Two-variable polynomials with dynamical Mahler measure zero

The (logarithmic) Mahler measure of a non-zero rational function $P\in\mathbb{C}(x_{1},\dots x_{n})^{\times}$ is defined by

where $\mathbb{T}^{n}=\{(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\,:\,|z_{1}|=\cdots=|z_{n}|=1\}$ is the unit torus.

In the particular case of a single variable polynomial, Jensen’s formula implies that if

then

The single variable polynomial version was first introduced by Lehmer [Leh33] as a useful tool in a method for producing large prime numbers by generalizing Mersenne’s sequences.

It is natural to ask the question about the range of possible values of Mahler measure. For $P\in\mathbb{Z}[x]$, Kronecker’s lemma [Kro57] implies that $\mathrm{m}(P)=0$ if and only if $P$ is (up to a sign) monic and a product of cyclotomic polynomials and a monomial.

From the context of finding large prime numbers, Lehmer posed the following question:

Given $\varepsilon>0$, can we find a polynomial $P\in\mathbb{Z}[x]$ such that $0<\mathrm{m}(P)<\varepsilon$?

Lehmer showed that

and declared that this was the polynomial with the smallest positive measure that he was able to find. To this day, Lehmer’s question remains unanswered and Lehmer’s polynomial remains the one with the smallest known positive Mahler measure. Some key progress on this question can be found, for example, in the works of Breusch [Bre51] and Smyth [Smy71], who proved that the Mahler measures of non-reciprocal polynomials have a lower bound; Dobrowolski [Dob79], who gave the first lower bound that approaches zero when the degree goes to infinity; and the recent proof of Dimitrov [Dim] of the Schinzel–Zassenhaus conjecture.

Mahler measure for multivariate polynomials was first considered by Mahler [Mah62] in connection to heights and their applications in transcendence theory. Later Smyth [Smy81, Boy81b] obtained the first exact formulas involving Dirichlet $L$-functions and the Riemann zeta function, such as

More recently, various formulas have been proven or conjectured involving $L$-functions associated to more complicated arithmetic-geometric objects such as elliptic curves. For example, the following formula was conjectured by Deninger [Den97] and Boyd [Boy98] and proven by Rogers and Zudilin [RZ14]

where $E_{15}$ is an elliptic curve of conductor 15.

We finish the discussion on classical Mahler measure by recalling a result due to Boyd [Boy81b] and Lawton [Law83]. If $P\in\mathbb{C}(x_{1},\dots,x_{n})^{\times}$, then

where

where $H({\bf s})=\max\{|s_{j}|:2\leq j\leq n\}$. Intituively, the above limit is taken while $k_{2},\dots,k_{n}$ go to infinity independently from each other.

This result expresses the Mahler measure of a multivariate polynomial in terms of the Mahler measure of a single variable polynomial. It sheds light into the Lehmer conjecture for multivariate polynomials, since the existence of any such polynomial with small Mahler measure would immediately imply the existence of infinitely many single variable polynomials with small Mahler measure.

In classical (holomorphic) dynamics, one studies topological and analytic properties of orbits of points under iteration of self-maps of $\mathbb{C}$ or, more generally, of a complex manifold. In the younger field of arithmetic dynamics, the self-maps act on number theoretic objects (for example, algebraic varieties) and the objects and maps are defined over fields of number theoretic interest (number fields, function fields, finite fields, non-archimedean fields, etc.).

We fix the following notation: Let $X$ be a set, possibly with additional structure. For $f:X\to X$ and $L\in\operatorname{Aut}(X)$, we write

This conjugation is a natural dynamical equivalence relation in that it respects iteration; that is, $(f^{L})^{n}=(f^{n})^{L}$. In the sequel, we are concerned with polynomial maps $f:\mathbb{C}\to\mathbb{C}$, so we consider equivalence up to conjugation by affine maps $L(z)=az+b\in\mathbb{C}[z]$ with $a\neq 0$.

Let $\alpha\in X$. The forward orbit of $\alpha$ under $f$ is the set $\mathcal{O}_{f}(\alpha)=\{f^{n}(\alpha)\,:\,n\geq 0\}$. Questions in arithmetic dynamics are often motivated by an analogy between arithmetic geometry and dynamical systems in which, for example, rational and integral points on varieties correspond to rational and integral points in orbits, and torsion points on abelian varieties correspond to preperiodic points (points with finite orbit). See [Sil07] for more comprehensive background and motivation and [BIJ⁺19] for a survey of recent progress in the field of arithmetic dynamics.

Two complementary sets play an important role in holomorphic dynamics: The Fatou set of $f$ is the maximal open set on which the family of iterates $\{f^{n}\,:\,n\geq 1\}$ is equicontinuous (considered as maps on $\mathbb{P}^{1}(\mathbb{C})$); its complement is the Julia set of $f$, which we denote $\mathcal{J}_{f}$. Informally, the Julia set is where the interesting dynamics happens. For a polynomial $f:\mathbb{C}\to\mathbb{C}$, we may define the Julia set a bit more concretely.

Let $f\in\mathbb{C}[z]$. The filled Julia set of $f$ is

The Julia set of $f$ is the boundary of the filled Julia set. That is, $\mathcal{J}_{f}=\partial\mathcal{K}_{f}$. It follows from these definitions that for a polynomial $f\in\mathbb{C}[z]$, both $\mathcal{K}_{f}$ and $\mathcal{J}_{f}$ are compact. In dynamical Mahler measure, the Julia set $\mathcal{J}_{f}$ for a general polynomial $f$ will play the role of the unit torus $\mathbb{T}$.

Two families of polynomials play an important role in this work: The pure power maps $f(z)=z^{d}$ and the Chebyshev polynomials. Both of these families have special dynamical properties that arise because they are related to endomorphisms of algebraic groups.

Consider the multiplicative group $\mathbb{G}_{m}$ where for a field $K$, the $K$-valued points are $\mathbb{G}_{m}(K)=K^{*}$. The endomorphism ring of $\mathbb{G}_{m}$ is $\mathbb{Z}$:

Iteration of pure power maps is particularly easy to understand:

For $d\geq 2$ and $K\subseteq\mathbb{C}$ we have three cases: If $|\alpha|>1$, then $|\alpha^{d^{n}}|\to\infty$ with $n\to\infty$. If $|\alpha|<1$, then $|\alpha^{d^{n}}|\to 0$ with $n\to\infty$. If $|\alpha|=1$, then $|\alpha^{d^{n}}|=1$ for all $n$. So for pure power maps, we can understand the Julia sets completely: $\mathcal{K}_{f}$ is the unit disc, and $\mathcal{J}_{f}$ is the unit circle.

To give another example, since the automorphism of $\mathbb{G}_{m}$ given by $z\mapsto z^{-1}$ commutes with the power map $z\mapsto z^{d}$, the power map descends to an endomorphism of $\mathbb{A}^{1}$, which we denote $T_{d}$, the $d^{\operatorname{th}}$ Chebyshev polynomial.¹¹1Classically the Chebyshev polynomials are normalized so that $\widetilde{T}_{d}(\cos t)=\cos(dt)$. The two normalizations satisfy $\widetilde{T}_{d}(w)=\frac{1}{2}T_{d}(2w)$.

Taking as a definition the fact that $T_{d}\in\mathbb{Z}[w]$ satisfies

one may prove existence and uniqueness of the Chebyshev polynomials along with a recursive formula:

The following rule for composition of Chebyshev polynomials follows directly from the definition in (1.4):

and it gives a simple form of iteration

For $d\geq 2$, we have two cases for $T_{d}:\mathbb{C}\rightarrow\mathbb{C}$: If $\alpha\in[-2,2]$, then $T_{d}(\alpha)\in[-2,2]$. If $\alpha\in\mathbb{C}\setminus[-2,2]$, then $|T_{d}^{n}(\alpha)|=|T_{d^{n}}(\alpha)|\to\infty$ with $n\to\infty$. Thus we understand the Julia set in this case as well: $\mathcal{K}_{T_{d}}=\mathcal{J}_{T_{d}}=[-2,2]$.

See [Sil07, Chapter 6] for more on the dynamics of pure power maps, Chebyshev polynomials, and other maps arising from algebraic groups, including proofs of some of the statements above.

An important tool in the study of arithmetic dynamics is the canonical height of a point in $\mathbb{P}^{1}$ relative to a morphism $f$, defined by Call and Silverman [CS93] and modeled after Tate’s construction of the canonical (Néron-Tate) height on abelian varieties. The standard (Weil) height of a point $\beta=\frac{a}{b}\in\mathbb{Q}$ (written in lowest terms) can be computed as $h(\beta)=\max\{|a|,|b|\}$, and, observing that $a$ and $b$ are (up to sign) the coefficients of the minimal polynomial of $\alpha$ over $\mathbb{Z}$, this definition extends easily to algebraic numbers $\alpha\in\bar{\mathbb{Q}}$. If $f\in\mathbb{Q}[z]$ is a polynomial and $\alpha\in\mathbb{P}^{1}_{\bar{\mathbb{Q}}}$, we then define:

where $f^{n}$ represents the $n$-fold composition of $f$ and $d=\deg(f)$. In analogy with the canonical height on abelian varieties, we find that this limit exists and that

• •

  $\hat{h}_{f}(f(\alpha))=(\deg f)\hat{h}_{f}(\alpha)$, and

• •

  $\hat{h}_{f}(\alpha)=0$ if and only if $\alpha$ is a preperiodic point for $f$ (that is, $f^{m}(\alpha)=f^{n}(\alpha)$ for some natural numbers $m>n$).

Let $\alpha\in\bar{\mathbb{Q}}$ with minimal polynomial $P\in\mathbb{Z}[z]$. The theory of heights and Mahler measure are connected via the formula

Let $\mathcal{K}\subset\mathbb{C}$ be compact. For any Borel probability measure $\nu$ on $\mathcal{K}$ the energy of $\nu$ is given by [Ran95, Definition 3.2.1]:

An equilibrium measure on $\mathcal{K}$ is a measure $\mu$ satisfying

where the supremum is taken over all Borel probability measures. An equilibrium measure always exists and is often unique [Ran95, Theorem 3.2.2].

Let

where $a_{d}\not=0$ and $d\geq 2$.

Brolin [Bro65], Lyubich [Lyu83], and Freire, Lopes, and Mañe [FLM83] construct a unique equilibrium measure $\mu_{f}$, supported on the Julia set $\mathcal{J}_{f}$, which is invariant under $f$. That is, $f_{*}\mu_{f}=\mu_{f}$, where the push-forward $f_{*}\mu$ is defined by $f_{*}\mu(B)=\mu(f^{-1}(B))$ for any Borel set $B$. We also have

and in particular, $I(\mu_{f})=0$ when $f$ is monic.

Suppose that $f\in\mathbb{Z}[z]$ is monic, and that $\beta=\frac{a}{b}\in\mathbb{Q}$ (written in lowest terms). Then one can show that

More generally, if $\alpha$ is an algebraic integer with monic minimal polynomial $P(z)$, then

Notice that we are associating the canonical height for the rational map $f$ to a generalized Mahler measure $\mathrm{m}_{f}$ for polynomials $P\in\mathbb{Z}[x]$. Taking $f(z)=z^{2}$ recovers the familiar Mahler measure from equation (1.7).

Piñeiro, Szpiro, and Tucker [PST05] have shown that, like heights, this generalized Mahler measure may be computed via an adelic formula:

where $K$ is a number field, $\alpha\in\mathbb{P}^{1}(K)$, $P$ is a minimal polynomial for $\alpha$ over $K$, and $\mathcal{J}_{f,v}$ denotes the Julia set of $f:\mathbb{P}^{1}(\mathbb{C}_{v})\to\mathbb{P}^{1}(\mathbb{C}_{v})$. For finite $v$, $\int_{\mathcal{J}_{f,v}}\log|P(z)|_{v}d\mu_{f,v}(z)=0$ unless $f$ has bad reduction at $v$ or all the coefficients of $P$ have nonzero $v$-adic valuation. Thus, equation (1.9) follows from (1.10) by specializing to $K=\mathbb{Q}$ and $\alpha$ algebraic integer.

The contributions at the finite places in equation (1.10) were shown to have the appropriate integral forms by Favre and Rivera-Letelier [FRL06, Proposition 1.3]. Thus, equation (1.9) can also be extended to the case where $f$ is not necessarily monic if the integral is replaced by a sum of integrals at different places.

Szpiro and Tucker [ST12] used diophantine approximation to show that this generalized Mahler measure of a polynomial $P$ at a place $v$ can be computed by averaging $\log|P|_{v}$ over the periodic points of $f$.

It is not easy to give precise examples of $\mathrm{m}_{f}(P)$. In the classical case when $f(z)=z^{d}$ with $d\geq 2$, then $\mathrm{m}_{f}(P)=\mathrm{m}(P)$ for $P\in\mathbb{C}(x)^{\times}$. For $T_{d}$ the $d^{\text{th}}$ Chebyshev polynomial with $d\geq 2$, normalized so that $T_{d}(z+z^{-1})=z^{d}+z^{-d}$, one can show that $\mathrm{m}_{T_{d}}(P)=\mathrm{m}\left(P\circ(z+z^{-1})\right)$. Because of the underlying group structure discussed above, these examples are straightforward to compute (see [CLMM]). For general polynomials, exact computation is much more difficult, and in fact these two families are the exceptions to many theorems stated in the sequel.

One can compute approximations to dynamical Mahler measure in some cases. For example, Ingram [Ing13] gave an asymptotic formula for $\mathrm{m}_{f}(x)$ for $f(z)=z^{2}+c$ as $c\rightarrow\infty$

Similarly for $f(z)=(z-c)^{2}$, one can find

It is natural to ask about higher-dimensional (multivariate) forms of this generalized Mahler measure.

We will later see that the above integral always converges and that in fact, $\mathrm{m}_{f}(P)\geq 0$ if $P\in\mathbb{Z}[x_{1},\dots,x_{n}]$ is nonzero.

It is an interesting to ask whether multivariable dynamical Mahler measure can also be interpreted as a height. We do not answer the question in this paper, but we suspect that the answer is yes. Zhang [Zha95], in the more general setting of a variety $X$ with a map $\Phi:X\to X$ and a line bundle $\mathcal{L}$ such that $\Phi^{*}\mathcal{L}=\mathcal{L}^{d}$, defined a canonical height $h_{\Phi,\mathcal{L}}(Y)$ of an arbitrary subvariety $Y\subset X$. Chambert-Loir and Thuillier [CLT09] showed that in the case where $X=\mathbb{P}^{n}$, $\Phi([x_{0},\dotsc x_{n}])=[x_{0}^{2},\dotsc,x_{n}^{2}]$ and $\mathcal{L}=\mathcal{O}(1)$, that, for a homogeneous polynomial $P\in\mathbb{Z}[x_{0},\dotsc,x_{n}]$ and $Y$ the hypersurface $\{P(x_{0},x_{1},\dotsc,x_{n})=0\}$, $h_{\Phi,\mathcal{L}}(Y)=\mathrm{m}(P)$ is the usual multivariable Mahler measure of $P$, which is also equal to the multivariable Mahler measure of the inhomogeneous polynomial $P(1,x_{1},\dotsc,x_{n})$.

We conjecture that the multivariable dynamical Mahler measure can be recovered in a similar way, using instead the map $\Phi=f\times\cdots\times f:(\mathbb{P}^{1})^{n}\to(\mathbb{P}^{1})^{n}$, which acts as $f$ on each of the $n$ components.

In this note, we prove the following results.

The above statement is expected to be true with an equality. This would correspond to a dynamical version of the Boyd–Lawton limit (1.2).

For the main result, we will assume the following conjecture.

Note that the second half of Theorem 1.5 is only a partial converse. We can guarantee that $\prod F_{j}\in\mathbb{Q}[x,y]$ by enlarging the set $\{F_{j}\}$ to contain all the Galois conjugates of all its members. However, as we do not know whether the polynomials $F_{j}$ have algebraic integer coefficients, we cannot guarantee $\prod F_{j}\in\mathbb{Z}[x,y]$. We conjecture however that it is always the case that the $F_{j}$ have algebraic integer coefficients, in which case we could strengthen this converse.

We remark that Theorem 1.5 complements a statement for classical Mahler measure. Indeed, it is shown in [EW99, Theorem 3.10] that for any primitive polynomial $P\in\mathbb{Z}[x_{1}^{\pm},\dots,x_{n}^{\pm}]$, $\mathrm{m}(P)$ is zero if and only if $P$ is a monomial times a product of cyclotomic polynomials evaluated on monomials. This case is fundamentally different from what is considered in Theorem 1.5, as the classical Mahler measure contains more structure given by the multiplicative nature of the function $f(z)=z^{d}$.

It is also interesting to compare our results with [Zha95, Conjecture 2.5], which says that a subvariety whose dynamical canonical height is $0$ must be preperiodic. If Zhang’s height agrees with the dynamical Mahler measure, as we expect to be the case, then Lemma 1.2 and Theorem 1.5 can be seen as special cases of Zhang’s Conjecture 2.5 for the dynamical maps $f:\mathbb{P}^{1}\to\mathbb{P}^{1}$ and $f\times f:\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$, respectively. (Note that Zhang’s conjecture is known not to hold in general: see [GTZ11] for a counterexample. However, it may be possible to use the arguments in [GNY19] to obtain a proof for the case of $f\times f:\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$.)

This article is organized as follows. Section 2 covers some background on complex dynamics. Section 3 is devoted to showing that the dynamical Mahler measure is well-defined in general, and non-negative for polynomials with integral coefficients, while Section 4 discusses the dynamical version of Kronecker’s Lemma 1.2. The Weak Dynamical Boyd–Lawton Proposition 1.3 is discussed and proven in Section 5. It serves as a preparation for the proof of our main result, Theorem 1.5, covered in Sections 6 and 7. Finally, in Section 8, we give a precise description of the polynomials $L$ and $\tilde{f}$ that appear in the statement of this theorem.

We are grateful to Patrick Ingram for proposing that we study the dynamical Mahler measure of multivariate polynomials and for many early discussions, and to the anonymous referees for their careful reading of the article and several helpful suggestions. This project was initiated as part of the BIRS workshop “Women in Numbers 5”, held virtually in 2020. We thank the workshop organizers, Alina Bucur, Wei Ho, and Renate Scheidler for their leadership and encouragement that extended for the whole duration of this project. This work has been partially supported by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant 355412-2013 to ML), the Fonds de recherche du Québec - Nature et technologies (Projets de recherche en équipe 256442 and 300951 to ML), the Simons Foundation (grant number 359721 to MM), and the National Science Foundation (grant DMS-1844206 supporting AC and grant DMS-1902772 to LM). This material is based upon work supported by and while the third author served at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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