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ZARISKI DENSE ORBITS FOR ENDOMORPHISMS OF A POWER OF
THE ADDITIVE GROUP SCHEME DEFINED OVER FINITE FIELDS
DRAGOS GHIOCA AND SINA SALEH
Abstract. We prove the Zariski dense orbit conjecture in positive characteristic for endo-
morphisms of GN
adefined over Fp.
1. Introduction
1.1. Notation. We let N0:= N∪ {0}denote the set of nonnegative integers.
For any morphism Φ on a variety Xand for any integer n≥0, we let Φnbe the n-th
iterate of Φ (where Φ0is the identity map id := idX, by definition). For a point x∈X, we
denote by OΦ(x) the orbit of xunder Φ, i.e., the set of all Φn(x) for n≥0. When Φ is only a
rational self-map of X, the orbit OΦ(x) of the point x∈Xis well-defined if each Φn(x) lies
outside the indeterminacy locus of Φ. For any self-map Φ on a variety X, we say that x∈X
is preperiodic if its orbit OΦ(x) is finite.
We denote by Mm,n(R) the set of m×n-matrices with entries in the ring R; we denote by
Imthe identity m×m-matrix.
1.2. The classical Zariski dense orbit conjecture. The following conjecture was moti-
vated by a similar question raised by Zhang [Zha06], and it was formulated independently by
Medvedev and Scanlon [MS14] and by Amerik and Campana [AC08].
Conjecture 1.1. Let Xbe a quasiprojective variety defined over an algebraically closed field
Kof characteristic 0and let Φ : X99K Xbe a dominant rational self-map. Then either there
exists α∈X(K)whose orbit under Φis well-defined and Zariski dense in X, or there exists
a non-constant rational function f:X99K P1such that f◦Φ = f.
There are several partial results known towards Conjecture 1.1 (for example, see [AC08,
BGRS17,BGR17,GH18,GSa,GS19,GS17,GX18,MS14]).
1.3. The picture in positive characteristic. If Khas characteristic p > 0, then Con-
jecture 1.1 does not hold due to the presence of the Frobenius endomorphism (see [BGR17,
Example 6.2] and also, [GS21, Remark 1.2]). Based on the discussion from [GS21], the authors
proposed the following conjecture as a variant of Conjecture 1.1 in positive characteristic.
2010 Mathematics Subject Classification. Primary 14K15, Secondary 14G05.
Key words and phrases. Zariski dense orbits, Medvedev-Scanlon conjecture, additive polynomials over fields
of positive characteristic.
1
2 DRAGOS GHIOCA AND SINA SALEH
Conjecture 1.2. Let Kbe an algebraically closed field of positive transcendence degree over
Fp, let Xbe a quasiprojective variety defined over K, and let Φ : X99K Xbe a dominant
rational self-map defined over Kas well. Then at least one of the following three statements
must hold:
(A) There exists α∈X(K)whose orbit OΦ(α)is Zariski dense in X.
(B) There exists a non-constant rational function f:X99K P1such that f◦Φ = f.
(C) There exist positive integers mand r, there exists a variety Ydefined over a finite
subfield Fqof Ksuch that dim(Y)≥trdegFpK+ 1 and there exists a dominant
rational map τ:X99K Ysuch that
τ◦Φm=Fr◦τ,
where Fis the Frobenius endomorphism of Ycorresponding to the field Fq.
Conjecture 1.2 has been proven in the case of algebraic tori in [GS21] and more generally
in the case of all split semiabelian varieties defined over Fpin [GSb]. For an illustration of the
trichotomy in the conclusion of Conjecture 1.2, we refer the reader to [GS21, Example 1.6].
Also, we note that one definitely requires the hypothesis that trdegFpK≥1 in Conjecture 1.2
since for any self-map Φ defined over Fp, each point of X(Fp) is preperiodic and therefore,
condition (A) cannot hold; on the other hand, there are plenty of examples of maps Φ defined
over Fpfor which neither condition (B) nor condition (C) would hold.
1.4. Our results. We prove the following more precise version of Conjecture 1.2 in the case
of group endomorphisms of GN
adefined over Fp.
Theorem 1.3. Let N∈Nand let Lbe an algebraically closed field of characteristic psuch
that trdegFpL≥1. Let Φ : GN
a−→ GN
abe a dominant group endomorphism defined over Fp.
Then at least one of the following statements must hold.
(A) There exists α∈GN
a(L)whose orbit under Φis Zariski dense in GN
a.
(B) There exists a non-constant rational function f:GN
a99K P1such that f◦Φ = f.
(C) There exist positive integers mand r, a positive integer N0greater than or equal to
trdegFpL+ 1 and a dominant group homomorphism τ:GN
a−→ GN0
asuch that
(1.3.1) τ◦Φm=Fr◦τ,
where Fis the usual Frobenius endomorphism of GN0
ainduced by the field automor-
phism x7→ xp.
1.5. Discussion of our proof. The strategy of our proof for Theorem 1.3 is as follows.
Suppose we have a group endomorphism Φ : GN
a−→ GN
adefined over Fqwhere q=p`for
some `∈N. As shown in [Pog17, Proposition 3.9], the endomorphism Φ is given by an N-
by-Nmatrix A(acting linearly on GN
a) whose entries are one-variable additive polynomials
in the variables x1, . . . , xN(corresponding to the Ncoordinate axes of GN
a), i.e.,
(1.3.2) A= (fi,j (xj))1≤i,j≤N,
ZARISKI DENSE ORBITS 3
where a polynomial f(x) is additive if it is of the form
(1.3.3)
r
X
k=0
cixpi;
furthermore, since Φ is defined over Fq, then each coefficient of each additive polynomial
fi(xj) belongs to Fq. We denote by Fthe Frobenius endomorphism (of L) corresponding to
the finite field Fp(i.e., x7→ xp). So, the action of Φ is given by a matrix of polynomials (with
coefficients in Fq) in the Frobenius operator, i.e., the entries of our matrix Alive in Fq[F].
We will study technical properties of Fq[F] and of the ring of matrices with entries in Fq[F]
in Section 3.
Now, for any given point α∈GN
a(L), the orbit OΦ(α) is contained in a finitely generated
Fp[F]-submodule Γ of GN
a. If we assume that OΦ(α) is not Zariski dense, then it must be
contained in some proper subvariety Vof GN
a. Then, we can describe the structure of V(L)∩Γ
using [Ghi08, Theorem 2.6]; for more details, see Section 3.3. Then we will use the fact that Φ
is integral over the commutative ring Fp[F`]; for more details, see Section 3. We also employ
several reductions discussed in Section 4which allow us to split the action of Φ into (Φ1,Φ2)
where Φ1is given by the diagonal action of powers of the Frobenius endomorphism, while the
minimal polynomial of Φ2over Fp[F`] has roots that are multiplicatively independent with
respect to F. This helps us reduce Theorem 1.3 into two separate extreme cases which are
much more convenient to deal with; the general case in Theorem 1.3 then follows.
1.6. Discussion of possible extensions. We note that our approach does not generalize to
the case the endomorphism Φ of GN
ais defined over an arbitrary field Lof characteristic p. The
reason is that in the case of general group endomorphisms of GN
a, the orbit will not necessarily
be contained in a finitely generated Fp[F]-submodule of GN
a, and thus, we cannot use the F-
structure theorem proven in [Ghi08, Theorem 2.6] anymore. In the special case when for each
1≤i, j ≤N, the linear term c0of fi(xj) (i.e., c0=f0
i(xj) from equations (1.3.2) and (1.3.3)) is
transcendental over Fp, then one can reformulate Conjecture 1.2 for (GN
a,Φ) in the context of
Drinfeld modules of generic characteristic. Even though there is a very rich arithmetic theory
for Drinfeld modules of generic characteristic built in parallel to the classical Diophantine
geometry questions in characteristic 0 (see, for example, [Bre05,CG20,GT07,GT08,Sca02]),
there are still several technical difficulties to overcome in this case alone. Furthermore, when
we deal with the most general case of an endomorphism of GN
a(defined over a field Lof
characteristic p), in which case some of the derivatives of the polynomials fi(xj) are in a finite
field, while others are transcendental over Fp, then there are significant more complications
since then we would be dealing with a mixed arithmetic structure coming from the action of
Drinfeld modules of both generic characteristic and also of special characteristic (see [Ghi05]
for a sample of difficulties arising in the context of Drinfeld modules of special characteristic).
1.7. Plan for our paper. In Section 4we use the technical results proven in Sections 2and 3
to show that instead of proving Theorem 1.3 for the dynamical system (GN
a,Φ), we can instead
prove it for the dynamical system (GN1
a×GN2
a,(Φ1,Φ2)) where N=N1+N2, Φ1is a group
endomorphism of GN1
agiven by the coordinate-wise action of powers of the Frobenius endo-
morphism, and the minimal polynomial of Φ2over Fp[F`] has roots that are multiplicatively
4 DRAGOS GHIOCA AND SINA SALEH
independent with respect to F. In other words, we reduce Theorem 1.3 to Proposition 5.2.
We conclude our paper by proving Proposition 5.2 in Section 5.
2. Technical Background
In this Section we gather some useful results for our proofs which come from two dif-
ferent sources: the theory of skew fields (see Section 2.1 and more generally [Coh95]) and
the F-structure theorem proven by Moosa-Scanlon [MS04] in order to describe intersections
of subvarieties of GN
mwith finitely generated subgroups of GN
mFp(t)(see Lemma 2.4 in
Section 2.2).
2.1. Some results about splitting matrices over skew fields. In this Section we state
useful results about splitting matrices over skew fields which will be used later in our proofs;
for more details about skew fields, we refer the reader to [Coh95].
Fact 2.1. Let Kbe a skew field with centre k. Suppose that Ais a matrix in Mn,n(K)which
is algebraic over k. Let f(x) = f0(x)f1(x)be the minimal polynomial of Aover kwhere
f0, f1∈k[x]are coprime. Then, Ahas a conjugate of the form
A0⊕A1
such that the minimal polynomials of A0and A1over kare equal to f0and f1, respectively.
Proof. Using the arguments after [Coh95, Corollary 8.3.4, p. 381–382], Amust have a conju-
gate of the form
B1⊕ · · · ⊕ Br,
where each Bicorresponds to an elementary divisor, say qi. Since f0and f1are coprime, each
qimust divide exactly one of f0and f1and be coprime with respect to the other one. So,
assume without loss of generality that for some i≥0 we have
qj|f0and (qj, f1)=1,
for all j≤iand
qj|f1and (qj, f0)=1,
for j > i. Letting
A0=B1⊕ · · · ⊕ Bi,and A1:= Bi+1 ⊕ · · · ⊕ Br,
gives us the desired conclusion.
Fact 2.2. Let Kbe a skew field with centre kand A∈Mn,n(K)be a matrix with a minimal
polynomial equal to (x−α)rfor some α∈kand r∈N. Then, there exist an invertible matrix
P∈Mn,n(K)such that
P−1AP =Jα,r1M· · · MJα,rm,
where Jα,s is the s-by-sJordan canonical matrix having unique eigenvalue αand its only
nonzero entries away from the diagonal being the entries in positions (i, i+1) (for i= 1, . . . , s−
1), which are all equal to 1.
Proof. This is a consequence of the the discussion after [Coh95, Corollary 8.3.4, p. 381–384];
also, see [Coh95, Theorem 8.3.6].
ZARISKI DENSE ORBITS 5
2.2. A special type of Diophantine equation. In this Section we prove Lemma 2.4 that
gives an asymptotic upper bound on the number of solutions of a special type of equation
given by equation (2.4.1). This bound will be instrumental in our proof Theorem 1.3. We
start with an easy result which will be used in our proof for Lemma 2.4. The result is actually
a consequence of Vandermonde determinants and still holds if we know that the equation
(2.3.1) holds for rconsecutive n’s.
Lemma 2.3. Let Kbe a field. Suppose that distinct non-zero elements λ1, . . . , λr∈Kare
given. Moreover, suppose that for some N > 0and c1, . . . , cr∈Kwe have
(2.3.1) c1λn
1+· · · +crλn
r= 0,
for every n≥N. Then, c1=· · · =cr= 0.
Before proving the main result of this Section, we recall that for a subset S⊆N, the (upper
asymptotic) density (also called natural density) of Sis defined as
µ(S) := lim sup
n→∞
#{m∈S:m≤n}
n.
Lemma 2.4. Let K=Fp(t), let r∈Nand let c0, c1, . . . , cr∈K. Suppose that λ∈K\ {0}
is multiplicatively independent with respect to t. Let Sbe the set of positive integers mfor
which there exist positive integers n1, . . . , nrsuch that
(2.4.1) λm=c0+
r
X
i=1
citni.
Then the natural density of Sis equal to zero.
Proof. Solving equation (2.4.1) is equivalent with analyzing the intersection of the hyperplane
V⊂G1+r
mgiven by the equation:
y=c0+
r
X
i=1
cixi
with the subgroup Γ of G1+r
mspanned by (λ, 1,...,1), (1, t, 1,...,1), . . . , (1,...,1, t). Then
by Moosa-Scanlon’s F-structure theorem (see [MS04, Theorem B]), we know the intersection
is a union of finitely many sets R1, . . . , Ru; furthermore, each set Rfrom the list R1, . . . , Ru
is of the form
R:= γ0·S(γ1, . . . , γs;δ1, . . . , δs)·H,
where His a subgroup of Γ, while γi∈G1+r
m(K) for each i= 0, . . . , s and δj∈Nfor each
j= 1, . . . , s, and the set S(γ1, . . . , γs;δ1, . . . , δs) is defined as follows:
S(γ1, . . . , γs;δ1, . . . , δs) :=
s
Y
j=1
γpkiδi
i:ki∈Nfor i= 1, . . . , s
.
So, the set Rconsists of points of the form
(2.4.2) γ0·
s
Y
j=1
γpkiδi
i·,
6 DRAGOS GHIOCA AND SINA SALEH
where is in the subgroup H; moreover, there exists a positive integer `such that γ`
i∈Γ
for each i= 0, . . . , m. For more details regarding the F-sets structure for the intersection of
a subvariety of a semiabelian variety with a finitely generated subgroup, we refer the reader
both to [MS04] and also to [Ghi08], for further refinements of Moosa-Scanlon’s original result.
Now, in order to show that the set of all positive integers mfor which equation (2.4.1)
is solvable has natural density equal to 0 (as a subset of N), it suffices to prove that the
projection of Hon the first coordinate of G1+r
mis trivial; this way, the set of those m’s that
satisfy equation (2.4.1) would be a sum of powers of p(due to equation (2.4.2)) and thus, it
would have natural density zero.
Let us assume, for the sake of contradiction, that Hprojects non-trivially on the first
coordinate of G1+r
m. This means that there exists some tuple
(m0, `1, . . . , `r)
with m06= 0 such that a coset of the cyclic subgroup H0spanned by
(λm0, t`1, . . . , t`r)
would be contained in the intersection of Vwith the subgroup Γ.
So, letting `0:= 0, there must exist some constants di∈Fp(t) (depending on the ci’s) such
that for all positive integers n, we have
(2.4.3) λm0n=
r
X
i=0
dit`in.
Combining the terms with the same exponent, we may assume without loss of generality that
the powers `1, . . . , `rare distinct. Using equation (2.4.3) and Lemma 2.3 we get that λm0
must be equal to t`ifor some i= 0,1, . . . , r (note that not all of the di’s could equal 0 since
λ6= 0), which means that λis multiplicative dependent with respect to t. This contradicts
our hypothesis and thus delivers the conclusion of Lemma 2.4.
3. Arithmetic and algebraic properties for rings involving the Frobenius
operator
From now on in this paper, we let pbe a prime number and let Fbe the Frobenius operator
corresponding to the field Fp.
3.1. Operators involving the Frobenius operator. We consider the polynomial ring
Fp[F] whose elements are operators of the form Pn
i=0 aiFiwhich act on any field Lof char-
acteristic pas follows:
(3.0.1) n
X
i=0
aiFi!(x) =
n
X
i=0
aixpifor x∈L.
Since Fleaves invariant each element of Fp, we can identify Fp[F] with a polynomial ring in
one variable over Fp; in particular, we can consider its fraction field, denoted Fp(F). For an
ZARISKI DENSE ORBITS 7
algebraically closed field L(of characteristic p), the action of an element uof Fp(F), which is
of the form
(3.0.2) u:= Pn
i=0 aiFi
Pm
j=0 bjFj
(for some non-negative integers mand nand moreover, the denominator in equation (3.0.2) is
nonzero and is coprime with respect to the numerator) can be interpreted as a finite-to-finite
map ϕu:L−→ Lwhich has the property that to each element x∈L, it associates the finitely
many elements y∈Lfor which
(3.0.3)
m
X
j=0
bjFj
(y) = n
X
i=0
aiFi!(x).
3.2. Linearly independent elements with respect to the maps from the polynomial
ring in the Frobenius operator. In this Section, we let Lbe an algebraically closed
field of positive transcendence degree over Fp. The following notion is used in our proof of
Proposition 5.2, which is a key technical step in deriving our main result (Theorem 1.3). First,
we note that similar to our construction of the ring of operators Fp[F] from Section 3.1, we
can construct the non-commutative ring of operators Fp[F].
Definition 3.1. Given elements δ1, . . . , δ`and γ1, . . . , γkin L, we say that γ1, . . . , γkare
linearly independent from δ1, . . . , δ`over Fp[F]if whenever there exist polynomial operators
P1(F), . . . , Pk(F)∈Fp[F]and Q1(F), . . . , Q`(F)∈Fp[F]such that
k
X
i=1
Pi(F)(γi) =
`
X
j=1
Qj(F)(δj),
then we must have that P1(F) = · · · =Pk(F)=0.
Moreover, in the special case `= 1 and δ1={0}, then the above condition simply translates
into asking that the points γ1, . . . , γkare linearly independent over Fp[F].
The following result will be used in our proof of Proposition 5.2.
Proposition 3.2. For any positive integers kand `, and any given elements δ1, . . . , δ`∈L,
there exist γ1, . . . , γk∈Lwhich are linearly independent from δ1, . . . , δ`over Fp[F].
Proof. We let L0⊂Lbe a finitely generated extension of Fpcontaining δ1, . . . , δ`. Then we
view L0as the function field of a projective, smooth variety Vdefined over Fp. We let ΩVbe
the set of inequivalent absolute values corresponding to the irreducible divisors of V. Since
there are only finitely many places of Vwhere the δj’s have poles, we can choose kelements
γi∈L0(for i= 1, . . . , k) such that there exist absolute values | · |vi∈ΩV(for i= 1, . . . , k)
satisfying the following properties:
(i) |δj|vi≤1 for each 1 ≤j≤`and each 1 ≤i≤k;
(ii) |γi|vi>1 for each i= 1, . . . , k; and
(iii) |γi|vj≤1 for each j6=i.
8 DRAGOS GHIOCA AND SINA SALEH
Conditions (i)-(iii) can be achieved since there exist infinitely many absolute values in ΩV
and so, we can proceed inductively on k, each time choosing an element γiwhich has a pole
at some place of Vwhere none of the δj’s and also none of the γ1, . . . , γi−1have poles.
Now, we claim that the elements γ1, . . . , γkare linearly independent from δ1, . . . , δ`over
Fp[F]. Indeed, if there exist polynomial operators P1(F), . . . , Pk(F)∈Fp[F] and Q1(F), . . . , Q`(F)∈
Fp[F] such that
(3.2.1)
k
X
i=1
Pi(F)(γi) =
`
X
j=1
Qj(F)(δj),
then we assume there exists some i0∈ {1, . . . , k}such that Pi0(F)6= 0 and we will derive a
contradiction. Indeed, using conditions (ii)-(iii) above, we get that
(3.2.2)
k
X
i=1
Pi(F)(γi)vi0
=|Pi0(F) (γi0)|vi0>1.
Note that in order to derive inequality (3.2.2), we use the fact that if |γ|v>1, then for any
nonzero polynomial operator P(F)∈Fp[F] of degree D≥0 (in the operator F), we have that
|P(F)(γ)|v=|γ|pD
v.
On the other hand, using condition (i) above, we get that
(3.2.3)
`
X
j=1
Qj(F)(δj)vi0
≤1.
Inequalities (3.2.2) and (3.2.3) yield a contradiction along with equality (3.2.1). This shows
that indeed, the elements γ1, . . . , γkmust be linearly independent from δ1, . . . , δ`over Fp[F],
which concludes our proof for Proposition 3.2.
3.3. A Mordell-Lang type theorem for the additive group scheme. We will employ in
our proof of Proposition 5.2 a Mordell-Lang type theorem for the additive group scheme, which
was proven in [Ghi08, Theorem 2.6]. Before stating our technical result (see Proposition 3.3),
we need to introduce some notation.
Let Nbe a positive integer and we extend the action of the ring of operators Fp[F] on
GN
aacting diagonally. Let Lbe an algebraically closed field of characteristic p. Inspired by
the definition of F-sets introduced by Moosa and Scanlon in [MS04], we define the following
subsets of GN
a(L). So, for points γ1, . . . , γr∈GN
a(L) and positive integers k1, . . . , kr, we define
(3.2.4) S(γ1, . . . , γr;k1, . . . , kr) := (r
X
i=1
Fniki(γi): ni∈Nfor i= 1, . . . , r).
The following result is proven in [Ghi08, Theorem 2.6].
Proposition 3.3. Let X⊆GN
abe an affine variety defined over an algebraically closed field
Lof characteristic p. Let Fbe the usual Frobenius map x7→ xpand we extend the action
ZARISKI DENSE ORBITS 9
of Fp[F]to GN
aacting diagonally. Let Γ⊂GN
a(L)be a finitely generated Fp[F]-submodule.
Then the intersection X(L)∩Γis a finite union of sets of the form
(3.3.1) γ0+S(γ1, . . . , γr;k1, . . . , kr) + H,
for some points γ0, γ1, . . . , γr∈GN
a(L)and positive integers k1, . . . , kr, where S(γ1, . . . , γr;k1, . . . , kr)
is defined as in equation (3.2.4), while His an Fp[F]-submodule of Γ.
Proposition 3.3 can be viewed as a Mordell-Lang type statement for the additive group
scheme, in the same spirit as Moosa-Scanlon’s main result from [MS04] (which is a Mordell-
Lang theorem for semiabelian varieties defined over finite fields). Actually, the proof of [Ghi08,
Theorem 2.6] followed the exact strategy employed by Moosa-Scanlon for proving [MS04,
Theorem 7.8]. Both [Ghi08, Theorem 2.6] and [MS04, Theorem B] (and also their common
generalization for arbitrary commutative algebraic groups proven in [BGM]) are extensions in
positive characteristic of the classical Mordell-Lang Theorem proven by Faltings [Fal94] (for
abelian varieties) and by Vojta [Voj96] (for semiabelian varieties).
Remark 3.4.We make a couple of important observations regarding Proposition 3.3. First, the
statement in [Ghi08, Theorem 2.6] assumed the points in Γ live in a finitely generated, regular
extension of some given finite field; this can always be achieved and also it is not essential for
the proof, as observed in [MS04, Remark 7.11] (and also noted before the statement of [BGM,
Theorem 2.2]).
Second, just as shown in [MS04, Lemma 2.7], one can prove that the points γ0, γ1, . . . , γr
corresponding to an F-set as appearing in the intersection of X(L)∩Γ from equation (3.3.1)
live in the Fp[F]-division hull of Γ, i.e., there exists some nonzero polynomial P(F)∈Fp[F]
such that
(3.4.1) P(F)(γi)∈Γ for i= 0,1, . . . , r.
3.4. A non-commutative ring of operators. From now on, we fix q=p`for some given
`∈N. Then the polynomial ring of operators Fq[F] acting as in equation (3.0.1) is no longer
a commutative ring since for some c∈Fq\Fp, we have that cp6=c.
From now on, we define K:= Fq[F]⊗Fp[F`]Fp(F`) and so, any element in Kcan be written
as
(3.4.2)
`−1
X
i=0
aiFi,
where ai∈Fq[F`]⊗Fp[F`]Fp(F`)⊆Fq(F`); note that Fq(F`) is a field since F`fixes each
element of Fq.
3.5. Matrices of operators. In this Section, we study matrices whose entries are themselves
operators from K(see the notation from Section 3.4).
Notation 3.5. Let Abe a matrix in Mn,n(K)for some n∈N. Using equation (3.4.2), we
can find unique matrices A0, . . . , A`−1∈Mn,n (Fq(F`)) such that
A=
`−1
X
i=0
AiFi.
10 DRAGOS GHIOCA AND SINA SALEH
From now on, we will use the matrices A0, . . . , A`−1each one of them belonging to Mn,n(Fq(F`))
in order to identify the matrix A∈Mn,n(K). Furthermore, for convenience, we will often use
the n×n`-matrix (A0, . . . , A`−1)∈Mn,n`(Fq(F`)) to identify the matrix A∈Mn,n (K).
The next result is a simple consequence of multiplying matrices from Mn,n(K) and keeping
track of the decomposition of their action as given in Notation 3.5.
Proposition 3.6. Given a matrix A∈Mn,n(K)there exists a unique matrix ˜
A∈Mn`,n`(Fq(F`))
such that for every matrix B∈Mn,n(K)we have
(3.6.1) ((BA)0,...,(BA)`−1)=(B0, . . . , B`−1)·˜
A.
Proof. The uniqueness of ˜
Ais obvious due to equation (3.6.1) since we can take each matrix
Bito be any arbitrary matrix in Mn,n(Fq(F`)).
Now, we identify as in Notation 3.5 the matrix A∈Mn,n(K) with the vector of matrices
(A0, . . . , A`−1), each one of them in Mn,n(Fq(F`)). We define the function red :{0, . . . , ` −
1}×{0, . . . , ` −1} −→ {0, . . . , ` −1}to be the map given by
(i, j)7→ (i−j) (mod `)
We define then ˜
A=FjAred(i,j)Fred(i,j )−i0≤i,j≤`−1,
which we view as an `×`-matrix whose entries are themselves matrices from Mn,n(Fq[F]).
The fact that the entries of ˜
Alie inside Fq(F`) is clear from the fact that
FjAred(i,j)Fred(i,j )−i=FjAred(i,j)F−jFj+red(i,j )−i.
Indeed, the entries of FjAred(i,j)F−jall lie inside Fq(F`) since the entries of Ared(i,j)lie inside
Fq(F`); furthermore, Fj+red(i,j)−iis a non-negative power of F`.
The following result is an immediate consequence of the definition of ˜
A∈Mn`,n`(Fq(F`))
for any given matrix A∈Mn,n(K) satisfying the conclusion of Proposition 3.6.
Proposition 3.7. The map Mn,n(K)−→ Mn`,n` (Fq(F`)) given by
A7→ ˜
A
is an embedding of Fp[F`]-algebras.
3.6. A skew field. Finally, in this Section we prove that Kis a skew field with center Fp(F`).
The next result is an easy consequence of Proposition 3.7.
Corollary 3.8. For every A∈Mn,n(Fq[F]) there exists a monic polynomial Q(x)∈Fp[F`][x]
such that Q(A)=0.
Proof. Since the n` ×n`-matrix ˜
Ahas its entries in the commutative ring Fq[F`], then the
classical Cayley-Hamilton’s theorem yields the existence of a monic polynomial with coeffi-
cients in Fq[F`] which kills the matrix ˜
A. Because Fq[F`] is itself integral over Fp[F`], then
we can find a monic polynomial Q(x)∈Fp[F`][x] such that Q(˜
A) = 0. Then, Proposition 3.7
yields that Q(A) = 0 as well.
ZARISKI DENSE ORBITS 11
Proposition 3.9. For any P(F)∈Fq[F]there exists a nonzero polynomial Q(F)∈Fq[F]
such that Q(F)P(F)∈Fp[F`].
Proof. We regard P(F) as a matrix in M1,1(Fq[F]). Since Pis non-zero, then Proposition 3.7
yields that ˜
Pmust be an invertible `×`-matrix. Therefore, there exists a vector (Q0, . . . , Q`−1)
with coordinates in Fq[F`] such that
(3.9.1) (Q0, . . . , Q`−1)·˜
P= (α, 0,...,0).
for some non-zero α∈Fq[F`]. If we let Qbe the (nonzero) polynomial in Fq[F`] corresponding
to (Q0, . . . , Q`−1), then equation (3.9.1) implies that Q1(F`) := Q(F)P(F)∈Fq[F`]. Since
Fqis a finite extension of Fp, there must exist another nonzero polynomial Q2∈Fq[F`] such
that Q2(F`)Q1(F`)∈Fp[F`]. So,
Q2(F`)Q(F)P(F)∈Fp[F`]
as desired.
Finally, the desired conclusion about Kbeing a skew field with center Fp(F`) follows as an
immediate consequence of Proposition 3.9.
Corollary 3.10. Fq[F]⊗Fp[F`]Fp(F`)is a skew field and Fp(F`)is its centre.
4. Reductions for our main result
Proposition 4.1. In order to prove Theorem 1.3 for the dynamical system (GN
a,Φ), it suffices
to prove Theorem 1.3 for the dynamical system (GN
a,Φn)for some n∈N.
Proof. It is clear that if condition (C) holds for an iterate of Φ then it also holds for Φ. The
fact that if conditions (A) and (B) hold for an iterate of Φ then they also hold for Φnfollows
from [BGRS17, Lemma 2.1].
Notation 4.2. Let hbe an element in Fp[F]and let Nbe a positive integer. We let [h]denote
the group endomorphism of GN
agiven by the coordinate-wise action of h.
Also, as a matter of notation thoughout our paper, we will often use ~x to denote the point
x∈GN
ajust so it would be more convenient when using a group endomorphism Φ of GN
a
corresponding to some matrix A∈MN,N (Fp[F]), because then we would write A~x to denote
Ψ(~x).
Definition 4.3. We call Ψ : GN
a−→ GN
aa finite-to-finite map (defined over Fp) if there
exists a nonzero element h∈Fp[F]with the property that [h]◦Ψis a group endomorphism of
GN
a. In other words, there exists a matrix B∈MN,N (Fp[F]) such that for each point x∈GN
a,
the finite-to-finite map Ψassociates to the point xthe finitely many points y∈GN
asuch that
[h](~y) = B~x.
Proposition 4.4. Let K=Fq[F]⊗Fp[F`]Fp(F`)where q=p`for some `∈N(see also
Section 3.4). Let Φ : GN
a−→ GN
abe a dominant group endomorphism of GN
adefined over
Fq. Then, there exists n∈Nand there exist non-negative integers N0and N1such that
12 DRAGOS GHIOCA AND SINA SALEH
N=N0+N1, along with a dominant group endomorphism Φ0:GN0
a−→ GN0
acorresponding
to the matrix
(4.4.1) A0:= JFn1`,m1M· · · MJFns`,ms,
and a finite-to-finite map Φ1:GN1
a−→ GN1
a(see also Definition 4.3) corresponding to a
matrix A1∈MN1,N1(K), where the minimal polynomial of A1over Fp[F`]has roots that are
multiplicatively independent with respect to F`, there exists a dominant group endomorphism
g:GN
a−→ GN0
a×GN1
adefined over Fq, and there exists a nonzero element h∈Fp[F`]such
that the next diagram commutes
(4.4.2)
GN
aGN
a
GN0
a×GN1
aGN0
a×GN1
a,
[h]◦Φmn
g g
[h]◦(Φm
0,Φm
1)
for all m∈N. In particular, [h]◦(Φm
0,Φm
1)is a well-defined group endomorphism for all
m∈N.
Proof. Suppose that Φ corresponds to a matrix A∈MN ,N (Fq[F]). For some suitable power Φn
of Φ we have that the roots of the minimal polynomial of Anover Fp(F`), say r(x)∈Fp(F`)[x],
are either a non-negative integer power of F`or multiplicatively independent with respect to
F. Indeed, note that the roots of the minimal polynomial of Aare integral over Fp[F`] (see
also Section 3.6) and so, if a root u0is multplicatively dependent with respect to F`, then a
power un
0(for n∈N) must be of the form F`j0for some non-negative integer j0.
So, with the above assumption regarding Anand its minimal polynomial r(x), then we can
write r(x) = r0(x)r1(x) where r0(x) is a polynomial whose roots are (non-negative integer)
powers of F`and r1(x) is a polynomial whose roots are multiplicatively independent with
respect to F. Using Facts 2.1 and 2.2 (see Section 2.1) along with Corollary 3.10, there must
exist an invertible matrix P∈MN,N (K) such that
(4.4.3) P AnP−1=A0⊕A1
where A0corresponds to a matrix of the form (4.4.1) and the minimal polynomial of A1over
Fp(F`) is r1. Using Equation (4.4.3) we have
(4.4.4) P AmnP−1=Am
0⊕Am
1,
for every positive integer m. Due to the definition of Kthere exists a nonzero u∈Fp[F`]
such that uP ∈Fq[F]; also, because A1is integral over Fp[F`], there exists a nonzero h∈
Fp[F`] such that each hAm
1(for m∈N) has entries in Fq[F]. Therefore, if we let gbe the
group endomorphism corresponding to the matrix uP , using equation (4.4.4) we will get a
commutative diagram of the form (4.4.2). This concludes our proof of Proposition 4.4.
The following result is an easy consequence of Proposition 4.4 and of the fact that for any
positive integer a, we have that pa
i= 0 in Fpwhenever 0 < i < pa.
Corollary 4.5. In Proposition 4.4, at the expense of replacing the positive integer nby a
multiple, we may assume without loss of generality that Φ0corresponds to a diagonal matrix
ZARISKI DENSE ORBITS 13
of the form
A0=Fn1`Im1M· · · MFns`Ims
where, n1, . . . , nsare distinct non-negative integers and m1, . . . , msare non-negative integers.
Proof. Let pabe a power of pthat is greater than all m1, . . . , msin the statement of Propo-
sition 4.4. Then, replacing nby npaand combining the Jordan blocks corresponding to the
same power of Fwill deliver the desired conclusion.
Let Φ be a dominant endomorphism of GN
a, let n∈N, let h∈Fp[F`] and Φ1:GN1
a−→ GN1
a
be as in the statement of Proposition 4.4, while Φ0:GN0
a−→ GN0
ahas the form as in
Corollary 4.5. With the above notation, we prove the next three technical lemmas.
Lemma 4.6. Suppose that there exists some i∈ {1, . . . , s}such that ni= 0, and also mi>0;
in particular, this means that N0≥1with the notation as in Corollary 4.5. Then there exists
a non-constant rational function f:GN
a−→ P1such that f◦Φn=f.
Proof. Suppose without loss of generality that n1= 0. Let π:GN
a−→ Gabe the projection
onto the first coordinate. Then, using Equation (4.4.2) we must have
π◦g◦[h]◦Φn= [h]◦π◦g.
However, since πand gare both defined over Fq, the map [h] commutes with both of them.
So, we have
π◦g◦[h]◦Φn=π◦g◦[h].
Hence, π◦g◦[h] defines a non-constant rational function that is left invariant by Φn.
Lemma 4.7. Suppose that the numbers n1, . . . , nsare all positive and max{m1, . . . , ms} ≥
trdegFpL+ 1. Then there exist integers r≥1and M≥trdegFpL+ 1, and there exists a
dominant group homomorphism τ:GN
a−→ GM
asuch that τ◦Φn=Fr◦τ.
Proof. Suppose without loss of generality that m1≥trdegFpL+ 1. Let πbe the projection
map onto the first m1coordinates of GN
a. Using the equation (4.4.2) we must have
π◦g◦[h]◦Φn= [h]◦Fn1`◦π◦g.
Since g, and πare defined over Fq, they must commute with [h]; also, they all commute with
Fn1`. Hence, we have
π◦g◦[h]◦Φn=Fn1`◦π◦g◦[h].
So, the map τ:= π◦g◦[h] has the desired property.
Lemma 4.8. Let Lbe an algebraically closed field of characteristic p. If there exists a point
α:= (α0, α1)with α0∈GN0
a(L)and α1∈GN1
a(L)such that
O:= {([h]◦(Φm
0,Φm
1)) (α0, α1) : m≥0}
is Zariski dense in GN0+N1
a, then there also exists a point β∈GN
a(L)such that OΦ(β)is
Zariski dense in GN
a.
14 DRAGOS GHIOCA AND SINA SALEH
Proof. Choose βsuch that g(β) = α(note that gis a dominant group endomorphism). Then
the commutative diagram (4.4.2) along with the fact that gand [h] are dominant group
endomorphisms yields that the orbit of βunder Φnmust be Zariski dense in GN
a. Since
OΦn(β)⊆ OΦ(β), we obtain the desired conclusion in Lemma 4.8.
Lemmas 4.6,4.7, and 4.8 along with Proposition 4.1 will reduce Theorem 1.3 to Proposition
5.2 stated and proved in the next Section.
5. Proof of Theorem 1.3
In this Section we conclude the proof of our main result. We work under the hypotheses of
Theorem 1.3. We start by stating a useful result, which is a special case of [GS21, Proposition
4.1].
Proposition 5.1. Let Lbe an algebraically closed field of transcendence degree d > 0over
Fp. Let Φ : GN
a−→ GN
abe a dominant group endomorphism corresponding to the matrix
A=Fn1Im1M· · · MFnsIms
where m1, . . . , ms, n1, . . . , nsare positive integers and n1, . . . , nsare distinct. Then there exists
a point α∈GN
a(L)such that every infinite subset of OΦ(α)is Zariski dense in GN
aif and
only if
(5.1.1) max{m1, . . . , ms} ≤ d.
Finally, we can state the technical reformulation of Theorem 1.3, which will allow us to
prove the desired conclusion in our main result.
Proposition 5.2. Let N0and N1be non-negative integers, let q:= p`, let Lbe an algebraically
closed field which has transcendence degree over Fpequal to d > 0, and let K=Fq[F]⊗Fp[F`]
Fp(F`). Let Φ0:GN0
a−→ GN0
abe a dominant group endomorphism corresponding to the
matrix
(5.2.1) A:= Fn1Im1M· · · MFnsIms,
(for some non-negative integers s, n1, . . . , ns, while N0=Ps
i=1 mi) and Φ1:GN1
a−→ GN1
abe a
finite-to-finite map corresponding to a matrix A1∈MN1,N1(K), where the minimal polynomial
of A1over Fp[F`]has roots that are multiplicatively independent with respect to F`. Suppose
there exists a non-zero element h∈Fp[F`]such that [h]◦(Φn
0,Φn
1)is a well-defined dominant
group endomorphism of GN0+N1
afor each n∈N. Then, one of the following statements must
hold:
(i)N0≥1and one of the numbers n1, . . . , nsis equal to zero.
(ii)The numbers n1, . . . , nsare all positive and max{m1, . . . , ms}> d.
(iii)There exists a point α:= (α0, α1)with α0∈GN0
a(L)and α1∈GN1
a(L)such that
(5.2.2) O:= {([h]◦(Φn
0,Φn
1)) (α0, α1) : n≥0}
is Zariski dense in GN0+N1
a.
ZARISKI DENSE ORBITS 15
As noted at the end of Section 4, lemmas 4.6,4.7,4.8 reduce Theorem 1.3 to Proposition
5.2, which we will prove next.
Proof of Proposition 5.2.First of all, as noted also in Section 4, for a point γ∈Gk
a(L) (for
some non-negative integer k), we will use the notation ~γ in order to emphasize that the point
~γ ∈Gk
a(L) is a vector consisting of kelements from L.
We will prove Proposition 5.2 by assuming that if conditions (i) and (ii) do not hold, then
condition (iii) must hold. If we assume that conditions (i) and (ii) do not hold, then by
Proposition 5.1 there must exist a point ~α0∈GN0
a(L) such that any infinite subset of
OΦ0(~α0) := {Φn
0(~α0) : n≥0},
is Zariski dense in GN0
a. Now, choose a point ~α1∈GN1
a(L) whose coordinates are linearly
independent with respect to the coordinates of ~α0over Fp[F] (see Proposition 3.2). Note that
if N0= 0, then our only requirement is that the coordinates of ~α1are linearly independent
over Fp[F] (see the second part of Definition 3.1).
We let ~α := (~α0, ~α1). Suppose for the sake of contradiction that the Zariski closure of O
(from equation (5.2.2)) in GN0+N1
ais a proper subvariety, say V. Let
Γ := {(B0~α0, B1~α1) : B0∈MN0,N0(Fq[F]) and B1∈MN1,N1(Fq[F])}.
Then Γ is a finitely generated Fp[F]-module that contains O. Therefore, according to Propo-
sition 3.3, the intersection V(L)∩Γ is contained in the union of finitely many sets of the
form
(5.2.3) ~
β+S(~γ1, . . . , ~γr;δ1, . . . , δr) + H,
where His an Fp[F]-submodule of Γ, and S(~γ1, . . . , ~γr;δ1, . . . , δr) is a sum of F-orbits of the
points ~γi∈GN0+N1
a(L) (for some given positive integers δi, as in equation (3.2.4)), i.e.,
S(~γ1, . . . , ~γr;δ1, . . . , δr) = (r
X
i=1
Fniδi(~γi) : ni∈Nfor i= 1, . . . , r).
Furthermore, as noted in Remark 3.4 (see equation (3.4.1)), there exists a nonzero polynomial
P(F)∈Fp[F] such that
(5.2.4) P(F)~
β:= (B0~α0, B1~α1),
and for each i= 1, . . . , r, we have
(5.2.5) P(F) (~αi) := (C0,i~α0, C1,i ~α1),
for some B0, C0,1, . . . , C0,r ∈MN0,N0(Fq[F]) and B1, C1,1, . . . , C1,r ∈MN1,N1(Fq[F]). Further-
more, since Fp[F] is a finite integral extension of Fp[F`], then (at the expense of multiplying
P(F) by a suitable nonzero element of Fp[F], which would only replace the matrices Biand
Ci,j by other matrices with entries in Fq[F]) we may assume that P(F)∈Fp[F`].
We let Ube a set of the form (5.2.3) that contains the subset
(5.2.6) OS:= {([h]◦(Φn
0,Φn
1)) (α0, α1) : n∈S},
of Owhere Sis a subset of Nthat has a positive natural density.
16 DRAGOS GHIOCA AND SINA SALEH
Now, since His an Fp[F]-submodule of GN0+N1
a, then its Zariski closure His an algebraic
subgroup of GN0+N1
adefined over Fp. So, let ~v = (~v0, ~v1) with ~v0∈Fp[F]N0and ~v1∈Fp[F]N1
such that
(5.2.7) ~vT
0~x0+~vT
1~x1= 0,
for all (~x0, ~x1)∈H(where always ~vTdenotes the transpose of ~v). Note that since His
an algebraic subgroup of GN0+N1
adefined over Fp, then His the zero locus of finitely many
equations of the form (5.2.7). Using both equations (5.2.7) and (5.2.3) along with equa-
tions (5.2.4) and (5.2.5) and with the fact that the operator P(F) leaves invariant the entries
in both ~v0and ~v1, we obtain that
(5.2.8)
1
X
i=0
P(F)~vT
i·([h]◦Φn
i) (~αi)=
1
X
i=0
~vT
iBi~αi+
r
X
j=1
~vT
iFnjδjCi,j ~αj
.
for all n∈S. Since for each j= 0,1, we have that Φjcorresponds to the matrix Aj∈
MNj,Nj(K), and then using that the set of the coordinates of ~α1are linearly independent
from the set of coordinates of ~α0over Fp[F], then writing h1:= P(F)·h∈Fp[F`], we get:
(5.2.9) ~vT
1
h1An
1−B1−
r
X
j=1
FnjδjC1,j
=~
0∈MN1,1(K) for all n∈S.
At the expense of replacing Swith a subset of Swith a positive natural density, we may
assume that each njδj(for j= 1, . . . , r) has the same remainder modulo `for all n∈S. This
allows us to rewrite equation (5.2.9) as an equation of the form
(5.2.10) ~vT
1
h1An
1−B1−
r
X
j=1
Fmj`C0
1,j
=~
0,
for some matrices C0
1,j ∈MN1,N1(Fq[F]) depending on the matrices C1,j . Let Vbe the N1×N1-
matrix whose rows are all equal to ~vT
1. So, we have
(5.2.11) V
h1An
1−B1−
r
X
j=1
Fmj`C0
1,j
= 0 ∈MN1,N1(K).
Applying the operator ∼(defined as in Notation 3.5 from Section 3.5) to equation (5.2.11)
and also using the fact that h1∈Fp[F`], we get
(5.2.12) ˜
V
h1˜
An
1−˜
B1−
r
X
j=1
Fmj`˜
C01,j
= 0.
Now suppose that Vis non-zero. Then we must have some nonzero row ~uTin ˜
V; so, we get
(5.2.13) ~uT
h1˜
An
1−˜
B1−
r
X
j=1
Fmj`˜
C01,j
= 0.
We note that equation (5.2.13) is similar to [GS21, Lemma 4.5, equation (4.5.1)] (after trans-
posing both sides). So, by proceeding exactly as in the proof of [GS21, Lemma 4.5], and
ZARISKI DENSE ORBITS 17
using the fact that the roots of the minimal polynomial of A1(and thus, also of ˜
A1) are
multiplicatively independent with respect to F`, equation (5.2.13) leads to an equation of the
form
(5.2.14) aλn=c0+
r
X
j=1
cjFmj`,
for some a, c0, c1, . . . , cr∈Fp(F`) with a6= 0 and some eigenvalue λof A1, which is thus
multiplicatively independent with respect to F`. Note that Fp(F`) is isomorphic to Fp(t) for
some transcendental element tsince Fp(F`) is naturally isomorphic to Fp(t). However, by
Lemma 2.4, the set of n’s for which equation (5.2.14) is solvable for some positive integers mj
must have a natural density equal to zero which contradicts our choice of the subset S.
This means that for any vector ~v = (~v0, ~v1) with ~v0∈Fp[F]N0and ~v1∈Fp[F]N1such that
~vT
0~x0+~vT
1~x1= 0,for all (~x0, ~x1)∈Hwe must have ~v1=~
0. Hence, H:= G0×GN1
afor some
algebraic subgroup G0⊆GN0
a.
Thus, U(the Zariski closure of the set Ucontaining the elements from equation (5.2.6)),
which is itself a subset of V, must be a set of the form W×GN1
afor some closed subset
W⊆GN0
asince {0} × GN1
ais contained in the stabilizer of U(because H=G0×GN1
a). On
the other hand, note that Wcontains
{([h]◦Φn
0) (~α0) : n∈S},
which must be Zariski dense in GN0
abecause of our choice of ~α0and the fact that [h] is a
dominant endomorphism of GN0
a. So, we conclude that U=GN0
a×GN1
awhich contradicts the
fact that Vis a proper subvariety of GN0
a×GN1
a. This contradiction completes our proof for
Proposition 5.2.
Since we proved that Theorem 1.3 reduces to Proposition 5.2, this concludes our proof for
Theorem 1.3.
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Dragos Ghioca, Department of Mathematics, University of British Columbia, Vancouver, BC
V6T 1Z2, Canada
Email address:dghioca@math.ubc.ca
Sina Saleh, Department of Mathematics, University of British Columbia, Vancouver, BC V6T
1Z2, Canada
Email address:sinas@math.ubc.ca
