A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED
AUTOMORPHISMS OF K-ALGEBRAS

arXiv:1312.5305v1 [math.AG] 18 Dec 2013

JASON P. BELL AND JEFFREY C. LAGARIAS
A BSTRACT. This paper proves a commutative algebraic extension of a generalized
Skolem-Mahler-Lech theorem due to the first author. Let A be a finitely generated
commutative K-algebra over a field of characteristic 0, and let σ be a K-algebra automorphism of A. Given ideals I and J of A, we show that the set S of integers m such
that σ m (I) ⊇ J is a finite union of complete doubly infinite arithmetic progressions in
m, up to the addition of a finite set. Alternatively, this result states that for an affine
scheme X of finite type over K, an automorphism σ ∈ AutK (X), and Y and Z any
two closed subschemes of X, the set of integers m with σ m (Z) ⊆ Y is as above. The
paper presents examples showing that this result may fail to hold if the affine scheme X
is not of finite type, or if X is of finite type but the field K has positive characteristic.

C ONTENTS
1. Introduction
1.1. The Skolem-Mahler-Lech theorem
1.2. Main result
1.3. Affine scheme version of main result
1.4. Generalizations
2. p-Adic Preliminaries
2.1. Evaluation maps and Jacobians for polynomial endomorphicms
2.2. p-adic approximate fixed points for automorphisms
2.3. p-adic reduction lemmas
3. Generalized p-adic Analytic Arc Theorem
4. Polynomial Ring Case
5. Proof of Main Result
6. Examples
7. Appendix: Dynamics of Endomorphisms–Geometric versus Algebraic
References

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Date: November 21, 2013, rev6.
2010 Mathematics Subject Classification. Primary: 11D45. Secondary: 14R10. 11Y55, 11D88.
Key words and phrases. automorphisms, endomorphisms, affine space, commutative algebras,
Skolem-Mahler-Lech theorem.
The research of the first-named author was supported by NSERC grant 611456.
The research of the second author was supported by NSF grants DMS-0801029 and DMS-1101373.
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JASON P. BELL AND JEFFREY C. LAGARIAS

1. I NTRODUCTION
The Skolem-Mahler-Lech theorem is a fundamental result, which characterizes the
structure of the set of zeros of a linear recurrence; we term the resulting structure the
SML property below. The paper is motivated by the question: “What is the maximal
level of generality of the Skolem-Mahler-Lech theorem, in which its conclusion holds?"
We view this question in the general framework of orbits of dynamical systems of algebraic type.
In 2006 the first author gave an algebro-geometric generalization of the SkolemMahler-Lech theorem which applied to orbits of an automorphism of an affine variety
acting on geometric points. The object of this paper is to show there is a further algebrogeometric generalization of the Skolem-Mahler-Lech theorem that applies to automorphisms on the coordinate ring of the variety acting at the level of ideals. The new result
may be interpreted as a result about automorphisms of affine schemes. To state it, we
first review the successive generalizations of the Skolem-Mahler-Lech theorem.
1.1. The Skolem-Mahler-Lech theorem. The Skolem-Mahler-Lech theorem (hereafter abbreviated to SML theorem) is a fundamental result which can be stated in several
apparently different forms. The original formulation of the SML theorem concerned the
zeros of power series coefficients of rational functions.
Theorem 1.1. (SML theorem-rational function form) Let K be a field of characteristic
zero, and let G(x) ∈ K(x) be a rational function that is finite at x = 0. If the Taylor
expansion G(x) ∈ K[[x]] at x = 0 is given as
∞
X
ak xk ,
G(x) =
k=0

then the set of its zero coefficients

S := {k ∈ N : ak = 0}

is a union of a finite number of arithmetic progressions {an + b : n ≥ 0}, with a, b > 0,
together with a (possibly empty) finite set.
This result was established by Skolem [30] in 1934 for the case of rational coefficients, using a p-adic method he introduced in [29]. It was extended by Mahler [21,
Sect. 6] in 1935 to series to coefficients in algebraic number fields, and much later in
1956 ([22]) to coefficients in the complex field C. In this last extension Mahler was
unaware of the 1953 work of Lech [20] discussed below, see [23].
There are now many different proofs and extensions of the Skolem-Mahler-Lech theorem in the literature [8, 18, 26, 27], some described in the book of Everest et al. [14,
Chap. 2.1]. The result is valid in any field of characteristic zero, however all known
proofs use p-adic methods.
The property of the set S formulated in the conclusion of all versions of the SkolemMahler-Lech theorem can be stated as follows.
Definition 1.2. A set of natural numbers S ⊂ N has the (one-sided) SML property if it
is a finite union of one-sided arithmetic progressions {an + b : n ≥ 0} with a, b > 0,
augmented by a finite set, possibly empty. (We allow the possibility b > a in the onesided arithmetic progressions, i.e. some initial terms of the complete progression in N
may be omitted.)

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A second form of the Skolem-Mahler-Lech theorem concerns zeros of recurrence
sequences. This version was first formulated by Lech [20] in 1953.
Theorem 1.3. (SML theorem-recurrence form) Let K be a field of characteristic zero
and let f : N → K be a K-valued sequence that satisfies a linear recurrence
f (n) = a1 f (n − 1) + a2 f (n − 2) + · · · + ar f (n − r), ai ∈ K.

for all n ≥ r. Then the set of zeros of the recurrence,
has the SML property.

S := {n ≥ 1 : f (n) = 0}.

We can view the recurrence version of the SML theorem as asserting a property of a
forward orbit of a discrete dynamical system given by the linear recurrence. Note that
the set S can also be interpreted as the intersection of the forward orbits of two different
dynamical systems, the first being the recurrence f (n) and the second being the constant
linear recurrence g(n) = 0.
The recurrence form of the SML theorem can be recast in a third equivalent form,
which concerns iteration of linear maps, and encodes containment relations of orbits of
invertible linear maps under iteration in a subspace ([4, Theorem 1.2 and Sect. 2]).
Theorem 1.4. (SML theorem-linear map form) Let K be a field of characteristic 0, and
let σ : K n → K n be an invertible linear map. If v ∈ K n , and W is a vector subspace
of K n of codimension 1, then the set
has the SML property.

S := {n ∈ N : σ n (v) ∈ W }

In 2006 the first author [4, 5] found a generalization of the SML theorem to algebraic
dynamics, which applies at the geometric level to iteration of automorphisms on affine
algebraic varieties. It concerns two-sided infinite sequences of iterates, and to state it
we extend the notion of SML property to this case.
Definition 1.5. A set of integers S ⊂ Z is said to have the two-sided SML property if it
is a finite union of complete arithmetic {an + b : n ∈ Z} progressions on Z augmented
by a finite set. (That is, we allow arithmetic progression with a = 0 which give one
element sets.)
The generalization of the SML theorem to automorphisms on affine varieties is as
follows.
Theorem 1.6. (Generalized SML Theorem for Affine Varieties) Let K be a field of
characteristic zero, and let X be an affine K-variety and let σ be an automorphism of
X. If x is a K-point of X and Y is a subvariety of X (Zariski closed subset of X), then
the set
S := {n ∈ Z : σ n (x) ∈ Y }
has the two-sided SML property.
Despite its restriction to two-sided infinite sequences, Theorem 1.6 recovers the original (one-sided) recurrence form of the Skolem-Mahler-Lech theorem above, via its implication of the linear map form (Theorem 1.4). The proof of Theorem 1.6 in [4, 5] again

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relies on a p-adic result, the p-adic analytic arc theorem, first established in [4, 5] and
then strengthened in [6]. This latter result was recently further strengthened by Poonen
[25].
We also note that since the conclusion of Theorem 1.6 is purely set-theoretic and is
preserved under extension of scalars, the result for a general field K follows easily from
the special that K is an algebraically closed field of characteristic zero.
Subsequent work of the first author with Ghioca and Tucker [6, Theorem 1.3], established an analogous theorem for forward orbits of étale endomorphisms of quasiprojective varieties defined over C. We also point out that Denis [11] had earlier proved a
special case of this result for étale self maps of PnC . Recently Sierra [28, Conjecture 5.15]
suggests a possible extension of these results that would have interesting consequences
for the algebras studied in noncommutative projective geometry.
1.2. Main result. Our starting point is the observation that Theorem 1.6 can be recast
in purely algebraic terms. To do this, suppose that K is an algebraically closed field of
characteristic zero. In classical affine algebraic geometry, we then have a contravariant
equivalence of categories
{Affine K-Varieties} ←→ {Reduced finitely generated K-algebras}
induced by the functor which takes an affine variety X to its coordinate ring K[X]
and which takes a morphism φ : X → Y of affine varieties to the homomorphism
φ∗ : K[Y ] → K[X] of K-algebras given by φ∗ (f ) = f ◦ φ ( [19, Corollaries 1.4 and
3.8]). The Nullstellensatz gives further information, showing that there is an inclusion
reversing bijection between the subvarieties of an affine variety X and the radical ideals
of its coordinate ring K[X]. In particular, for a morphism φ : X → X the inclusion
φ(V (I)) ⊆ V (J) of zero sets of radical ideals I, J corresponds to the reversed algebraic
inclusion φ⋆ (I) ⊇ J. It follows that Theorem 1.6 restricted to algebraically closed fields
K may be recast in algebraic terms as: If A is a finitely generated reduced commutative
K-algebra with a K-algebra automorphism σ and M and J are ideals of A with M a
maximal ideal and J a radical ideal, then the set of integers n for which σ n (M) ⊇ J
has the two-sided SML property.
From this algebraic perspective, it is natural to ask whether a similar result holds for
more general ideal inclusions in finitely generated commutative K-algebras. Our main
result gives an affirmative answer to this question.
Theorem 1.7. (Generalized SML theorem for ideal inclusions) Let K be any field of
characteristic zero and let A be a finitely generated commutative K-algebra. If σ :
A → A is a K-algebra automorphism and I and J are ideals of A, then
S = S(I, J) := {n ∈ Z : σ n (I) ⊃ J}

has the two-sided SML property. Here σ(I) := {σ(a) : a ∈ I}.

The two-sided SML property of this result also holds for the set of integers n for
which σ n (J) ⊆ I, because this condition holds if and only if σ −n (I) ⊇ J, where
σ −1 (I) = {b : σ(b) ∈ I}; apply Theorem 1.7 to the automorphism σ −1 .
Theorem 1.7 implies Theorem 1.6 and hence all the earlier forms of the SML theorem
stated above. In the special case that I is a maximal ideal, Theorem 1.7 is itself deducible
from the algebraic form of Theorem 1.6. The main content of Theorem 1.7, and also the

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source of the difficulty in proving it, revolves around relaxing the maximality condition
on I. For example, if A = C[x, y], then a polynomial p(x, y) ∈ A is in the maximal
ideal (x, y) if and only if p(0, 0) = 0. On the other hand, p(x, y) is in the ideal (x, y 2 ) if
and only if p(0, 0) = 0 and ∂p(x, y)/∂y vanishes at (0, 0). In this sense, the ideal I =
(x, y 2 ) encodes additional “infinitesimal” information. It seems easier to understand the
behavior of the iterates of an element p(x, y) of A under an automorphism σ with respect
to evaluation at a given point than to understand how the iterates behave with respect
to more complicated conditions involving the vanishing of partial derivatives at a point.
This difficulty and, more generally, the possible occurrence of nil ideals and non-radical
ideals in A adds an extra level of complication to the problem. (see Examples 6.2 and
6.3).
In Section 6 we examples showing that this result is strictly more general than Theorem√1.6. √
In particular, there exist two ideals I1 , I2 having the same radical ideal
I = I1 = I2 and J such the sets S(I1 , J) 6= S(I2 , J), see 6.1.
As with previous work, our method to prove Theorem 1.7 makes use of a p-adic result. We present a p-adic interpolation result that strengthens the p-adic analytic arc
theorem established in [5], which we call the generalized p-adic analytic arc theorem
(Theorem 3.1). This result concerns polynomial automorphisms σ : Zp [x1 , . . . , xd ] →
Zp [x1 , . . . , xd ] and applies for p ≥ 5. It asserts that for a Zp -algebra S that is finitely
generated and is a torsion-free Zp -module, the induced (nonlinear) map fσ : S d → S d
has the property that iterates of a suitable initial point s = (s1 , . . . , sd ) ∈ S d under σ
can be embedded in a p-adic analytic arc. Note that S is a free Zp -module of finite rank
r > 0, and one might think that this extension can be obtained by fixing a Zp -module
isomorphism between S and (Zp )r and then applying the analytic arc theorem of [4]
separately to each coordinate. This is not the case, however, as application of the map σ
involves expressions that include combinations of elements from different coordinates.
Our innovation to get around this obstacle consists of working with a larger ring of functions than usual; namely, the subset R(S) of S ⊗Zp Qp [z] that consists of all polynomials
which map Zp into S, and obtaining analytic maps via successive approximations via
functions in this ring. The commutative ring R(S) is rather pathological, it is not Noetherian and is not of finite type over Zp . Our proof requires establishing a nice property
of certain subalgebras of R(S), given in Lemma 3.7. As in the case of the p-adic analytic arc theorem treated in [4], the generalized p-adic analytic arc theorem fails to hold
for p = 2, and its truth for p = 3 remains open.
The proof of Theorem 1.7 is commutative algebraic, aside from the p-adic interpolation result. A key idea is to first establish a result for the ring A = Zp [x1 , . . . , xd ],
treating only for the special case that I, J are p-reduced ideals (Definition 4.2) when
p ≥ 5 (Theorem 4.5). The p-reduced condition is needed in order to apply the generalized p-adic analytic arc theorem. Here we also use an idea of Amitsur (Lemma 4.4),
to handle the case of p-reduced ideals I where (A/I) ⊗ Qp is an infinite-dimensional
Qp -vector space. We next deduce the result for arbitrary ideals I, J in a polynomial ring
A = K[x1 , . . . , xd ] over a field of characteristic 0 (Theorem 4.1), as follows. We first
show that K can be taken to be finitely generated over Q; next, we use the fact that such
a field embeds into infinitely many p-adic fields Qp via the Chebotarev density theorem;
finally, passing from Qp to Zp -coefficients, we show that one can pass from ideals to
suitable p-reduced ideals. We then treat the general case that A is a finitely-generated

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commutative K-algebra, by using a theorem of Srinivas [31, Theorem 2] to reduce this
case to that of a polynomial ring K[x1 , . . . , xd ].
An interesting feature of this proof is that while Theorem 1.7 concerns finitelygenerated commutative K-algebras A which are Noetherian rings, the proof itself currently requires a detour using the non-Noetherian ring R(S).
1.3. Affine scheme version of main result. We can formulate the main result in the
category of affine schemes. There is a standard correspondence of categories
{Affine Schemes} ←→ {Commutative rings with identity} .

and using it Theorem 1.7 may be restated as follows.

Theorem 1.8. Generalized SML theorem for affine schemes of finite type) Let K be
a field of characteristic zero and let X be an affine scheme of finite type over K. If
σ ∈ AutK (X) and Y and Z are closed subschemes of X, then
S = S(Z, Y ) := {n ∈ Z : σ n (Z) ⊆ Y }

has the two-sided SML property.

From the scheme-theoretic viewpoint Theorem 1.6 corresponds to the special case
that Y and Z are reduced closed subschemes and Z = {x} is a point. The main difficulty
in establishing Theorem 1.8 is to omit the requirement that the source Z and target Y be
reduced, while allowing a source Z of arbitrary dimension is less of a difficulty.
We show in Section 6 via examples that for automorphisms one cannot generalize this
result to arbitrary affine schemes without altering the conclusion that the set is a twosided SML set. Example 6.4 shows the need for the scheme to be of finite type over K,
while example 6.5 shows the need to work over a field K of characteristic zero. There
remains a possibility of generalizing the result to arbitrary schemes of finite type over
characteristic zero fields.
1.4. Generalizations. The ultimate level of generalization possible for mappings having the SML property is not clear.
A natural question is: “Is there an SML theorem for endomorphisms of algebraic
varieties?" For endomorphisms that are not automorphisms, the maps τ m with m < 0
are not defined, and extensions of Theorem 1.7 to endomorphisms must be formulated in
terms of one-sided infinite arithmetic progressions. Recent work described below shows
that the SML property does hold for some classes of endomorphisms at the geometric
level, while endomorphisms at the algebraic level have not yet been studied. At present
in the characteristic 0 case no counterexamples to the (one-sided) SML property are
known for any endomorphism at either the geometric and algebraic level.
The dynamics of endomorphisms in the geometric setting is currently a very active
area of study. In 2009 Ghioca and Tucker [16] conjectured that the forward iterates of
a point P under an endomorphism τ : X → X of a quasiprojective variety over C
should satisfy the SMLproperty for intersecting a closed subvariety V . They term this
assertion the dynamical Mordell-Lang conjecture. This conjecture fits in the general
framework of dynamical conjectures of S. Zhang [33, Sect. 4]. It is now known that onesided versions of the Skolem-Mahler-Lech theorem are valid for some special classes
of endomorphisms—see for example Benedetto, Ghioca, Kurlberg and Tucker [7]. The
dynamical Mordell-Lang conjecture currently appears to be difficult in the general case.

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When endomorphisms are viewed at the ideal level, as in this paper, there are two
different questions to consider: the first concerns upward inclusions and the second
concerns downward inclusions of ideals under preimages of endomorphisms of polynomial rings. Given an endomorphism τ : A → A of a commutative K-algebra A, and a
nonzero ideal I of A the preimages τ −1 (I) are ideals of A, but the image τ (I) is usually
not an ideal of A but is an ideal of the image ring A′ := τ (A). To formulate inclusions
purely in terms of A-ideals, these concern the sets
S(I, J) := {m ≥ 0 | I ⊇ (τ m )−1 (J)}
and
S ′ (I, J) := {m ≥ 0 | (τ m )−1 (I) ⊇ J},
respectively.
Our methods in this paper do not give information about either of the sets S(I, J)
and S ′ (I, J). However a number of results in this paper partially extend to the endomorphism case, see Remarks 2.2 and 3.3. New ideas are certainly needed for general
endomorphisms, because the generalized analytic arc theorem is known not to hold in
the p-adic case near some superattracting fixed points.
Acknowledgments. We thank Harm Derksen for comments on the nonzero characteristic case and Brian Conrad for comments on the scheme theory viewpoint. We thank the
reviewer for helpful comments.
2. p-A DIC P RELIMINARIES
Much of our study of automorphisms of algebras relies on reductions to simpler cases.
We will eventually reduce the proof of Theorem 1.7 to the special case that one is working with a polynomial ring over a p-adic field.
For the analysis of this special case we establish preliminary results on Jacobian matrices of p-adic mappings and some results from p-adic analysis. Section 2.1 allows
S-algebra endomorphisms, while all later sections specialize to the automorphism case.
2.1. Evaluation maps and Jacobians for polynomial endomorphicms. Let R =
S[x1 , . . . , xd ] be a polynomial ring over an integral domain S. Any S-algebra endomorphism τ : R → R is uniquely determined by its values {τ (xi ) : 1 ≤ i ≤ d}, on the
monomials xi , and any assignment of values
τ (xi ) := Fi (x1 , . . . , xd ) ∈ S[x1 , . . . , xd ],

1 ≤ i ≤ d,

uniquely extends to an endomorphism τ : R → R that acts as the identity on S, given
by
τ (P (x1 , . . . , xd )) := P (τ (x1 ), . . . , τ (xd )) ∈ S[x1 , . . . , xd ].

(2.1)

Composition of endomorphisms will be denoted τ2 ◦ τ1 (P ) := τ2 (τ1 (P )). In what follows we use τ to denote a general S-algebra endomorphism, while symbols σ, ρ are
reserved for S-algebra automorphisms.

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The Jacobian matrix J(τ ; x) ∈ Md×d (S[x1 , . . . , xd ]) of the map τ at x is given by

 ∂
∂
F
F · · · ∂x∂ d F1
∂x1 1 ∂x2 1
 ∂ F2 ∂ F2 · · · ∂ F2 
∂x1
∂x2
∂xd
.
J(τ ; x) := 


···
···
∂
∂
∂
F
F · · · ∂xd Fd
∂x1 d ∂x2 2

These matrices satisfy the polynomial identity under composition
J(τ2 ◦ τ1 ; x) = J(τ2 ; τ1 (x))J(τ1 ; x),

in the ring Md×d (S[x1 , . . . , xd ]).
For s = (s1 , s2 , . . . , sd ) ∈ S d , we define the evaluation map evs : S[x1 , x2 , . . . , xd ] →
S, which assigns xi 7→ si , i.e.,
evs (F )(x1 , . . . , xd ) := F (s1 , s2 , . . . , sd ).
(For constants c ∈ S we have evs (c)(x1 , .., xd ) = c.) Using this map, we define the
Jacobian matrix of a polynomial map τ with its entries evaluated at the point s ∈ S d as:
 

∂
J(τ ; s) := evs (J(τ ; x)) = evs
Fi (x1 , . . . , xd )
,
∂xj
1≤i,j≤d
with J(τ ; s) ∈ Md×d (S).
In the same vein, given an endomorphism τ of S[x1 , . . . , xd ] we obtain a map fτ :
S d → S d defined for each s ∈ S d , acting coordinatewise, by
fτ (s) := evs (τ (x1 ), τ (x2 ), · · · , τ (xd )) = (F1 (s1 , . . . , sd ), F2 (s1 , . . . , sd ), . . . , Fd (s1 , . . . , sd )).
We call fτ the dynamical evaluation map associated to τ . The important property it has
is compatibility with composition of maps, given as:
fτ2 ◦τ1 (s) = (fτ2 ◦ fτ1 )(s).

(2.2)

In particular, for iteration of the map fτ one has fτ m = (fτ )m . A consequence of this
compatibility is that if the endomorphism is an automorphism σ, then the dynamical
evaluation map fσ : S d → S d is a bijection, because fσ ◦ fσ−1 = fid , the identity map.
A second consequence is the identity
J(τ2 ◦ τ1 ; s) = J(τ2 ; fτ1 (s))J(τ1 ; s),

s∈S

(2.3)

in the ring Md×d (S).
The dynamical evaluation map fτ is analogous to a map acting an affine variety as
in Theorem 1.6. In general it is a nonlinear map, and it usually does not respect either addition or coordinatewise multiplication on S d , i.e. one may have fτ (s1 + s2 ) 6=
fτ (s1 ) + fτ (s2 ) and fτ (s1 s2 ) 6= fτ (s1 )fτ (s2 ). Furthermore one may have fτ (0) 6= 0,
where 0 := (0, 0, .., 0) ∈ S d .
The above definitions are stated for endomorphisms τ but in the remainder of the paper we will restrict to the case of automorphisms σ, unless specified otherwise. In the
Appendix to this paper we prove a result that clarifies the differences between the geometric and algebraic view of dynamics of endomorphisms τ acting on a variety (Proposition 7.1).

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2.2. p-adic approximate fixed points for automorphisms. Now we specialize to the
case that the integral domain S is a Zp -algebra. We consider an automorphism σ of R =
S[x1 , . . . , xd ]. and study the dynamical evaluation map fσ . The following lemma asserts
the existence of approximate fixed points (mod p) with invertible Jacobian matrices for
dynamical evaluation maps of polynomial automorphisms.
Lemma 2.1. Let p be a prime, let S be a Zp -algebra that is finitely generated as a Zp module and let σ = (F1 , . . . , Fd ) : Zp [x1 , . . . , xd ] → Zp [x1 , . . . , xd ] be an Zp -algebra
automorphism. Then there is an integer m such that for every point s0 = (s1 , . . . , sd ) ∈
S d with the property that the dynamical evaluation map fσ : S d → S d at s0 satisfies
fσ (s1 , . . . , sd ) ≡ (s1 , . . . , sd ) (mod pS),

the Jacobian J(σ m ; x) evaluated at (x1 , x2 , . . . , xd ) = (s1 , . . . , sd ) is the identity matrix
(mod pS); that is, evs0 (J(σ m ; x)) ≡ I(mod pS).
Proof. The ring S/pS is a finite ring since S is a finitely generated Zp -module. Furthermore, J(σ; s) has inverse mod p given by J(σ −1 ; s) and hence it is invertible mod p.
Take m to be the order of GLd (S/pS). We let
J(σ; s) := evs (J(σ; x)) ∈ Md×d (S)

denote the Jacobian matrix of a polynomial map σ with its entries evaluated at the point
s ∈ S d . Since σ is a Zp -algebra automorphism, we have for s1 , s2 ∈ S d that

s1 ≡ s2 (mod pS) ⇒ fσ (s1 ) ≡ fσ (s2 ) (mod pS).
It follows that the quotient map f¯σ : (S/pS)d → (S/pS)d is well-defined. Similarly,
since J(σ; x) has entries given by polynomials with coefficients in Zp , then
s1 ≡ s2 (mod pS) ⇒ J(σ; s1 ) ≡ J(σ; s2 ) (mod pS),

where ≡ is taken entry-wise.
Then for any point s = (s1 , . . . , sd ) ∈ S d we have

J(σ m ; s) = J(σ; fσm−1 (s))J(σ; fσm−2 (s)) · · · J(σ; s)
= J(σ; (fσ )m−1 (s))J(σ; (fσ )m−2 (s)) · · · J(σ; s).
Hence if s0 is a fixed point of the quotient map f¯σ , then
for all j ≥ 1, so that

(fσ )j (s0 ) ≡ s0 (mod pS),
J(σ; (fσ )j (s0 )) ≡ J(σ; s0 ) (mod pS).

Substituting these in the formula above yields

J(σ m ; s0 ) ≡ J(σ; s0 )m (mod pS).

By our choice of m, M := J(σ m ; s0 ) is congruent to the identity mod pS, as required.

Remark 2.2. The argument of Lemma 2.1 extends to Zp -algebra endomorphisms τ , but
yields only the weaker conclusion: there exists a fixed point s0 of the evaluation function
fτ (mod pS) at which the Jacobian evs0 (J(τ m ; x)) ≡ M(mod pS) where M 2 = M
is an idempotent in Md×d (S/pS).

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2.3. p-adic reduction lemmas. We start with an embedding theorem due to Lech [20,
§4–5]. His result can be regarded as a p-adic analogue of the Lefschetz principle.
Lemma 2.3. Let K be a finitely generated extension of Q and let S be a finite subset of
K. Then there exist infinitely many primes p such that K embeds in Qp ; moreover, for
all but finitely many of these primes every nonzero element of S is sent to a unit in Zp .
Proof. This is shown in [4, Lemma 3.1]. The Chebotarev density theorem is used to
show there exists a positive density set of primes p having the required property.

Strassman’s theorem [32] asserts that if a power series f ∈ Qp [[z]] converges in the
closed p-adic unit disc
BQp (0; 1) := {z ∈ Qp : |z|p ≤ 1} = Zp .
and has infinitely many zeros in this disc, then it is identically zero. We will use the
following variant of Strassman’s theorem.
Theorem 2.4. (Extended Strassman’s Theorem) Let p be a prime and let R be a finitedimensional Qp -algebra. Suppose that the formal power series f (z) ∈ R[[z]] is absolutely convergent for all z ∈ Zp and has infinitely many zeros in Zp . Then f (z) is
identically zero.
Proof. Let n denote the dimension of R as a Qp -vector space. We write
R = Qp v1 + Qp v2 + · · · + Qp vn ,
with v1 = 1. Then
P∞

f (z) = v1 f1 (z) + · · · + vn fn (z)

with each fi (z) = k=0 ck z k ∈ Qp [[z]], converging for all z ∈ Zp . Linear independence
of the vi over Qp implies that f (z) = 0 requires that each fi (z) = 0 separately. Since
f (z) has infinitely many zeros on z ∈ Zp , by Strassman’s theorem ([32], cf. Cassels [10,
Theorem 4.1]) applied to fi (z), each fi (z) is identically zero. Thus f (z) is identically
zero.

Finally, we will need a finiteness result for reduction (mod p) in a Zp -module.
Lemma 2.5. Let p be prime and let M be a Zp -module that is isomorphic to a submodule of Qdp for some natural number d, then M/pM is a finite-dimensional Z/pZ-vector
space.
Proof. Note that M/pM is a Z/pZ-vector space. We claim that its dimension is at most
d. To see this, let θ1 , . . . , θd+1 be elements of M. If we regard M as a submodule of Qdp ,
then we see that 0 is a non-trivial Qp -linear combination of the images of θ1 , . . . , θd+1
in Qdp . Clearing denomiators, we see that there exist a1 , . . . , ad+1 ∈ Zp , not all of
P
which are in pZp , such that d+1
i=1 ai θi = 0. Reducing mod pM, we see that the images
of θ1 , . . . , θd+1 in M/pM are linearly dependent over Z/pZ and thus any set of size
d + 1 in M/pM is linearly dependent. Hence M/pM is at most d-dimensional and in
particular it is a finite-dimensional Z/pZ-vector space.


A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 11

3. G ENERALIZED p- ADIC A NALYTIC A RC T HEOREM
In this section we prove a generalization of the p-adic analytic arc theorem given in
[5, Theorem 1.1] that applies to a larger class of rings. The generalization applies to any
Zp -algebra S that is finitely generated and torsion-free as a Zp -module with the added
hypothesis that p ≥ 5. Examples given in [4] show this theorem does not hold for p = 2;
the case p = 3 of the result below is currently open.
Theorem 3.1. (Generalized p-adic analytic arc theorem) Let p ≥ 5 be prime, and suppose
σ = (H1 , H2 , . . . , Hd ) is a polynomial automorphism of Zp [x1 , . . . , xd ]. Let S be a
Zp -algebra that is finitely generated and torsion-free as a Zp -module, and let fσ :
S d → S d denote the induced dynamical evaluation map. Suppose that an initial value
s0 = (s1 , . . . , sd ) ∈ S d satisfies the following two conditions:
(i) Hi (s1 , . . . , sd ) ≡ si (mod pS) for 0 ≤ i ≤ d;
(ii) The Jacobian matrix M = J(σ; x)|x=s0 evaluated at x = s0 is the identity
matrix (mod pS).
Then there exist power series f1 (z), . . . , fd (z) ∈ (S ⊗Zp Qp )[[z]] which converge for all
z ∈ Zp and which satisfy:
(1) fi (z + 1) = Hi (f1 (z), . . . , fd (z)), for 1 ≤ i ≤ d;
(2) fi (0) = si , for 1 ≤ i ≤ d.

Remark 3.2. The associated analytic arc in S d constructed in Theorem 3.1 is
C = C(σ, s0 ) := {(f1 (z), . . . , fd (z)) : z ∈ Zp } ⊂ S d .
(m)

The result implies that the arc C contains all the iterates {fσ (s0 ) : m ≥ 0} of the initial
(m)
value s0 , since (1) shows fσ (s0 ) = (f1 (m), . . . , fd (m)) holds. IT covers cases such
as S = Z5 with σ(x) = x + 5 on Z5 [x], where one may take s0 = 0, and nevertheless
the induced dynamical evaluation map fσ has no fixed points. The proof follows the
basic plan of the result in [4], but involves more complications. It shows that conditions
(pm )
(i), (ii) imply that fσ (s0 ) converges rapidly to s0 as m → ∞, and uses this to show
analyticity of the constructed maps f1 (z), . . . , fd (z).
Remark 3.3. Theorem 3.1 cannot be extended to cover all general algebra endomorphisms of Zp [x1 , ..., xd ]. For general endomorphisms there may be no point where condition (b) holds, and Remark 2.2 asserts one can only guarantee there exists a point
where the Jacobian, when evaluated (mod pS), is an idempotent matrix. The worst
case is where the Jacobian vanishes identically (mod pS). Consider d = 1 with
H1 (x) = x10 , where we take p = 5 and s0 = 5, a case where the Jacobian vanishes idenn
tically. In this case, we have that σ n (s0 ) = 510 for n ≥ 0. Suppose that it were possible
to find an arithmetic progression {an + b}n∈Z and a 5-adic analytic map f : Z5 → Z5
with the property that f (n) = σ an+b (s0 ) for all integers n. Then we would necessarily
have f (5k ) → 0 as k → ∞. But f (0) = σ b (5) 6= 0 and so it is impossible to find even a
continuous map f : Z5 → Z5 that has the property (1).
To begin the proof, we first introduce a ring of polynomials which takes “integral”
values when evaluated at values in Zp .

12

JASON P. BELL AND JEFFREY C. LAGARIAS

Definition 3.4. Given a prime p and a Zp -algebra S that is finitely generated and torsionfree as a Zp -module, let R(S) denote the subring of the polynomial ring (S ⊗Zp Qp )[z]
that consists of those polynomials which under evaluation at each z ∈ Zp belong to S.
The Zp -algebra R(S) offers a crucial advantage over the polynomial ring S ⊗Zp Q[z],
which is that the following closure property holds: if Q(z) ∈ R(S) then the equation
F (z + 1) − F (z) = Q(z)

(3.1)

has a solution with F (z) ∈ R(S). We use this in proving our general analytic arc
theorem (Theorem 3.1). We note however that the ring R(S) has some pathological
properties. It is a non-Noetherian ring, by a general criterion of Cahen and Chabert [9,
Prop. VI.2.4]. (In fact, for S = Zp , the ring R(S) = Zp [fn (z) = z(z−1)·(z−(n−1))
:n≥
n!
0] as a linear space over Zp —see Lemma 3.5 below. An example of an infinite ascending
chain of ideals in R(S) that does not stabilize is: (z), (z, fp (z)), (z, fp (z), fp2 (z)), . . ..)
Secondly, it can be shown that R(S) is not of finite type over Zp , i.e., it is not finitely
generated as a Zp -algebra. Finally, R(S) is not a UFD, in general: for S = Z2 , then
u := z(z − 1)/2, v := (z − 2)(z − 3)/2, u′ := z(z − 3)/2, v ′ := (z − 1)(z − 2)/2
are irreducible elements of R(S) and uv = u′ v ′ ; there are similar examples for S = Zp ,
p ≥ 3.

Lemma 3.5. Given a prime p and a Zp -algebra S that is finitely generated and torsionfree as a Zp -module, the ring R(S) is given by
)
(
 
m
X
z
: m ≥ 0 and bi ∈ S .
(3.2)
bi
R(S) = f (z) =
i
i=0

Proof. In the special case S = Zp the lemma asserts that the polynomials kz =
z(z−1)···(z−k+1)
for k ≥ 0 are a basis of the polynomials in Qp [z] that map Zp into itk!
self. This is a theorem of Mahler [24, p. 49–50]. For the general case, since S is finitely
generated and torsion-free, S ⊗Zp Qp [z] is a finite direct sum of copies of Qp [z]. The
result then follows from the special case applied to each copy separately.

Lemma 3.6. Let p be prime and let S be a Zp -algebra that is finitely generated and
torsion-free as a Zp -module. Suppose Q1 (z), . . . , Qd (z) ∈ R(S) are given polynomials
of degree at most N. Then there exists a solution [h1 (x), . . . , hd (x)]T ∈ R(S)n to the
equation



 

h1 (z + 1)
h1 (z)
Q1 (z)
..
..

 ≡  ...  − 
 (mod pR(S))
(3.3)
.
.
hn (z + 1)
hd (z)
Qd (z)
such that h1 (0) = · · · = hd (0) = 0 and h1 , . . . , hd have degree at most N + 1.

Proof. By assumption, each Qi (z) ∈ R(S) is of degree at most N, and so
 
N
X
z
Qi (z) =
ci,k
k
k=0

A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 13

with each ci,k ∈ S. We define
hi (z) := −

N
X
k=0

cj,k




z
,
k+1

(3.4)

which implies hi ∈ R(S) and is of degree at most N + 1. It is easy to check that this
gives a solution to equation (3.3), using the identity
  
 

z
z
z+1
.
=
−
k
k+1
k+1
Furthermore, hi (0) = 0 for 1 ≤ i ≤ d.



To create analytic maps in the modified version of the p-adic analytic arc lemma, we
will use the following lemma about subalgebras of R(S).
Lemma 3.7. Let p be a prime and let S be a Zp -algebra that is finitely generated and
torsion-free as a Zp -module. Let N be a natural number, and let
(
)
N
X
SN = c +
pi hi (z) | c ∈ S, hi (z) ∈ R(S), deg(hi ) ≤ 2i − 1
i=1

and

TN = SN +

(

c+

M
X
i=1

)

pi hi (z) | M ≥ 1, c ∈ S, hi (z) ∈ R(S), deg(hi ) ≤ 2i − 2 .

Then the Zp -subalgebra of R(S) generated by SN is contained in TN .
Proof. Since SN and TN are both closed under addition and SN ⊆ TN , it is sufficient to
show that SN TN ⊆ TN . To do this, suppose
H(z) = c +

N
X

pi hi (z) ∈ SN

M
X

pi gi (z) ∈ TN ,

i=1

and
G(z) = d +

i=1

where c, d ∈ S and hi (z), gi (z) ∈ R(S) with deg(gi ) ≤ 2i − 2 for i > N and
deg(gi ), deg(hi ) ≤ 2i − 1 for i ≤ N. We must show that H(z)G(z) ∈ TN . Notice
that
N
M
X
X
i
H(z)G(z) = cG(z) + dH(z) − cd +
p hi (z)
pj gj (z).
i=1

j=1

Then cG(z), dH(z), and cd are all in TN and since TN is closed under addition, it is
sufficient to show that
N
X
i=1

pi hi (z)

M
X
j=1

pj gj (z) =

NX
+M
k=2

pk

k−1
X
i=1

gi (z)hk−i (z)

14

JASON P. BELL AND JEFFREY C. LAGARIAS

is in TN . But since gi (z) has degree at most 2i − 1 and hk−i (z) has degree at most
2(k − i) − 1, we see that
k−1
X
gi (z)hk−i (z)
i=1

has degree at most 2k − 2. It follows that
N
X

i

p hi (z)

i=1

The result follows.

M
X
j=1

pj gj (z) ∈ TN .


Proof of Theorem 3.1. We construct (f1 (z), . . . , fd (z)) by successive approximation
(mod pj R(S)). The approximations will be denoted gi,j (z) for 1 ≤ i ≤ d. We initialize with
gi,0 (z) := si for 1 ≤ i ≤ d.
We prove by induction on j that one can recursively pick
gi,j (z) := si +

j
X

pk hi,k (z),

k=1

such polynomials hi,j (z) ∈ R(S) (1 ≤ i ≤ d) satisfy the three conditions:
(1) hi,j (0) = 0 for 1 ≤ i ≤ d;
(2) hi,j (z) has degree at most 2j − 1 for 1 ≤ i ≤ d;
(3) there holds
gi,j (z + 1) ≡ Hi (g1,j (z), . . . , gd,j (z)) (mod pj+1R(S)).

The base case of the induction is j = 0. Conditions (1) and (2) are vacuous, and (3)
holds using hypothesis (i), observing that R(S) ∩ S = S.
Let j ≥ 1 and assume that we have defined hi,k for 0 ≤ i ≤ d and k < j so that
conditions (1)–(3) hold. Our object is now to construct
gi,j+1(z) := gi,j (z) + pj hi,j (z),
in which polynomials hi,j (z) ∈ R(S) are to be determined, so that conditions (1)–(3)
hold. By assumption
gi,j−1(z + 1) − Hi (g1,j−1(z), . . . , gd,j−1(z)) = pj Qi,j (z),

(3.5)

with Qi,j ∈ R(S) for 1 ≤ i ≤ d. Using the notation of the statement of Lemma 3.7, we
see that conditions (2) and (3) show that g1,j−1 (z), . . . , gd,j−1(z) are in Sj−1 . Thus by
Lemma 3.7 we see that
pj Qi,j (z) = gi,j−1(z + 1) − Hi (g1,j−1 (z), . . . , gd,j−1(z))

is in the algebra generated by Sj−1 and hence is in Tj−1 . It follows that we can write
P
k
pj Qi,j (z) = ci,j + M
k=1 p qi,j,k (z) for some ci,j ∈ S and polynomials qi,j,k (z) ∈ R(S)
such that deg(qi,j,k ) ≤ 2k − 1 for k ≤ j − 1 and deg(qi,j,k ) ≤ 2k − 2 for k ≥ j.
Consequently, pj Qi,j (z) is equivalent modulo pj+1 R to the polynomial
ci,j +

j
X
k=1

pk qi,j,k (z),

A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 15

a polynomial of degree at most 2j − 2. Hence Qi,j (z) is congruent to a polynomial in
R(S) of degree at most 2j − 2 mod pR(S). To satisfy property (3) for j it is sufficient
to find {hi,j (z) ∈ R(S) : 1 ≤ i ≤ d} such that
gi,j−1(z + 1) + pj hi,j (z + 1) − Hi (g1,j−1 (z) + pj h1,j (z), . . . , gd,j−1(z) + pj hd,j (z))
is in pj+1R(S) for 1 ≤ i ≤ d. This expression becomes
j

j

p Qi,j (z) + p hi,j (z + 1) − p

j

d
X

hℓ,j (z)

ℓ=1

∂Hi
(x1 , . . . , xd )
∂xℓ

x1 =g1,j (z),...,xd =gd,j (z)

modulo pj+1 R(S). It therefore suffices to solve for 1 ≤ i ≤ d the system
Qi,j (z) + hi,j (z + 1) −

d
X
ℓ=1

hℓ,j (z)

∂Hi
(x1 , . . . , xd )
∂xℓ

x1 =g1,j (z),...,xd =gd,j (z)

(3.6)

(mod pR(S)), where we may assume that Qi,j is of degree at most 2j − 2. Now
consider the Jacobian matrix M (j) (z) ∈ Md×d (R(S)) with polynomial entries
M (j) (z)iℓ :=

∂Hi
(x1 , . . . , xd )
∂xℓ

x1 =g1,j (z),...,xd =gd,j (z)

.

Property (1) for j yields for 1 ≤ i ≤ d that
gi,j (z) ≡ si (mod pR(S)).
It follows that
M (j) (z) ≡ J(σ; s0 )(mod pR(S)).

By hypothesis (ii) the matrix M = J(σ; s0 ) ∈ Md×d (S) is congruent to the identity
(mod pS), and we have
M (j) (z) ≡ M(mod pR(S)).
Now equation (3.6) can be rewritten in the form

 



Q1,j (z)
h1,j (z)
h1,j (z + 1)
..
..
..
 (mod pR(S)).
−
≡M

.
.
.
Qd,j (z)
hd,j (z)
hd,j (z + 1)

The hypotheses of Lemma 3.6 are satisfied, so we conclude that there exists a solution
[h1,j (z), . . . , hd,j (z)] ∈ R(S)d with hi,j (0) = 0 for 1 ≤ i ≤ d and hi (z) of degree at
most 2j − 1. Thus conditions (1)–(3) are satisfied for j, completing the induction step.
We now set
∞
X
fi (z) := si +
pj hi,j (z).
j=1

Now each hi,j (z) ∈ R(S) is of degree at most 2j − 1 and hence
 
2j−1
X
z
,
ci,j,k
hi,j (z) =
k
k=0

16

JASON P. BELL AND JEFFREY C. LAGARIAS

with ci,j,k ∈ S. We find that



 
Nj
X
z 
fi (z) = si +
pj 
ci,j,k
k
j=1
k=0
 
∞
X
z
= s0 +
bi,k
k
k=0
∞
X

in which

bi,k :=

∞
X

(3.7)

pj ci,j,k

j=1

is absolutely convergent p-adically, since each ci,j,k ∈ S. To show the series (3.7)
converges to an analytic map on z ∈ Zp , we must establish that |bi,k |p /|k!|p → 0 as
k → ∞, i.e. that for any j > 0 one has bi,k /k! ∈ pj Zp for all sufficiently large k. To do
this, we note that ci,j,k = 0 if j < (k + 1)/2. Hence
X
bi,k :=
pj ci,j,k .
j≥(k+1)/2

It follows that |bi,k |p < p−(k+1)/2 . Since 1/|k!|p < pk/(p−1) , we see that bi,k /k! → 0
since p > 3. Hence f1 , . . . , fd are analytic maps on Zp .
The argument above also showed that
fi (z) ≡ gi,j (z) (mod pj R(S)).
It then follows from property (3) above that
fi (z + 1) ≡ Hi (f1 (z), . . . , fd (z))(mod pj R(S))
Since this holds for all j ≥ 1, we conclude that
fi (z + 1) = Hi (f1 (z), . . . , fd (z)).
This establishes (1). Finally, we have
fi (0) = si +

∞
X

pj hi,j (0) = si ,

j=1



which establishes (2).

4. P OLYNOMIAL R ING C ASE
In this section, we consider automorphisms of a polynomial ring over a field. We
prove the following theorem.
Theorem 4.1. Let K be a field of characteristic zero and let A = K[x1 , . . . , xd ]. If
σ : A → A is a K-algebra automorphism and I and J are ideals of A, then
{n ∈ Z : σ n (I) ⊇ J}

has the two-sided SML property.

A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 17

In Section 5 we will deduce the main Theorem 1.7 from this result. We establish
Theorem 4.1 using a reduction to the ring Zp [x1 , . . . , xd ], given below (Theorem 4.5).
A key ingredient in the proof of Theorem 4.5 will be the p-adic analytic arc theorem
proved earlier, see Proposition 4.3.
In restricting ideals from a ring with Qp -coefficients to one with Zp -coefficients, we
only need to consider the following special subclass of ideals in Zp [x1 , x2 , . . . , xd ], as
follows.
Definition 4.2. Let p be a prime and let A be a Zp -algebra. We say that a proper ideal I
of A is p-reduced if whenever x ∈ A has the property that px ∈ I, we necessarily have
x ∈ I.
The condition that an ideal I of a Zp -algebra A be p-reduced is imposed to ensure
that the quotient ring A/I is torsion-free as a Zp -module.
Proposition 4.3. Let p be prime, and let I and J be two p-reduced ideals in
Zp [x1 , . . . , xd ]. Suppose that (Zp [x1 , . . . , xd ]/I) ⊗Zp Qp is finite dimensional as a
Qp -vector space. If σ is a Zp -algebra automorphism of Zp [x1 , . . . , xd ], then the set
{n ∈ Z | σ n (I) ⊇ J} has the two-sided SML property.
Proof. For a polynomial automorphism σ n (I) ⊇ J if and only if σ −n (J) ⊆ I. Thus,
replacing σ with σ −1 , it is sufficient to show that
{n ∈ Z : σ n (J) ⊆ I}

is a finite union of complete arithmetic progressions and a finite set. Let S :=
Zp [x1 , . . . , xd ]/I. The ring S is a finitely generated and torsion-free Z-algebra, since
I is p-reduced. Write σ = (F1 , . . . , Fd ) ∈ Zp [x1 , . . . , xd ]d . The automorphism
σ induces the dynamical evaluation map fσ : S d → S d , given by (s1 , . . . , sd ) 7→
(F1 (s1 , . . . , sd ), . . . , Fd (s1 , . . . , sd )), for si ∈ S, which is a (nonlinear) bijection on
S d . Since S is a torsion-free Zp -algebra and since, by hypothesis, S ⊗Zp Qp is finitedimensional, we have S d /pS d is a finite ring. Although fσ is nonlinear, we have
fσ (s + pS d ) ⊆ fσ (s) + pS d ,

hence fσ induces a well-defined bijective map of S d /pS d . Since S/pS is finite, there
exists a positive integer a such that fσa (s) ≡ s (mod pS d ) for all s ∈ S d .
Now set ρ = σ ma , where m is chosen as in the statement of Lemma 2.1, and write
ρ = (H1 , . . . , Hd ). Then for any s ∈ S d , we have fρ (s) ≡ s (mod pS d ) and by
Lemma 2.1, the Jacobian matrix of ρ is the identity matrix mod pS when evaluated at
any point in S d . For any polynomial P (x1 , . . . , xd ) ∈ Zp [x1 , . . . , xd ] and for any k with
0 ≤ k ≤ ma − 1, the fact that σ and ρ are endmorphisms gives (via (2.1))
ρn (σ k (P (x1 , . . . , xd ))) ∈ I if and only if P ◦ fρn ◦ fσk (s1 , . . . , sd ) = 0,

where si := xi + I. We now treat each k separately. By construction,
fρ ◦ fσk (s1 , . . . , sd ) ≡ fσk (s1 , . . . , sd ) (mod pS),

and the Jacobian matrix J(ρ, fσk (s1 , . . . , sd )) is congruent to the identity mod pS. Now
the generalized p-adic analytic arc theorem (Theorem 3.1) applies to show there exist
power series
f1 (z), . . . , fd (z) ∈ (S ⊗Zp Qp )[[z]]

18

JASON P. BELL AND JEFFREY C. LAGARIAS

which are analytic on Zp and which satisfy (f1 (0), . . . , fd (0)) = fσk (s1 , . . . , sd ) and
fi (z + 1) = Hi (f1 (z), . . . , fd (z))

for 1 ≤ i ≤ d.

By construction, fρn ◦ fσk (s1 , . . . , sd )) = (f1 (n), . . . , fd (n)) for all n ∈ N. Next, select
a generating set for the ideal J,
J = hP1 (x1 , . . . , xd ), . . . , Pm (x1 , . . . , xd )i.

Then for 1 ≤ i ≤ m, define

gi (z) := Pi (f1 (z), . . . , fd (z)) ∈ (S ⊗Zp Qp )[[z]].
Since f1 , . . . , fd are analytic on Zp and P1 , . . . , Pm ∈ Zp [x1 , . . . , xd ], each gi (z) is a
power series which converges on Zp , for 1 ≤ i ≤ d. Moreover, gi (n) = 0 if and only if
ρn ◦ σ k (Pi (x1 , . . . , xd )) ∈ I. Notice if gi (n) = 0 for infinitely many integers n, then it
is identically zero by Theorem 2.4. Thus
{n ∈ Z | ρn (σ k )(J) ⊆ I}
is either a finite set or is all of Z. Since this holds for 0 ≤ k ≤ ma1 , result now follows.

We must also treat the case that (Zp [x1 , . . . , xd ]/I) ⊗ Qp is an infinite-dimensional Qp vector space. This is handled by a reduction to the preceding proposition. To do this
reduction, we use the following lemma, based on an idea of Amitsur [2, Lemma 4, p.
41].
Lemma 4.4. Let K be an uncountable field and let A be a finitely generated commutative K-algebra. Suppose that there exists a countable set of ideals {Ii | i ≥ 1} of A such
that each ideal L of A of finite codimension contains one of the Ii . Then there exists a
finite set j1 , . . . , jd such that
Ij1 ∩ · · · ∩ Ijd = (0).
Proof. Let J(A) denote the Jacobson radical of A. Since A is a finitely generated Kalgebra, A is a Jacobson ring (that is, every prime ideal of A is the intersection of the
maximal ideals containing it) [13, Theorem 4.19]. Thus J(A) is the intersection of the
prime ideals of A (that is, it is the nilradical of A). By the Hilbert basis theorem A
is Noetherian and hence some power of the nilradical of A is zero [3, Corollary 7.15].
Hence there is some m ≥ 1 such that J(A)m = (0). Given a maximal ideal M of A, note
that there is some i such that Ii ⊆ M m . Let Qi denote the intersection of all ideals of the
form M m , where M is a maximal ideal of A with the property that M m ⊇ Ii . Similarly,
we let Pi denote the intersection of all ideals maximal ideals M with the property that
M m ⊇ Ii . Then Pim = Qi ⊇ Ii for each i ≥ 1.
Each maximal ideal M of A necessarily contains some Pi . A result of Amitsur (cf.
[2, Corollary 4, p. 40]) shows that there exist j1 , . . . , jd such that Pj1 ∩ Pj2 ∩ · · · ∩ Pjd
is contained in the Jacobson radical J(A) of A. Hence
Ij1 ∩ · · · ∩ Ijd ⊆ Qj1 ∩ · · · ∩ Qjd = Pjm1 ∩ · · · ∩ Pjmd ⊆ J(A)m = (0),
as required.



A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 19

Theorem 4.5. Let p ≥ 5 be prime, let I and J be two p-reduced ideals in Zp [x1 , . . . , xd ]
with I ∩ Zp = J ∩ Zp = (0). If σ is a Zp -algebra automorphism of Zp [x1 , . . . , xd ], then
the set
{m ∈ Z | σ m (I) ⊇ J}
is the union of a finite number of complete doubly-infinite arithmetic progressions, up to
the addition of a finite set.
Proof. Let T denote the collection of subsets of Z which are finite union of complete doubly-infinite arithmetic progressions along with a finite set. Then T is countable. Suppose first that I ⊆ Zp [x1 , . . . , xd ] is a p-reduced ideal with the property that
(Zp [x1 , . . . , xd ]/I) ⊗Zp Qp is finite-dimensional as a Qp -vector space. Then by Proposition 4.3, we have
{n ∈ Z | σ n (I) ⊇ J} ∈ T .

We now treat the general case. Given T ∈ T , let IT denote the intersection of all
p-reduced ideals L that contain I and such that:
(1) L ∩ Zp = (0);
(2) (Zp [x1 , . . . , xd ]/L) ⊗Zp Qp is a finite-dimensional Qp -vector space;
(3) T = {n ∈ Z | σ n (L) ⊇ J}.

Suppose first that T ∈ T is such that IT is a non-empty intersection. We first note
that if n ∈ T , then σ n (IT ) ⊇ J; moreover IT is an intersection of ideals which contain
I and hence it contains I. If n 6∈ T , then σ n (I) cannot contain J, since by definition
σ n (IT ) 6⊇ J and I ⊆ IT . Thus if IT is a non-empty intersection, then {n | σ n (IT ) ⊇
J} = T .
We next note that the collection of ideals
{IT | T ∈ T , IT is a nonempty intersection}
in Zp [x1 , . . . , xd ] is countable. Consider the ideals IT′ = IT Qp [x1 , . . . , xd ]. By Lemma
4.4 one of the following holds; either:
(1) there exist T1 , . . . , Td ∈ T such that IT′ 1 ∩ · · · ∩ IT′ d ⊆ IQp [x1 , . . . , xd ]; or
(2) there is an ideal L ⊃ IQp [x1 , . . . , xd ] of Qp [x1 , . . . , xd ] of finite codimension
which does not contain any of the non-empty IT′ .
In the first case, note that T0 = T1 ∩· · ·∩Td ∈ T and by definition, IT′ 0 ⊆ IT′ 1 ∩· · ·∩IT′ d ⊆
IQp [x1 , . . . , xd ]. Since IQp [x1 , . . . , xd ] ⊆ IT0 , we see that IQp [x1 , . . . , xd ] = IT0 . It
follows that the set of integers n for which σ n (I) ⊇ J is exactly T0 .
If, on the other hand, (2) holds, then there exists an ideal L of finite codimension which contains I but doesn’t contain any of the non-empty IT ; then L1 = L ∩
Zp [x1 , . . . , xd ] is p-reduced and satisfies Zp [x1 , . . . , xd ]/L1 ⊗Zp Qp ∼
= Qp [x1 , . . . , xd ]/L
is finite-dimensional. Hence IT ⊆ L1 for some T ∈ T , and so IT′ ⊆ L, a contradiction.
We thus obtain the result.

Now we can prove Theorem 4.1.
Proof of Theorem 4.1. Let K be a field of characteristic 0 and let A = K[x1 , . . . , xd ],
with σ : A → A a K-algebra automorphism, and let I and J be two proper ideals
of A. Then there exist polynomials F1 , . . . , Fd , G1 , . . . , Gd in A such that σ(xi ) =

20

JASON P. BELL AND JEFFREY C. LAGARIAS
−1

Fi (x1 , . . . , xd ) and σ (xi ) = Gi (x1 , . . . , xd ) for i ∈ {1, . . . , d}. Next there exist polynomials C1 , . . . , Cs in A that generate I as an ideal and polynomials D1 , . . . , Dt in A
that generate J.
We let S denote the set of all nonzero elements of K that occur as a coefficient of
one of F1 , . . . , Fd , G1 , . . . , Gd and C1 , . . . , Cs , D1 , . . . , Ds and we let R0 = [S] denote
the finitely generated Z-algebra generated by the elements of S inside K and we let K0
denote the field of fractions of R0 . By construction, σ(R0 [x1 , . . . , xd ]) ⊆ R0 [x1 , . . . , xd ]
and σ −1 (R0 [x1 , . . . , xd ]) ⊆ R0 [x1 , . . . , xd ], and so σ restricts to an R0 -algebra automorphism of R0 [x1 , . . . , xd ] and to a K0 -algebra automorphism of K0 [x1 , . . . , xd ].
We let I0 := I ∩ K0 [x1 , . . . , xd ]) and J0 = J ∩ K0 [x1 , . . . , xd ]), which are both
K0 [x1 , x2 , . . . , xd ]- ideals. We claim that:
σ n (I) ⊇ J

if and only if σ n (I0 ) ⊇ J0 .

To see this, first observe that if σ n (I) ⊇ J, then we have σ n (I0 ) ⊇ J0 , since
σ n (K0 [x1 , . . . , xd ]) = K0 [x1 , . . . , xd ]. Next suppose that n is an integer for which
σ n (I0 ) ⊇ J0 . Note that σ n (I) is generated by σ n (C1 ), . . . , σ n (Cs ) and each of these
generators is in K0 [x1 , . . . , xd ]. Since K[x1 , . . . , xd ] is a free K0 [x1 , . . . , xd ]-module, we
have that σ n (I0 ) is generated by σ n (C1 ), . . . , σ n (Cs ) as an ideal in K0 [x1 , x2 , . . . , xd ].
By construction of K0 , the generators Di of J lie in J0 for i ∈ {1, . . . , t} and so by
assumption
t
X
Di ∈
K0 [x1 , . . . , xd ]σ n (Ci ) ⊆ σ n (I)
i=1

for i = 1, . . . , t. It follows that J ⊆ σ n (I), proving the claim.
By Lemma 2.3, there exists a prime p ≥ 5 such that R0 embeds in Zp as a Z-algebra.
We identify R0 with its image in Zp and we similarly identify K0 with its image in Qp .
We then see that the restriction of σ to K0 [x1 , . . . , xd ] lifts to a Qp -algebra automorphism
ρ of Qp [x1 , . . . , xd ], by identifying Qp [x1 , . . . , xd ] with (K0 [x1 , . . . , xd ]⊗K0 Qp ) and then
taking ρ to be the map σ ⊗ id. In addition, the restriction of σ to R0 [x1 , . . . , xd ] lifts
to a Zp -algebra automorphism ρ0 of Zp [x1 , . . . , xd ] ∼
= (R0 ⊗R0 Zp )[x1 , . . . , xd ]. Note
that Qp [x1 , . . . , xd ] is a free K0 [x1 , . . . , xd ]-module and hence the ideals I0 and J0 lift
to ideals I ′ and J ′ of Qp [x1 , . . . , xd ] respectively. Moreover, since any basis for Qp over
K0 is fixed by ρ, by freeness we have
{m ∈ Z : σ m (I0 ) ⊇ J0 } ≡ {m ∈ Z : ρm (I ′ ) ⊂ J ′ }.
Next, we let
and

Ie = I ′ ∩ Zp [x1 , . . . , xd ]

Je = J ′ ∩ Zp [x1 , . . . , xd ].
e Je are necessarily p-reduced. Since
Since I ′ , J ′ are Qp [x1 , . . . , xd ]-ideals, the ideals I,
ρ restricts to the automorphism ρ0 of Zp [x1 , . . . , xd ], a similar argument to the one employed earlier in the proof gives
e
e
{m ∈ Z : ρm (I ′ ) ⊂ J ′ } ≡ {m ∈ Z : ρm
0 (I) ⊂ J }.

Theorem 4.5 now applies to show that the set on the right is the union of a finite number
of complete-doubly infinite arithmetic progressions plus a finite set; this equals the set

A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 21

of n with σ n (I0 ) ⊇ J0 . Finally, using the claim above, this set is identical to the set of
integers n for which σ n (I) ⊇ J, giving the result.

5. P ROOF

OF

M AIN R ESULT

In this section we prove Theorem 1.7. This is established by reduction of the general
case to the polynomial ring case using the following result of Srinivas, and then using
Theorem 4.1.
Theorem 5.1. (Srinivas) Let A be a finitely generated algebra over an infinite field K.
Then there exists a natural number n = n(A) such that for all N > n, if
f : K[x1 , . . . , xN ] → A and g : K[x1 , . . . , xN ] → A

are two surjective K-algebra homomorphisms, then there is an elementary K-algebra
automorphism φ : K[x1 , . . . , xN ] → K[x1 , . . . , xN ] such that f = g ◦ φ.

Remark. Here “elementary K-algebra automorphism” means one that is a finite composition of automorphisms of the types:
(i) (Translations) xi 7→ xi + ci , with
Pci ∈ K;
(ii) (Linear transformations) xi 7→ j ci,j xj where (ci,j ) ∈ GLN (K);
(iii) (Triangular automorphisms) xi 7→ xi for 1 ≤ i ≤ N − 1 and xN 7→ xN +
F (x1 , x2 , . . . , xN −1 ).
For our application we only need the fact that φ is a K-algebra automorphism.
Proof. See Srinivas [31, Theorem 2, p. 126].



We obtain the following result as an immediate corollary.
Proposition 5.2. Let A be a finitely generated algebra over a field K and let φ : A → A
be a K-algebra automorphism. Then there exists a natural number N, a surjective Kalgebra homomorphism ν : K[x1 , . . . , xN ] → A, and a K-algebra automorphism φ̃ of
K[x1 , . . . , xN ] such that:
(1) ν ◦ φ̃n = φn ◦ ν for all integers n;
(2) If I and J are ideals in A, then φn (I) ⊇ J if and only if
φ̃n (ν −1 (I)) ⊇ ν −1 (J).

Proof. We pick n(A) as in the statement of Theorem 5.1. Since A is finitely generated,
by f1 , . . . , fr , say, we can take N = max(r, n(A) + 1) and construct a surjective map
ν : K[x1 , . . . , xN ] → A sending xi 7→ fi for 1 ≤ i ≤ r and, if r < n(A) + 1, sending
any extra xj to 0. Now, since ν and φ ◦ ν are surjective, by Theorem 5.1 there exists an
automorphism φ̃ of K[x1 , . . . , xN ] satisfying ν ◦ φ̃ = φ◦ν. By construction ν ◦ φ̃ = φ◦ν.
Note that by composing on the left by φ−1 and on the right by φ̃−1 , we see
ν ◦ φ̃−1 = φ−1 ◦ ν.

We now prove property (1) by induction on |n|, noting the base case |n| = 1 is
established (for both n = 1, −1). Assume that it is true for all integers n with 0 < |n| <
m. Then
φm ◦ ν = φ ◦ (φm−1 ◦ ν) = φ ◦ (ν ◦ φ̃m−1 ) = (φ ◦ ν) ◦ φ̃m−1 = ν ◦ φ̃m .

A similar argument shows that φ−m ◦ ν = ν ◦ φ̃−m , and so (1) follows by induction.

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JASON P. BELL AND JEFFREY C. LAGARIAS

To verify (2), suppose that I and J are ideals of A. Since ν is surjective, we have
ν(ν −1 (L)) = L for every ideal of A. If φn (I) ⊇ J, then φn ◦ ν(ν −1 (I)) ⊇ J. Consequently, ν ◦ φ̃n (ν −1 (I)) ⊇ J. Thus
φ̃n (ν −1 (I)) ⊇ ν −1 (ν ◦ φ̃n (ν −1 (I))) ⊇ ν −1 (J).
Also, if φ̃n (ν −1 (I)) ⊇ ν −1 (J), then ν(φ̃n (ν −1 (I))) ⊇ ν(ν −1 (J)) = J. Since ν ◦ φ̃n =
φn ◦ ν, we see φn (I) = φn (ν(ν −1 (I)) ⊇ J. Thus (2) is established.

Proof of Theorem 1.7. We are given a field K of characteristic 0 and a finitely generated
commutative K-algebra A with a K-algebra automorphism σ. We wish to show that
the set of integers n such that σ n (I) ⊇ J is a finite union of complete doubly-infinite
arithmetic progressions along with a finite set.
By Proposition 5.2 there exists a polynomial algebra K[x1 , . . . , xN ] with an automorphism φ and ideals I ′ and J ′ such that
φn (I ′ ) ⊇ J ′ ⇐⇒ σ n (I) ⊆ J.


The result now follows from Theorem 4.1.
6. E XAMPLES

We now give several examples which show that results for √
general
√ ideals I, J can
change the answers compared to their associated radical ideals I, J: allowing nonradical ideals can change the structure of infinite arithmetic progressions, or eliminate
them entirely. We recall that in a commutative ring R, the radical is the ideal I consisting of all nilpotent elements of R. In particular, R/I is reduced.
Example 6.1. Let K be an algebraically closed field of characteristic 0 and let A =
K[x, y, z, t, u]. We define a K-algebra automorphism σ by
x 7→ y − (x − y + yz)t + (x − y + yz),
t 7→ −t,

y 7→ x − y + yz,
u 7→ u.

Then σ is an automorphism. Let
J = (x2 , xy, y 2, y − x, zt − 1, x − u),
I = (x2 , xy, y 2, y − x, zt − 1, x(z − 1), x − u).
Then

√
√
σ n ( I) ⊇ J for all n ∈ Z.

However we have
σ n (I) ⊇ J if and only if n ≡ 0, 3 (mod 4)
Also
σ n (I) ⊇

√

J never holds.

z 7→ −z,

A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 23

Proof. We note that σ is an automorphism, since it is a composition of automorphisms
σ3 ◦ σ2 ◦ σ1 , where
σ1 (x) = x + yt − y,
σ2 (x) = y,
σ3 (x) = x − y + yz
σ1 (y) = y,
σ2 (y) = x,
σ3 (y) = y
σ1 (z) = z,
σ2 (z) = z,
σ3 (z) = −z
σ1 (t) = t,
σ2 (t) = t,
σ3 (t) = −t
σ1 (u) = u,
σ2 (u) = u,
σ3 (u) = u
√
We note
that
J ⊇ (x, y, zt − 1, u), as x2 , y 2 ∈ I and zt − 1, x − u ∈ J. We claim
√
that J = (x, y, zt − 1, u). To see this, observe that J ⊆ (x, y, zt − 1, u) and so A/J
√
−1
] is reduced, we see that J
surjects onto A/(x, y, zt−1, u) ∼
= K[z, z −1 ]. Since K[z, z√
must have zero image under this homomorphism and so J ⊆ (x, y, zt − 1, u). Thus
√
√
J = (x, √
y, zt − 1,√u), which is a σ-invariant ideal. Also J ⊂ (x, y, zt − 1, xz, u) ⊂ I
hence σ n ( J) ⊆ I for all n ∈ Z. Let L = (x2 , xy, y 2, y − x, zt − 1). We note that
L ⊆ J is σ-invariant. An easy induction shows that
σ n (x) ≡ (−1)(

n−1
2

) xz n (mod L)

for n ∈ Z, where we interpret z −i as being ti (mod L). Thus σ n (J) ⊆ I if and only if
(−1)(

n−1
2

) xz n − u ∈ L + A(x(z − 1)) + A(x − u).

This occurs if and only if
(−1)(

n−1
2

) x − u ∈ L + A(x(z − 1)) + A(x − u).

If n ≡ 0, 1 (mod 4) then this occurs. Note that

L + A(x(z − 1)) + A(x − u) ⊆ (z − 1, t − 1, x2 , y, x − u)

and
A/(z − 1, t − 1, x2 , y, x − u) ∼
= K[x]/(x2 ).
n−1
n−1
The image of (−1)( 2 ) x−u under this homomorphism is (−1)( 2 ) x−x+(x2 ), which
has nonzero image for n ≡ 2, 3 (mod 4). Hence we have two arithmetic progressions
n ≡ 0 (mod 4) and n ≡ 1 (mod 4) for which σ n (J) ⊆ √
I. Equivalently,
σ n (I) ⊇ J if
√
and only if n ≡ 0, 3 (mod 4).
We already saw
that σ n ( I) ⊇ J for all n. Finally,
√
√
note that we always have σ n ( J) 6⊆ I, since J is σ-stable and x 6∈ I.

We next consider finitely generated commutative K-algebras A having a nontrivial
radical. We give two examples of automorphisms of algebras with nonzero radicals for
which the radical affects the dynamics non-trivially. We begin with a simple example
that shows that an automorphism of a ring whose action is trivial on the reduced ring
can still produce non-trivial dynamics.
Example 6.2. Let K be a field of characteristic 0 and let A = K[x, y, z]/(x, y)3 . Let
σ : A → A be the automorphism given by σ(x) = y, σ(y) = x, σ(z) = z. Let I = (x)
and J = (y). Then
σ n (I) ⊇ J if and only if n ≡ 1 (mod 2).
.

24

JASON P. BELL AND JEFFREY C. LAGARIAS

Proof. This holds by inspection. The point of this example is that σ induces the trivial
automorphism on K[z] ∼
= A/J(A), where J(A) denotes the radical of the zero ideal.

We next give an example of an automorphism of a non-reduced ring A with suitable
ideals exhibiting nontrivial dynamics, consisting of infinite arithmetic progressions, on
both A and its reduction A/J(A), but the dynamics differ.
Example
6.3.
Let
K
be
a
field
of
characteristic
0,
−1
−1
3
and let A = K[u, v, y, y , z, z ]/(u, v) . We let σ : A → A be the automorphism
given by σ(u) = uy, σ(v) = vz, σ(y) = −y, σ(z) = z. Let
J = (u + v)
I = (y − 1, z − 1, u + v).
Then
σ n (I) ⊇ J if and only if n ≡ 0, 3 (mod 4).
Proof. We have



n−1
σ n (J) = (−1)( 2 ) uy n + vz n ,

which is contained in I if and only if n−1
≡ 0 (mod 2). This occurs exactly when
2
n ≡ 0, 1 (mod 4). The result follows.

In Example 6.3 the ring A is non-reduced and the arithmetic progressions that occur each
have “gaps” of length 4. However when one studies the action of the automorphism σ on
the reduced ring A/J(A) ≃ K[y, y −1, z, z −1 ] we see that it has order 2, so that its action
on the reduced ring for any ideals I, J will have orbits which decompose into arithmetic
progressions whose gaps have length 1 or 2 along with a finite set; in Example 6.3 it is
all integers n since J ⊆ J(A). An interesting question is whether the “gaps” between
the two cases can be bounded in terms of the size of the gaps that occur from the induced
action on the reduced ring and the degree of nilpotency of the radical ideal.
We conclude with examples showing that the hypotheses that the commutative Kalgebra A must be finitely generated over K, and that K must have characteristic zero,
are both needed for the truth of the theorems above.
Example 6.4. Let K be a field and let S be an arbitrary subset of the integers. Then
there exists a commutative K-algebra A = AS which is not Noetherian and which is
infinitely generated over K, having the property that it has an automorphism σ and
ideals I and J such that σ i (I) ⊇ J holds if and only if i ∈ S.
Proof. Let A = K[xn : n ∈ Z] be a polynomial ring in infinitely many variables and
let σ be the two-sided shift automorphism defined by σ(xi ) = xi+1 for i ∈ Z. Given a
subset S of integers, we let PS = (xi | − i ∈ S). Then σ i ((x0 )) ⊆ PS if and only if
−i ∈ S and so
σ n (PS ) ⊇ (x0 )

precisely when n ∈ S.



A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 25

If we impose the extra requirement that A be Noetherian, but allow it to be infinitely
generated, there are nontrivial restrictions on the allowable sets S ⊂ Z giving ideal
inclusions. A result of Farkas [15, Theorem 8] (applied with G = Z) shows that if A is
Noetherian and φ is an endomorphism of A, then the set of natural numbers i such that
φi (I) ⊆ J must either be the entire set of natural numbers, or else must have a syndetic
complement; that is, there exists a natural number d such that if m is in the complement
then there exists j with 1 ≤ j ≤ d such that m + j is also in the complement. For
K a field of characteristic 0, we do not know any example of an infinitely generated
Noetherian commutative K-algebra A with ideals such that the set S is not a finite
union of arithmetic progressions, possibly augmented by a finite set.
Our final example shows that the hypothesis that the ground field K have characteristic zero is necessary for Theorem 1.7 to hold. In 1953 Lech [20] gave a counterexample
in positive characteristic p, and we observe it applies at the level of ideals.
Example 6.5. (Lech) Let p be a prime, let K = Fp (t) for the finite field Fp , and let
A = K[x, y]. Define σ : A → A by σ(x) = tx and σ(y) = (1 + t)y. Take

Then

J = (x + y − 1)
I = (x − 1, y − 1).

σ n (I) ⊇ J if and only if n ∈ {−1, −p, −p2 , . . . }.
Proof. We show the equlvalent assertion σ n ((x − y + 1)) ⊆ (x − 1, y − 1) if and only if
n ∈ {1, p, p2, . . .}. Note that σ n (x − y + 1) = tn x − (1 + t)n y + 1, whence we have the
ideal inclusion σ n ((x − y + 1)) ⊆ (x − 1, y − 1) if and only if tn − (1 + t)n + 1 = 0.
This equation holds if and only if n is a power of p.

In the case K has characteristic p > 0, Derksen [12] has further shown that if σ is a
linearizable endomorphism of A = K[x1 , . . . , xd ], then for ideals I and J of A, the set
of m such that σ m (I) ⊆ J can be classified; in particular this set is always a p-automatic
set, as defined in Allouche and Shallit [1].
7. A PPENDIX : DYNAMICS

OF

E NDOMORPHISMS –G EOMETRIC
A LGEBRAIC

VERSUS

This appendix presents a result addressing the difference between the geometric and
algebraic formulations of dynamics of endomorphisms. This result concerns the difference between the geometric action of endomorphisms acting as maps on S d by
forward iteration versus the algebraic action of endomorphisms acting on ideals in
R = S[x1 , . . . , xd ] by backward iteration. It shows that the two actions differ in some
circumstances.
To state it, suppose that S is an integral domain, take R = S[x1 , . . . , xd ] and let U be
a subset of R. The set of S-points cut out by U in S d is
V (U) := VS (U) = {s = (s1 , . . . , sd ) ∈ S d : evs (p(x)) = 0 for all p(x) ∈ U}.

Of course V (U) = V (I(U)) where I(U) is the smallest ideal containing U. Note that
for endomorphisms τ and ideals I that τ −1 (I) is an R-ideal while τ (I) := {τ (p(x)) :
p(x) ∈ I} need not be an R-ideal.

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JASON P. BELL AND JEFFREY C. LAGARIAS

Proposition 7.1. Let S be an integral domain of any characteristic, and consider the
polynomial ring R = S[x1 , . . . , xd ]. Let τ : R → R be a S-algebra endomorphism of
R. Consider the following possible relations between ideals I and J of R, the S-algebra
endomorphism τ , and the dynamical evaluation map fτ :
(a) τ −1 (I) ⊇ J, in the ring R.
(b) I ⊇ τ (J), in the ring R.
(c) V (I) ⊆ V (τ (J)) in S d .
(d) fτ (V (I)) ⊆ V (J) in S d .
(e) V (I) ⊆ (fτ )−1 (V (J)) in S d .
Then:
(1) There holds (a) ⇔ (b) and (c) ⇔ (d) ⇔ (e). In addition (b) ⇒ (c), however in
general (c) 6⇒ (b).
(2) If S = K is an algebraically closed field of any characteristic and I is a radical
ideal then (c) ⇔ (b). In this case all five relations (a)–(e) are equivalent.
This proposition shows that for endomorphisms τ the obstruction to the equivalence
of all five of these properties is (c) 6⇒ (b), which concerns non-radical ideals I. This
difference matters at the level of the generalized SML theorem for ideal inclusion (Theorem 1.7) in that it can change the allowed arithmetic
√
√progressions for inclusion relations
′
of ideals I, I having the same radical ideal I = I ′ with a fixed J, cf. Example 6.1.
Proof of Proposition 7.1. Let τ be an S-algebra endomorphism of R = S[x1 , ..., xd ].
(1). (a) ⇔ (b). For any U ⊆ R, set τ −1 (Y ) := {r(x) ∈ R : τ (r(x)) ∈ U}. Then we
have the inclusions
τ ◦ τ −1 (U) ⊆ U ⊆ τ −1 ◦ τ (U).

Furthermore if τ := σ is an automorphism, then equality holds in both inclusions.
Suppose τ −1 (I) ⊇ J. We apply τ to both sides to obtain
I ⊇ τ ◦ τ −1 (I) ⊇ τ (J).

Conversely, given I ⊇ τ (J), applying τ −1 to both sides gives
τ −1 (I) ⊇ τ −1 ◦ τ (J) ⊇ J.

(b) ⇒ (c). We are given I ⊃ τ (J). Now suppose s ∈ V (I) = VS (I) and we are
to show s ∈ V (τ (J)). The hypothesis asserts evs (q(x)) = 0 for all q(x) ∈ I. The
conclusion asserts that
τ (p)(s) := evs (τ (p)(x)) = 0 for all

p(x) ∈ J.

To verify this, the inclusion τ (J) ⊆ I gives
τ (p)(x) = p(τ (x1 ), . . . , τ (xd )) =: q(x),
for some q(x) ∈ I. Now
evs (τ (p))(x) = p(evs (τ (x1 )), . . . , evs (τ (xd )))
and by definition evs (τ (x1 ), . . . , τ (xd )) = fτ (s). We conclude
evs (τ (p)(x)) = evfτ (s) (p(x)).

A SKOLEM-MAHLER-LECH THEOREM FOR ITERATED AUTOMORPHISMS OF K-ALGEBRAS 27

Now s ∈ V (I) gives q(s) = 0 whence

0 = q(s) := evs (q(x)) = evs (p(τ (x1 ), . . . , τ (xd )))
= p(evs (τ (x1 ), . . . , τ (xd )) = evfτ (s) (p(x))

Since this holds for all p(x) ∈ J, we have s ∈ V (τ (J)).
(c) ⇔ (d). Let s ∈ S d . The key property is the identity, valid for all r(x) =
r(x1 , . . . , xd ) ∈ R, that
evs (τ (r(x1 , . . . , xd ))) = evs (r(τ (x1 ), . . . , r(τ (xd ))) = evfτ (s) (r(x)).

In what follows we abbreviate r(s) := evs (r(x1 , . . . , xd )).
Suppose V (I) ⊆ V (τ (J)). If s ∈ V (I) then q(s) = 0 for all q(x) ∈ I. We are to
show that fτ (V (I)) ⊆ V (J), which asserts that for all p(x) ∈ J there holds p(fτ (s)) :=
evfτ (s) (p(x)) = 0. Here, using the equality above
evfτ (s) (p(x)) = evs (τ (p)(x)) =: τ (p(s)).
Now τ (p(s)) = 0 because V (I) ⊂ V (τ (J)).
Conversely, suppose fτ (V (I)) ⊆ V (J) in S d . We must show V (I) ⊂ V (τ (J)).
Given s ∈ V (I). then we must show that for each p(x) ∈ J that
τ (p)(s) := evs (τ (p)(x)) = 0.

Using the identity above evs (τ (p)(x)) = evfτ (s) (p(x)). But by hypothesis fτ (s) ∈
V (J), whence
p(fτ (s)) := evfτ (s) (p(x)) = 0,
as required.
(d) ⇔ (e). For any subset N ⊂ S d , set fτ−1 (N ) = {s ∈ S d : fτ (s) ∈ N }. Then we
have the inclusions
fτ ◦ fτ−1 (N ) ⊆ N ⊆ fτ−1 ◦ fτ (N ).
Furthermore if τ := σ is an automorphism, we have equality in both inclusions, for
in this case fσ is a bijection with inverse fσ−1 . The argument is similar to (a) ⇔ (b).
Suppose fτ (V (I)) ⊂ V (J). Applying fτ−1 yields
V (I) ⊆ fτ−1 ◦ fτ (V (I)) ⊆ fτ (V (J))

Conversely, suppose V (I) ⊆ fτ−1 (V (J)). Applying fτ−1 yields
fτ (V (I)) ⊆ fτ ◦ fτ−1 (V (J)) ⊆ V (J)

(c) 6⇒ (b). This is well known. Take R = K[x], for K a field. and τ (x) = x3 . Take
I = (x4 ) and J = (x). Then x3 ∈ τ (J) ⊂ (x3 ), so that V (I) = V (τ (J)) = {0}, and
V (I) ⊆ V (τ (J)). But x3 6∈ I, so I 6⊇ τ (J). (Note also that τ (J) = {τ (p)(x) : p(x) ∈
J} is not an R-ideal.)
(2). The assertion (c) ⇒ (b) when S = K is an algebraically closed field (of any
characteristic) and I is a radical ideal is the Nullstellensatz. This gives all the equalities
(a)–(e). No condition is imposed on the ideal J to get the equality.

Remark 7.2. The failure of the methods of this paper to handle general endomorphisms
arises not from Proposition 7.1, but rather from the failure of the generalized p-adic
analytic arc theorem to apply to certain endomorphisms.

28

JASON P. BELL AND JEFFREY C. LAGARIAS

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D EPARTMENT OF M ATHEMATICS , U NIVERSITY
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E-mail address: jpbell@uwaterloo.ca
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