In addition to these challenges, mmWave technology is also faced with the
challenge of high power consumption. Because mmWave signals have a high
bandwidth, they require a large amount of power to transmit and receive. This
can be a significant challenge for mobile devices, which have limited battery
life and are not always able to support the high power requirements of mmWave
technology.

To overcome this challenge, several solutions have been proposed, including
the use of power-efficient hardware and software design. Using power-efficient
hardware and software, it is possible to reduce the power consumption of
mmWave systems, making it possible to support mmWave communication on mobile
devices with limited battery life. Additionally, the use of advanced power
management techniques, such as dynamic voltage and frequency scaling, can also
help reduce the power consumption of mmWave systems.

##  3 Potential Opportunities

This high-frequency communication technology offers many benefits, including
high data rates, ultra-low latency, and the ability to support massive amounts
of data transmission. As such, mmWave technology has the potential to open up
many new opportunities in the fields of telecommunications, healthcare, and
beyond.

One of the main opportunities offered by mmWave technology is the ability to
support enhanced mobile broadband services. With its high data rates and wide
bandwidth, mmWave technology is ideally suited to support the growing demand
for data-intensive applications and services, such as streaming video and
online gaming. This could lead to the development of new and innovative mobile
services, such as virtual reality and augmented reality, which require high-
speed, high-capacity communication.

Another potential opportunity offered by mmWave technology is the ability to
support the Internet of Things (IoT). The IoT is a network of connected
devices, sensors, and other objects that can communicate with each other and
with the wider internet. With its high data rates and low latency, mmWave
technology is well-suited to support the high-speed, low-latency communication
required by the IoT, making it possible to connect more devices and sensors
than ever before. This could lead to the development of new IoT-based
applications and services, such as smart cities and connected healthcare.

In addition to these opportunities, mmWave technology also has the potential
to transform the field of healthcare. With its ability to support high-speed,
high-capacity communication, mmWave technology could be used to connect
medical devices and sensors, enabling real-time monitoring and remote
diagnostics. This could lead to the development of new and innovative
healthcare applications, such as remote surgery and telemedicine, which
require high-speed, low-latency communication.

Overall, mmWave technology offers many potential opportunities for the future
of communication and beyond. As this technology continues to develop and
expand, it has the potential to open up new and exciting possibilities in a
wide range of fields, from telecommunications and healthcare to the IoT and
beyond.

###  3.1 Applications of Artificial Intelligence on mmWare

Artificial intelligence (AI) and machine learning (ML) have been increasingly
applied to 5G and mmWare communication in order to improve performance, reduce
costs, and increase efficiency. Some examples of how AI is being used in these
areas include:

  1. 1.

Network optimization: AI algorithms can be used to analyze network traffic and
identify patterns that can be used to optimize the allocation of network
resources. This can help to improve network coverage, reduce latency, and
increase capacity.

  2. 2.

Predictive maintenance: AI can be used to analyze data from sensors and other
sources to predict when equipment is likely to fail, allowing maintenance to
be scheduled before problems occur.

  3. 3.

Self-organizing networks: AI algorithms can be used to help networks adapt to
changing conditions, such as changes in traffic patterns or the addition of
new devices. This can help to improve network reliability and performance.

  4. 4.

Traffic prediction: AI can be used to analyze traffic patterns and predict
future demand, allowing networks to be configured to meet expected demand.

  5. 5.

Security: AI can be used to detect and respond to security threats in real-
time, helping to protect networks from cyber attacks.

  6. 6.

Resource allocation: AI can be used to optimize the allocation of resources,
such as bandwidth and power, in order to improve overall network performance.
For example, AI algorithms can be used to predict which users are likely to
require more resources at any given time and allocate those resources
accordingly.

  7. 7.

Network slicing: AI can be used to create and manage virtual networks, known
as "slices," which can be customized to meet the specific requirements of
different types of users or applications. For example, a slice might be
created for a group of connected vehicles that need low latency and high
reliability, or for a group of mobile users who need high-bandwidth data
services.

  8. 8.

Edge computing: AI can be used to process and analyze data at the edge of the
network, close to the source of the data, rather than in a central location.
This can reduce latency and improve the efficiency of data processing.

  9. 9.

Interference management: AI can be used to predict and mitigate interference
in mmWave communication systems, which can be particularly challenging due to
the high frequency and limited propagation range of these signals.

  10. 10.

Spectrum management: AI can be used to optimize the use of available spectrum,
for example by allocating different frequencies to different users or
applications based on their requirements and the conditions of the
environment. This can help to improve the efficiency of the network and reduce
interference.

Overall, the use of AI in 5G and mmWare communication is still in the early
stages, but it has the potential to significantly improve the performance,
efficiency, and cost-effectiveness of these networks. AI can be used to
optimize various aspects of network design, operation, and management, such as
network optimization, predictive maintenance, self-organizing networks,
traffic prediction, security, resource allocation, network slicing, edge
computing, interference management, and spectrum management.

As AI technologies continue to advance and become more widely adopted, it is
likely that they will play an increasingly important role in the development
and deployment of 5G and mmWare communication systems. However, it is also
important to consider the potential risks and challenges associated with the
use of AI in these contexts, such as the need to ensure privacy, security, and
ethical considerations.

##  4 Discussion and Conclusion

In a comprehensive review of the research status and development of millimeter
wave communication technology, based on the development trend, the millimeter
wave communication system is analyzed as a technical challenge. Multiplexing
Gain and Diversity from Realizing Millimeter Wave Communication Gain Angle,
Analyzing Digital-Analog Hybrid Antennas for mmWave Communications The
formation of the architecture is really applied to the millimeter wave
technology in the actual communication system technical challenges and
problems that still need to be solved, and introduces the current. In
implementing the IEEE 802.11aj LAN protocol specification, prototyping
progress and research results obtained.

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