NORMS EXTREMAL WITH RESPECT TO THE MAHLER MEASURE

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1 NORMS EXTREMAL WITH RESPECT TO THE MAHLER MEASURE PAUL FILI AND ZACHARY MINER Abstract. In this paper, we introduce and study seeral norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer s conjecture and in at least one case that the conerse is true as well. We ealuate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We proe that the infimum in the construction is achieed in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures. 1. Introduction 1.1. Background. Let K be a number field with set of places M K. For each M K lying oer a rational prime p, let be the absolute alue on K extending the usual p-adic absolute alue on Q if is finite or the usual archimedean absolute alue if is infinite. Then for α K, the absolute logarithmic Weil height h is gien by h(α) = [K : Q ] log + α [K : Q] M K where log + t = max{log t, 0}. As the right hand side aboe does not depend on the choice of field K containing α, h is a well-defined function mapping Q [0, ) which anishes precisely on the roots of unity Tor(Q ). Related to the Weil height is the logarithmic Mahler measure, gien by m(α) = (deg α) h(α), where deg α = [Q(α) : Q]. Perhaps the most important open question regarding the Mahler measure is Lehmer s conjecture that there exists an absolute constant c such that (1.1) m(α) c > 0 for all α Q \ Tor(Q ). The question of the existence of algebraic numbers with small Mahler measure was first posed in 1933 by D.H. Lehmer [8]. The current best known lower bound, due to Dobrowolski [4], is of the form ( ) log log deg α 3 m(α) for all α Q \ Tor(Q ) log deg α where the implied constant is absolute. Date: May 2, Mathematics Subject Classification. 11R04. Key words and phrases. Weil height, Mahler measure, Lehmer s problem. 1

2 2 FILI AND MINER The Weil height h naturally satisfies the conditions of being a metric on the space G = Q / Tor(Q ) of algebraic numbers modulo torsion, and in fact, iewing G as a ector space oer Q written multiplicatiely (see the paper of Allcock and Vaaler [1]), it is easy to see that h is a ector space norm. The study of the Mahler measure on the ector space of algebraic numbers modulo torsion presents seeral difficulties absent for the Weil height, first of which is that while m also anishes precisely on Tor(Q ), unlike h, it is not well-defined on the quotient space modulo torsion. To get around that difficulty, Dubickas and Smyth [6] first introduced the metric Mahler measure, which gae a well-defined metric on G satisfying the additional property of being the largest metric which descends from a function bounded aboe by the Mahler measure on Q. Later, the first author and Samuels [10, 12] defined the ultrametric Mahler measure which satisfies the strong triangle inequality and gies a projectie height on G. It is easy to see that the metric and ultrametric Mahler measures each induce the discrete topology on G if and only if Lehmer s conjecture is true. In this paper we will introduce ector space norms on G which satisfy an analogous extremal property with respect to the Mahler measure as the metric Mahler measure does. Before presenting our constructions, let us fix our notation. We denote the L p Weil heights for 1 p < by ( h p (α) = [K : Q ] [K : Q] M K log α p ) 1/p for α K, noting that the classical Weil height satisfies 2h = h 1 (see [1]) and is well-defined, independent of choice of K. For p =, we let h (α) = sup M K log α for α K, noting that this height seres as a generalization of the (logarithmic) house of an algebraic integer (see Section 1.3 below for more details). Analogously, we hae the L p Mahler measure defined on Q to be m p (α) = (deg α) h p (α), where m 1 = 2m is twice the usual Mahler measure. For an algebraic number α Q, we let α G denote its equialence class modulo torsion. Let d : G N be gien by d(α) = min deg ζα, ζ Tor(Q ) where ζα ranges oer all representaties of the equialence class α. The minimal logarithmic Mahler measure is defined to be the function m : G [0, ) gien by 1 m(α) = d(α)h(α). (Recall that h p is constant on cosets modulo torsion, so that h p (α) = h p (α) for all α Q.) More generally, we define the minimal logarithmic L p Mahler measure m p (α) = d(α)h p (α). 1 The usual Mahler measure is not defined on G, thus our use of m for the minimal Mahler measure should result in no confusion.

3 EXTREMAL NORMS 3 This function is called minimal because it yields, for any element α G, the minimal logarithmic L p Mahler measure amongst all of the representaties in Q of our α G, that is, m p (α) = min ζ Tor(Q ) m p (αζ). Let us now recall the construction of the metric Mahler measure m : G [0, ) of Dubickas and Smyth [6]. This construction may be applied to any height function as in [7] and will in general produce metrics defined on G modulo its zero set. The (logarithmic) metric Mahler measure is defined by m(α) = inf α=α 1 α n n m(α i ), where the infimum is taken oer all possible ways of writing any representatie of α as a product of other algebraic numbers. This construction is extremal in the sense that any other function g : G [0, ) satisfying (1) g(α) m(α) for all α G, and (2) g(αβ 1 ) g(α) + g(β) for all α, β G, (Triangle Inequality) is then smaller than m, that is, g(α) m(α) for all α G. Equialently, lifting to Q in the natural way, it is easy to see that m satisfies the same extremal property with respect to the logarithmic Mahler measure. This extremal property is characteristic of the metric construction for height functions [6, 7, 10] Main results. The space G has a ector space structure oer Q (written multiplicatiely), so we might ask if there exists a ector space norm satisfying the same extremal property with respect to the Mahler measure. We define the extremal norm m p associated to m p to be: m p (α) = inf α=α r 1 1 α rn n n r i m p (α i ), where the infimum is taken oer all ways of writing α as a linear combination of ectors α i G with r i Q. (Obsere that m p (α r ) r m p (α) for general α G and r Q, so that in general the metric construction m p and the norm construction m p will not agree.) We proe that m p is a well-defined ector space norm on G which is extremal amongst all seminorms with respect to the Mahler measure, in the sense that if g : G [0, ) is a function satisfying (1) g(α) m p (α) for all α G, (2) g(αβ 1 ) g(α) + g(β), and (3) g(α r ) = r g(α) for all α G, r Q, then g m p, that is, g(α) m p (α) for all α G. Our main result is a finiteness theorem for the extremal norm m 1 analogous to the main result of [12] for the infimum of the metric Mahler measure. Let K be a number field and let V K = {α r : r Q and α K / Tor(K )}

4 4 FILI AND MINER be the ector subspace inside G spanned by elements of K / Tor(K ). Notice that α V K if and only if for any coset representatie α Q we hae α n K for some n N. Let M K be the set of places of K, and let S M K be a finite set of places of K, including all archimedean places. Then for any field extension L/K, define V L,S = {α V L : α w = 1 for w M K \ S}. Obsere that by Dirichlet s S-unit theorem, V L,S is a finite dimensional ector space. Our main result is the following: Theorem 1. Let α V K, where K is Galois. Then there exists a finite set of rational primes S, containing the archimedean place, such that m 1 (α) = F K[F : Q] h 1 (α F ) where α F V F,S, α = F K α F. Furthermore, for each pair of fields E F K, h 1 (α F ) = inf h 1 (α F /β). β V E,S We may interpret the last sentence of Theorem 1 as saying the norm of each α F is equal to the quotient norm of α F with respect to any subfield. For p > 1 we are able to show that m p attains its infimum on roots of rationals by computing m p (α) = h p (α) directly for such numbers in Proposition 3.7. (See Samuels and Jankauskas [13] for analogous results on the p-metric construction. 2 ) In order to proe our main result on the infimum of m 1, we proe in Section 4.1 seeral results related to heights of algebraic numbers modulo multiplicatie group actions ery much related to the results of A.C. de la Maza and E. Friedman [11], which we interpret as results about quotient norms. In particular, for L/K and α V L,S, we use essentially the same proof used in [11] to show in Theorem 4, that the infimum inf β VK,S h(αβ 1 ) is attained in the closure V K,S. In Theorem 5, we show that under certain extra conditions we may find the infimum inf β VK,S h(αβ 1 ) within V K,S, extending a result of [11]. In our final result on quotient norms, Theorem 6, we show that for α V L,S we can find an element η V K,S of minimal height which satisfies both (1) h 1 (αη 1 ) = inf β V K,S h 1 (αβ 1 ), and (2) h 1 (η) + [L : K]h 1 (αη 1 ) = inf β V K,S ( h1 (β) + [L : K]h 1 (αβ 1 ) ). We then construct an S-unit projection which allows us to reduce to finite dimensions, which we beliee is new and of interest in itself as it is a nonincreasing map with respect to the height. 2 The p-metric construction studied in [13] is ery different from our metric construction f f. The p-metric Mahler measure M p is the infimum oer all representations α = α 1 α n of the l p norm of the ector (m(α 1),..., m(α n)). One notable difference between the two constructions is that m p satisfies the triangle inequality m p(αβ) m p(α)+ m p(β), while M p satisfies a p-metric triangle inequality M p(αβ) p M p(α) p + M p(β) p. Howeer, when p = 1, they are essentially the same: m 1(α) = M 1(α).

5 EXTREMAL NORMS Applications to Lehmer s problem. Gien that the norms m p are extremal with respect to the Mahler measure, it is natural to ask what applications these norms hae to the Lehmer problem. Define A G to be the set of 1 α G which hae a representatie α satisfying the following properties: (1) α is an algebraic unit. (2) [Q(α n ) : Q] = [Q(α) : Q] for all n N. (3) For any proper subfield F of K = Q(α), Norm K F (α) Tor(F ). The conditions of the set A are exactly, in the terminology of [9], that α be a unit, representable 3, and projection irreducible, respectiely. Then in [9, Theorem 4] it is proen that for any 1 p there exists a constant c p such that (1.2) m p (α) = (deg α) h p (α) c p > 0 for all α Q \ Tor(Q ) if and only if (1.3) m p (α) = d(α) h p (α) c p > 0 for all α A. We note that equation (1.2) is equialent to the Lehmer conjecture for p = 1 and the Schinzel-Zassenhaus conjecture for p = [9, Proposition 4.1]. Therefore we formulate the following conjecture: Conjecture 1. For each 1 p, there exists a constant c p such that (1.4) m p (α) c p > 0 for all α A. Theorem 2. If Conjecture 1 is true, then (1.2) holds. In particular, for p = 1 (1.4) implies that Lehmer s conjecture is true, and for p = equation (1.4) implies that the Schinzel-Zassenhaus conjecture is true. For p 2, we are unable to proe the conerse to Theorem 2. We neertheless expect the result is true and make the following conjecture: Conjecture 2. If (1.2) holds, then Conjecture 1 is true. Howeer, when p = 2 we are able to proe that: Theorem 3. There exists a constant c 2 such that if and only if m 2 (α) = (deg α) h 2 (α) c 2 > 0 for all α Q \ Tor(Q ) m 2 (α) c 2 > 0 for all α A. Proof. In [9], we construct a norm m,2, and proe in [9, Theorem 4] that bounding m,2 away from zero on A is equialent to bounding m 2 away from zero on Q \Tor(Q ). Further, in [9, Theorem 6 et seq.] we proe that α m,2 m 2 (α) for all α G. It follows by the extremal property for m 2 that for all α G, and the claim now follows. α m,2 m 2 (α) m 2 (α) 3 Such numbers are called Lehmer irreducible in earlier drafts.

6 6 FILI AND MINER The format of this paper is as follows. In Section 2 we proe basic results about degree functions on G and projections onto subspaces. In Section 3 we construct the extremal norms m p arried at by the infimum process, proe a ery useful alternatie formulation of m p in Proposition 3.6, and examine explicit classes of algebraic numbers for which we can compute the alue of the norms (for example, on surds and in the p = 1 case on Salem and Pisot numbers). Lastly in Section 4 we study m 1 in particular and proe our main result that for any gien class of an algebraic number the infimum in the construction of m 1 is attained in a finite dimensional ector space. The authors would like to acknowledge Jeffrey Vaaler for many helpful conersations in general, and specifically for his contributions to Lemma 4.3, as well as Clayton Petsche and Felipe Voloch for helpful remarks regarding this same lemma. We also thank the referee of this paper for many helpful suggestions. 2. Preliminary Lemmas 2.1. Subspaces associated to number fields. We will now proe some lemmas regarding the relationship between certain subspaces determined by number fields. Let G = Gal(Q/Q) and let us define K = {K/Q : [K : Q] < } and K G = {K K : σk = K σ G}. Let us briefly recall the combinatorial properties of the sets K and K G partially ordered by inclusion. Recall that K and K G are lattices, that is, partially ordered sets for which any two elements hae a unique greatest lower bound, called the meet, and a least upper bound, called the join. Specifically, for any two fields K, L, the meet K L is gien by K L and the join K L is gien by KL. If K, L are Galois then both the meet (the intersection) and the join (the compositum) are Galois as well, thus K G is also a lattice. Both lattices hae a minimal element, namely Q, and are locally finite, that is, between any two fixed elements we hae a finite number of intermediate elements. For each K K, let V K = {α r : r Q and α K / Tor(K )}. Then V K is the subspace of G spanned by elements of K / Tor(K ). We call a subspace of the form V K for K K a distinguished subspace. Suppose we fix an algebraic number α G. Then the set {K K : α V K } forms a sublattice of K, and by the finiteness properties of K this set must contain a unique minimal element. Definition 2.1. For any α G, the minimal field is defined to be the minimal element of the set {K K : α V K }. We denote the minimal field of α by K α. Note that the action of G = Gal(Q/Q) on G is well-defined (see [1]). Lemma 2.2. For any α G, we hae Stab G (α) = Gal(Q/K α ) G. Notation 2.3. By Stab G (α) we mean the σ G such that σα = α. As this tacit identification is conenient we shall use it throughout with no further comment.

7 EXTREMAL NORMS 7 Proof. Clearly Gal(Q/K α ) Stab G (α), as α l K α for some l N by definition of V Kα. To see the reerse containment, obsere that K α = Q(α l ) for some l N. Now, for σ Stab G (α), we hae σα = ζα for ζ Tor(Q ). Then if σ(α l ) α l, there would exist an m N such that σ(α lm ) = α lm. Thus, σ is contained in a proper supergroup of Gal(Q/K α ), so there would be a proper subfield of K α containing α lm, contradicting the definition of K α Representability. Obsere that the action of the absolute Galois group G = Gal(Q/Q) is well-defined on the ector space of algebraic numbers modulo torsion G, and in fact it is easy to see that each Galois automorphism gies rise to a distinct isometry of G in the h p norm (see [9, 2.1] for more details). Let us denote the image of the class α under σ G by σα. In order to associate a notion of degree to a subspace in a meaningful fashion so that we can define our norms associated to the Mahler Measure we define the function δ : G N by (2.1) δ(α) = #{σα : σ G} = [G : Stab G (α)] = [K α : Q] to be the size of the orbit of α under the Galois action, with the last equality aboe following from Lemma 2.2. Obsere that since taking roots or powers does not affect the Q-ector space span, and in particular the minimal field K α, the function δ is inariant under nonzero scaling in G, that is, δ(α r ) = δ(α) for all 0 r Q. In order to better understand the relationship between our elements in G and their representaties in Q, we need to understand when an α V K has a representatie α K (or is merely a root of an element α n K for some n > 1). Naturally, the choice of coset representatie modulo torsion affects this, and we would like to aoid such considerations. Therefore we define the function d : G N by (2.2) d(α) = min{deg ζα : α Q, ζ Tor(Q )}. In other words, for a gien α G, which is an equialence class of an algebraic number modulo torsion, d(α) gies us the minimum degree amongst all of the coset representaties in Q modulo the torsion subgroup. A number α G can then be represented by an algebraic number in K α if and only if d(α) = δ(α). We therefore make the following definition: Definition 2.4. We define the set of representable elements of G to be the set (2.3) R = {α G : δ(α) = d(α)}. The set R consists precisely of the α G that can be represented by some α Q of degree equal to the degree of the minimal field K α of α. We recall the terminology from [5] that a number α Q is torsion-free if α/σα Tor(Q ) for all distinct Galois conjugates σα. Thus, torsion-free numbers gie rise to distinct elements σα G for each distinct Galois conjugate σα of α in Q. Lemma 2.5. We hae the following: (1) For each α G, there is a unique minimal exponent l(α) N such that α l(α) R. (2) For any α Q, we hae δ(α) deg α.

8 8 FILI AND MINER (3) α R if and only if it has a representatie in Q which is torsion-free. Proof. For α G, choose a representatie α Q and let l = lcm{ord(α/σα) : σ G and α/σα Tor(Q )} where ord(ζ) denotes the order of an element ζ Tor(Q ). Then obsere that α l is torsionfree. Now if a number β Q is torsion-free, then each distinct conjugate σβ determines a distinct element in G, so we hae deg β = [G : Stab G (β)] = [K β : Q] = δ(β). Thus deg α l = δ(α l ). This proes existence in the first claim, and the existence of a minimum alue follows since N is discrete. To proe the second claim, obsere that Q(α l ) Q(α), so with the choice of l as aboe, we hae δ(α) = [Q(α l ) : Q] [Q(α) : Q] = deg α for all α Q. The third now follows immediately. It is proen in [9] that in fact, the minimal alue l(α) satisfies d(α) = l(α)δ(α) Projections to distinguished subspaces. Suppose β V K for a number field K. In our proof of Theorem 1, it will be necessary to replace an arbitrary representation β = β 1 β n with another representation β = β 1 β n where each β i belongs to V K and satisfies δh 1 (β i ) δh 1(β i ). To this end, we define an operator that projects an element α G onto the subspace V K. Let H = Gal(Q/K) G, and let σ 1,..., σ k be right coset representaties from Stab H (α) H, with k = [H : Stab H (α)]. Define the map P K on elements of G ia ( k 1/k (2.4) P K (α) = σ i (α)). Lemma 2.6. Let α G. Then: (1) P K (α r ) = P K (α) r. (2) P K (α) V K. (3) P K (α) = α for all α V K. Proof. (1) P K (α r ) = P K (α) r follows from its definition in terms of the Galois action on G. (2) By scaling if necessary, we may assume α R. Choose a torsion-free representatie α Q. Then, the result will follow from N K(α) K = k σ i(α), since N K(α) K (α) K. To see this, note that for α torsion-free, α/σ(α) is neer a nontriial torsion element, so its orbit in Q and its G orbit coincide. (3) Again, we may assume α R, so that a torsion-free representatie α K. Then N K(α) K = α k, and P K (α) = α follows. Proposition 2.7. Let K be a number field. Then P K is a projection onto V K of norm 1 with respect to the L p norms for 1 p.

9 EXTREMAL NORMS 9 Proof. By Lemma 2.6, it follows that PK 2 = P K is a projection onto V K. Then for 1 p, h p (P K α) = h p (σ 1 (α) σ k (α)) 1/k 1 k h p (σ i α) = 1 k h p (α) = h p (α), k k since the Weil p-height is inariant under the Galois action. This proes P K has operator norm P K 1, and since V Q is fixed for eery P K, we get P K = 1. As a corollary, if we let G p denote the completion of G under the Weil p-norm h p and extend P K by continuity, we obtain: Corollary 2.8. The subspace V K G p is complemented in G p for all 1 p. As G 2 = L 2 (Y, λ) is the L 2 space for a certain measure space (Y, λ) constructed explicitly in [1], and thus a Hilbert space, more is in fact true: Proposition 2.9. For each K K, P K is the orthogonal projection onto the subspace V K G 2. Proof. Obsere that P K is idempotent and has operator norm P K = 1 with respect to the L 2 norm, and any such projection in a Hilbert space is orthogonal (see [15, Theorem III.1.3]). We now explore the relationship between the Galois group and the projection operators P K for K K. Lemma For any field K Q and σ G, Equialently, P K σ = σp σ 1 K. σp K = P σk σ. Proof. We proe the first form, the second obiously being equialent. Let H = Gal(Q/K), and note that if τ H, then στσ 1 Gal(Q/σK). Then by the definition of P K : σp K α = σ (σ 1 α σ k α) 1/k = (σσ 1 α σσ k α) 1/k = ( σσ 1 (σ 1 σ)α σσ k (σ 1 σ)α ) 1/k = ( (σσ 1 σ 1 )σα (σσ k σ 1 )σα ) 1/k = P σk (σα). We will be particularly interested in the case where the projections P K, P L commute with each other (and thus P K P L is a projection to the intersection of their ranges). To that end, let us determine the intersection of two distinguished subspaces: Lemma Let K, L Q be extensions of Q of arbitrary degree. Then the intersection V K V L = V K L. Proof. Let α m K and α n L for some m, n N. Then α mn K L, so α V K L. The reerse inclusion is obious.

10 10 FILI AND MINER Lemma Suppose K K and L K G. Then P K and P L commute, that is, P K P L = P K L = P L P K. In particular, the family of operators {P K : K K G } is commuting. Proof. It suffices to proe P K (V L ) V L, as this will imply that P K (V L ) V K V L = V K L by the aboe lemma, and thus that P K P L is itself a projection onto V K L, implying P K P L = P K L, and since any orthogonal projection is equal to its adjoint, we find that P K L = P L P K as well. To proe that P K (V L ) V L, obsere that for α V L, P K (α) = (σ 1 α σ k α) 1/k where the σ i are right coset representaties of Stab H (α) in H = Gal(Q/K). Howeer, σ(v L ) = V L for σ G since L is Galois, and thus, P K (α) V L as well. But P K (α) V K by construction and the proof is complete. From these facts, we derie the following useful lemma: Lemma If K K G, then δ(p K α) δ(α) for all α G. Proof. Let F = K α. Since K K G, we hae by Lemma 2.12 that P K α = P K (P F α) = P K F α. Thus, P K α V K F, and so δ(p K α) [K F : Q] [F : Q] = δ(α). 3. Extremal metric heights and norms 3.1. Construction. The aim of this section is to construct norms extremal with respect to the minimal Mahler measure. Let us begin by recalling the metric construction, as applied in [6]: Definition 3.1. For f : G [0, ), the metric height associated to f is defined to be the function f : G [0, ) gien by n f(α) = f(α i ), inf α=α 1 α n where the infimum ranges oer all possible factorizations α = α 1 α n in G. Proposition 3.2. Suppose f(α 1 ) = f(α) for all α G. Then the function f satisfies: (1) f(α) f(α) for all α G. (2) f(α 1 ) = f(α) for all α G. (3) f(αβ 1 ) f(α) + f(β) for all α, β G. (4) The zero set Z( f) = {α G : f(α) = 0} is a subgroup of G, and f is a metric on G/Z( f). It is the largest function that does so, that is, for any other function g which satisfies the aboe conditions, we hae g(α) f(α) for all α G. In particular, if f already satisfies the triangle inequality, then f = f. This last property of being the largest metric less than or equal to f we call the extremal property. The construction of metric heights only uses the group structure of G, and ignores the ector space structure. If we wish to respect scaling in G as well, we arrie at the notion of a norm height:

11 EXTREMAL NORMS 11 Definition 3.3. Let f : G [0, ) be a gien function. We define the norm height associated to f to be the function f : G [0, ) gien by n f(α) = r i f(α i ), inf α=α r 1 1 α rn n where the infimum ranges oer all possible factorizations α = α r 1 1 αrn n in G with α i G and r i Q. Proposition 3.4. The norm height f satisfies the properties: (1) f(α) f(α) for all α G. (2) f(α r ) = r f(α) for all r Q and α G. (3) f(αβ 1 ) f(α) + f(β) for all α, β G. (4) The zero set Z( f) = {α G : f(α) = 0} is a ector subspace of G. Thus, f is a seminorm on G, and a norm on G/Z( f). It is the largest function on G that satisfies the aboe properties, that is, for any other function g which satisfies the aboe conditions, we hae g(α) f(α) for all α G. In particular, if f is already a seminorm on G, then f = f. The proof of Proposition 3.4 follows easily from the definitions. As aboe, we refer to the last part of the proposition as the extremal property of the norm height construction. Obsere that if f satisfies the scaling property f(α r ) = r f(α), then f = f and the construction is the same. Proposition 3.5. m p is a ector space norm on G. Proof. It only remains to show that the m p anishes precisely on the zero subspace of the ector space G, which is {1}. Obsere that h p m p, and therefore, by the extremal property, h p (α) m p (α) for all α G. In particular, we see that m p (α) = 0 if and only if h p (α) = 0, which occurs precisely when α = 1. Note that as the degree function δ is inariant under nonzero scaling, we hae that δh p (α r ) = r δh p (α) for rational r. Therefore, the metric construction δh p δh p results in a norm on G. The following result shows that this norm is exactly m p, which is computationally ery useful and will be tacitly used seeral times in our proofs below: Proposition 3.6. The norm m p extremal with respect to the Mahler measure m p is precisely δh p, that is, m p (α) = δh p (α) for all α G. Proof. By Lemma 2.5, there is a unique minimal l N such that d(α l ) = δ(α). Then it is easy to see that for α G, the expression s/r m p (α r/s ) = s/r d(α r/s )h p (α r/s )

12 12 FILI AND MINER is minimized for r/s = l, and for that alue, We may then conclude that m p (α) = inf α=α r 1 1 α rn n which is the desired result. 1/l d(α l )h p (α l ) = δ(α)h p (α). n r i m p (α i ) = inf α=α 1 α n n δ(α i )h p (α i ) = δh p (α) 3.2. Explicit alues. We will now compute the alues of the norms m p on certain classes of algebraic numbers. Recall that a surd is an algebraic number α Q such that α n Q for some n N. Call α G a surd if one (and therefore all) coset representaties of α are surds. Proposition 3.7. If α G is a surd, then m p (α) = h p (α). Proof. Obsere that δ(α) = 1 for any surd. Since h p m p is a norm, we hae by the extremal property of m p that h p (α) m p (α) m p (α) for all α G. But then m p (α) = δh p (α) δ(α)h p (α) = h p (α), and therefore we hae equality. For a comparison of computations of m p on surds with the p-metric Mahler measures, see [6, 13]. We now consider a class of numbers analogous to the CPS numbers of [6]. Lemma 3.8. Suppose that α G satisfies m p (α n ) = n m p (α) for all n N. m p (α) = m p (α). Proof. Using Theorem 1 we may choose a factorization α = α 1 α n so that n δh p (α) = δ(α i )h p (α i ). Then By Lemma 2.5 we hae an exponent l i = l(α i ) N such that α l i i R for 1 i n. Let k = lcm{l 1,..., l n }. Then obsere that k δh n n p (α) = δ(α k i )h p (α k i ) = d(α k i )h p (α k i ) m p (α k ) = k m p (α). By the extremal property, δh p (α) m p (α), so we must hae equality, as claimed. Definition 3.9. Call τ G a Pisot/Salem number if it has a representatie τ Q that can be written as τ = τ 1 τ k where each τ i > 1 is a Pisot number (that is, an algebraic integer with all of its conjugates strictly inside the unit circle) or a Salem number (an algebraic integer with all of its conjugates on or inside the unit circle, and at least one on the unit circle).

13 EXTREMAL NORMS 13 Proposition Eery Pisot/Salem number τ is representable, that is, τ R. Proof. It is easy to see that for a Pisot/Salem number τ G and its gien representatie τ > 1 that all other Galois conjugates τ τ hae τ < τ. Therefore τ is torsion-free, since otherwise there would be a conjugate τ = ζτ for some 1 ζ Tor(Q ), haing the same modulus as τ, a contradiction. It follows by Lemma 2.5 that τ R. For a Pisot/Salem number τ G, it is shown in [6, Theorem 1(c)] that m 1 (τ n ) = 2 log τ n for all n N. Thus, by Lemma 3.8 aboe, we hae the following result: Proposition Let τ G be a Pisot/Salem number with gien representatie τ Q. Then m 1 (τ ) = m 1 (τ ) = 2 log τ. Since there exist Pisot and Salem numbers of arbitrarily large degree, and for a Pisot or Salem number τ > 1 we hae h 1 (τ) = (2/ deg τ) log τ, we easily see that the norms h 1 and m 1 are inequialent. 4. The infimum in the m 1 norm 4.1. S-unit subspaces and quotient norms. Let K K be a number field with places M K. Let S M K be a finite set of places of K, including all archimedean places. Then for any field L, let (4.1) V L,S = {α V L : α w = 1 for w M K \ S}. Then V L,S is the Q-ector space span inside V L of the S -units of L, where S is the set of places w of L such that w S. Since we always require that S include the archimedean places, V L,S will always include the ector space span of the units of L. Dirichlet s S-unit theorem and, in particular, the non-anishing of the S-regulator, imply the following result: Proposition 4.1. If S M K as aboe, then the Q-ector space V K,S and its completion V K,S hae finite dimension #S 1. For L K, the space V L,S has dimension #S 1 where S is the set of places w of L such that w S. In what follows below, we will primarily require S to be a set of rational primes, including the infinite prime. Notice that under these definitions, if K L, then V K,S V L,S. One of the goals of this section will be to determine the properties of the quotient norm of V L,S /V K,S, in a manner inspired by the initial work of A.M. Bergé and J. Martinet [2, 3] and in particular the more recent work of A.C. de la Maza and E. Friedman [11]. The main result of this section is the following theorem, which is essentially an analogue for the norm m 1 of the main result of [12] for the infimum of the metric Mahler measure: Theorem 1. Let α V K, where K is Galois. Then there exists a finite set of rational primes S, containing the archimedean place, such that m 1 (α) = F K[F : Q] h 1 (α F )

14 14 FILI AND MINER where α F V F,S, α = F K α F. Furthermore, for each pair of fields E F K, h 1 (α F ) = inf h 1 (α F /β). β V E,S In other words, the norm of each α F is equal to the quotient norm of α F with respect to any subfield. In contrast to the main result of [12], we are unable to proe that this infimum is in fact attained in the ector space of classical algebraic numbers G, rather than the completion. Howeer, our result is strengthened by the fact that the S-unit spaces in which the infimum is attained are finite dimensional real ector spaces. Therefore, if we must pass to the completion, we know that the terms in the infimum are limits of the form lim n α rn where α G and r n is a sequence of rational numbers tending to a real limit r as n. Before we can proe Theorem 1, we must first proe seeral quotient norm results ery much related to the results of [11], and then we will construct an S-unit projection which will allow us to reduce to the specified situation. Let S M K be a finite set of places to be specified later, and consider two number fields K L. Again let V K,S denote the ector subspace of V K spanned by the S-units of K and let V L,S denote the corresponding subspace of V L. Now for each S let d = [K : Q ] be the local degree. Rather than following the usual conention and considering the places of L which lie aboe the places S of K, we will consider the [L : K] absolute alues which restrict to each place (that is, we will not consider equialence on L nor weight such by local degrees). Thus we get #S [L : K] absolute alues on L. Let us fix the α V L,S \ V K,S for which we want to compute the quotient norm modulo V K,S. For a gien S, order the [L : K] absolute alues on L which extend so that α,1 α,2 α,[l:k]. Now we associate to α a ector a R S [L:K] ia ϕ : V L,S R S [L:K] α a = (d log α,i ) S, 1 i [L:K] Note that by the product formula and our normalization aboe, the sum of the components of a is zero. By the ordering aboe, we also hae a,i a,i+1 for all S and 1 i < [L : K]. The goal of this section is to proe the following results which will be needed below: Theorem 4 (de la Maza, Friedman 2008). For α V L,S and the ector a = ϕ(α) R S [L:K] with indices ordered as aboe, [L:K] inf h 1 (αβ 1 1 ) = β V K,S [L : Q] a,i. Equialently, α VL,S /V K,S = [L:K] S a,i, S

15 EXTREMAL NORMS 15 where we are loosely using the notation V L,S /V K,S for the quotient space of the ector spaces ϕ(v K,S ) ϕ(v L,S ) R S [L:K] endowed with the L 1 norm. Remark 4.2. When the authors of [11] claimed the infimum appearing in Theorem 4 occurs in V K,S, they use in eqn. (2.19) that for a dense open subset of V L,S = V L,S R (the real span of the image of the logarithmic embedding) the components a,i for a gien may be assumed distinct. Howeer, this is not always true as the distinct places of L lying aboe might not be [L : K] in number (for example, if is a finite place which ramifies or has inertia). Thus we might always hae a certain number of equalities amongst the {a,i : 1 i [L : K]} for a gien. Howeer, if we make the ery minor modification of working inside of R S [L:K] rather than V L,S, then we hae no such number theoretic restrictions and may assume [L : K] distinct places for a gien. After adjusting for this, the remainder of the proof in [11] carries through to show the slightly weaker result that infimum actually occurs in the completion V K,S. We make a slight extension of another result of [11]: Theorem 5. Let α V K hae nonzero support at only the infinite places and one finite place of K. Let W denote the subspace of V K spanned by the units of K. Then there exists β W such that h 1 (αβ 1 ) = inf γ W h 1(αγ 1 ) = ( 1 [K : Q] d log α + d w log α w ). We conclude with a new theorem that will be used to describe the infimum of m 1 : Theorem 6. For a gien α V L,S, there exists η V K,S such that the following conditions all hold: (1) h 1 (αη 1 ) = inf β V K,S h 1 (αβ 1 ), and (2) h 1 (η) + [L : K]h 1 (αη 1 ) = inf β V K,S ( h1 (β) + [L : K]h 1 (αβ 1 ) ). We now proide the proofs for the aboe results. Proof of Theorem 4. Notice that a,1 a,2 S S S Let k be an index such that a,k+1 a,[l:k]. a,k 0 S S where we let k = 0 or k = [L : K] if S a,1 0 or S a,[l:k] 0, respectiely. We will assume for the moment that 1 k < [L : K] and defer the proof for the extreme cases for the moment. Let X denote the set of x ϕ(v K,S ) R S [L:K] which satisfy the conditions: a,k x a,k+1 for all S and x = 0, S w

16 16 FILI AND MINER where we use x to denote the common alue of x,i, which must be equal for all i since x arises from V K,S. It is easy to see that X is nonempty as it contains, for example, x = a,k + where s i = S a,i. Notice that a x 1 = [L:K] s k s k+1 s k (a,k+1 a,k ) a,i x = ( [L:K] k ) (a,i x ) (a,i x ) S S i=k+1 [L:K] = a,i ([L : K] 2k) x = S S [L:K] a,i. Since x = ϕ(η) for some η V K,S and [L : Q] h 1 (αη 1 ) = a x 1, the result will be proen if we can show that the aboe alue is minimal for the function F a : R S R gien by y a y 1 where we again iew y as a ector in R S [L:K] ia y,i = y. The function F a is clearly conex. As remarked earlier, we must work inside of R S [L:K] in order to be assured of our ectors haing distinct components. Obsere that for any ɛ > 0 we may find an a R S [L:K] such that its components a,i for a gien are distinct and further that a a 1 < ɛ. In order that the aboe computation remains unchanged for a x 1, we will construct a from a by adding sufficiently small ɛ i < 0 to each component a,i for 1 i k, and adding sufficiently small ɛ i > 0 for k + 1 i [L : K] in such a way that i ɛ i = 0. We will determine the minimum of F a and this will in turn tell us the minimum of F a. Let Y denote the subset of R S defined by a,k < y < a,k+1 for all S, and S y = 0. Obsere that by our choice of a, we hae that X Y. For our ector x from aboe, obsere that by the triangle inequality, we hae and hence x a 1 x a 1 a a 1 < ɛ, F a (x) F a (x) a a 1 < ɛ. Thus, by allowing ɛ to approach 0 we see that in showing F a (x) is the minimum alue on X, it suffices to show F a (x) is the minimum alue on Y. Notice that Y is an open set of R S and that our ector x lies in Y so it is nonempty. Notice further that by construction of a the aboe computation at x works out still to gie the same alue for F a at any y Y. Therefore, as F a is a conex function of R S which is constant on the open set Y, we conclude that F a is minimal on Y, as any conex function which is constant on an open set attains its minimum on that set, which completes the proof for all 1 k < [L : K]. For the remaining cases where k = 0 or k = [L : K] we make some triial modifications to our set X. For the case k = 0, we let X ϕ(v K,S ) R [L:K] S be gien by x < a,1 for all S S

17 and EXTREMAL NORMS 17 x = 0, S where we again use x to denote the common alue of x,i. Now we demonstrate that X is nonempty by constructing x = a,1 s 1 #S. where s i = S a,i. In the case k = [L : K] likewise we take S x = 0 and x > a,[l:k] for all S, to define our set X and obsere that we hae a point gien by (noting that s [L:K] 0 in this case). aboe. x = a,[l:k] s [L:K] #S The remainder of the proof continues exactly as Proof of Theorem 5. This is in essence an application of the aboe theorem with W substituted as the subspace; the primary difference is that we wish to show that in this instance, the infimum claimed is in fact attained in W, rather than W. Suppose without loss of generality that d log α < 0 so that a < 0 (for otherwise we may replace α by α 1 and the height is unaffected). Then s = w d w log α w = w a w > 0. Let X ϕ(w ) R S (where S = {w M K : w } {}) be the set of x satisfying and The set X is nonempty as it contains x = 0, x w < a w, for all w, x w = 0. w x w = a w s/n for all w, where n = #{w M K : w }. But then a x 1 = a + a w x w = a + w x w ) = a + a w, w w (a w and the claim will follow if we can show that this alue is minimal, since [K : Q]h 1 (γ) = ϕ(γ) 1 for γ V K. But X is nonempty and is open as a subspace of the hyperplane ϕ(w ), where the conex function F : ϕ(w ) R gien by y a y 1 is constant, therefore, it is the minimum of this function. Since we hae an open subset of ϕ(w ) clearly we hae a β W such that y = ϕ(β) X and the proof is complete.

18 18 FILI AND MINER Proof of Theorem 6. With notation as in Theorem 4, recall that we defined X to be the set of all ectors x ϕ(v K,S ) R S [L:K] satisfying: and a,k x a,k+1 for all S x = 0, S where we use x to denote the common alue of x,i, which must be equal for all i since x arises from V K,S. It was shown in Theorem 4 that for η ϕ 1 (X) V K,S, we hae the first condition that h 1 (αη 1 ) is minimized. Our goal will be to show that if we choose η ϕ 1 (X) of minimal height, then the remaining two conditions will be satisfied. Let us determine then what the minimal height of x = ϕ(η) R S can be. For a real number t we will denote t + = max{t, 0} and t = max{ t, 0}, so that t = t + t and t = t + + t. Assume for the moment that 1 k < [L : K] and let (4.2) ɛ = a,k x and ɛ = x + a +,k. Note that ɛ, ɛ 0 since a,k x. It follows that we may write (4.3) x = a,k + ɛ + ɛ, and (4.4) x = a,k ɛ + ɛ. To minimize x 1 we want to let ɛ be as large as possible, and it is easy to see that we must hae 0 ɛ min{a,k, a,k+1 a,k }. Obsere that min{a,k, a,k+1 a,k } = a,k a,k+1, for suppose the minimum is a,k. Then a,k+1 = 0, and min{a,k, a,k+1 a,k } = a,k = a,k a,k+1. Now, suppose min{a,k, a,k+1 a,k } = a,k+1 a,k. Then we must hae a,k a,k+1 0, and so min{a,k, a,k+1 a,k } = a,k+1 a,k = a,k a,k+1. Thus in general min{a,k, a,k+1 a,k } = a,k a,k+1. Define C = ( ) a,k a,k+1 S to be the largest possible alue for ɛ. Our proof will break into two cases. First, assume that C a,k, and note that this condition is equialent to (4.5) a +,k a,k+1. Recall that a,k 0 and obsere that this is equialent to (4.6) a +,k a,k.

19 EXTREMAL NORMS 19 Then by equations (4.5) and (4.6) we may subtract from a,k a real number b satisfying (4.7) b a,k+1 and b = a +,k, and since a,k a,k+1 implies that a,k+1 a,k, the alue b may further be chosen to satisfy a,k b 0 for each. Thus, when C a,k, we define ɛ to be a,k b and ɛ to be 0. It then follows that x = a,k + ɛ [a,k, a,k+1 ] for each and x = a+,k b = 0, giing us x 1 = a,k ɛ = 2a +,k, and x 1 is minimal since ɛ is maximized. (Our choices of ɛ and ɛ agree with our preious definitions (4.2), by obsering that x = a,k +ɛ = a +,k b. Thus, if a,k = a,k, then x = b 0, so that ɛ = a,k x = x a,k, and from (4.3) we get ɛ = 0. While if a,k = a +,k, we hae 0 b a,k = 0, so that x = a +,k 0, implying ɛ = 0, and ɛ = x a = a +,k a+,k = 0 = a,k x.) Now for the second case, assume that C < a,k. Again, in order to minimize x 1 we want to let ɛ be as large as possible; which by construction is equal to C. But we require (ɛ + ɛ ) = a,k ( 0) in order to hae x = 0, so this implies that we will need ɛ, precisely such that ɛ = a,k ɛ = a,k C. Then clearly x 1 = a,k ɛ + ɛ = a,k a,k 2C = 2a,k 2 (a,k a,k+1 ) = 2a,k+1. So, by (4.5) we may express the minimal height of x in both cases as { x 1 = max 2a +,k, } 2a,k+1. Using such a minimal η = ϕ 1 (x) V K,S we see that the first two claims are satisfied. It remains to show that the third claim is true, specifically, that h 1 (η) + [L : K]h 1 (αη 1 ) [L : K]h 1 (α). Translated into the appropriate L 1 -norms, this claim is equialent to: x L 1 (R S ) + a x L 1 (R [L:K] S ) a L 1 (R [L:K] S ).

20 20 FILI AND MINER Where in the term a x L 1 (R [L:K] S ) we iew x as a ector in R[L:K] S ia x,i = x for all i. Writing this expression out, we hae [L:K] [L:K] x + a,i a,i, equialently, rearranging these terms, { (4.8) 2 max a +,k, a,k+1 } 2 ( k [L:K] a +,i + i=k+1 which is clearly true and completes the proof for the cases 1 k < [L : K]. For the remaining cases, obsere that for k = 0 we hae x < a,1 and thus it is easy to see that our minimal height is x = 2a,1 and since the right hand side of (4.8) holds for k = 0, the inequality still holds. The k = [L : K] case is similar, as a,[l:k] < x implies our minimal height is x = 2a +,[L:K] S-unit projections and proof of Theorem 1. Let K be a finite Galois extension of Q with set of places M K. We normalize our absolute alues by letting be the absolute alue which extends p for the rational prime p such that p, and let = [K:Q]/[K:Q]. Denote by S a finite set of places to be fixed later which includes all of the archimedean places. Let O K be the ring of algebraic integers of K and let U S be the group of S-units of K. Since S is finite and contains the archimedean places, we know by Dirichlet s S-unit theorem that U S is a free abelian group of finite rank s = #S 1. Recall that the class group is the group of nonzero fractional ideals of K modulo principal ideals. It is wellknown that for number fields, the class group of a number field has a finite order, and we will denote the order of the class group of K by h. It follows immediately that if for some finite place M K the ideal a,i ), is not principal, then P = {α K : α < 1} O K (4.9) P h = (α) O K is a principal ideal of O K, since the class of P h is triial in the class group. The goal of this section is to construct a projection P S : V K V K,S, with V K,S defined by equation (4.1), which will be instrumental in the proof of the main theorem. Let S consist of the following places of K: (1) The archimedean places of K. (2) The support of α (all places where α has nontriial aluation). (3) The Galois conjugates of the aboe places under the natural action σ = σ 1 ( ).

21 EXTREMAL NORMS 21 It is clear that S is finite. We now proceed to associate a generator to each place outside of S: Lemma 4.3. Let K/Q be galois. For any M K \ S, we can find α V K such that (1) α < 1, (2) α w = 1 for all w M K \ S with w, and (3) α w 1 for all w S. (4) h 1 (α ) = inf β VK,S h 1 (α /β). Proof. If P = {α K : α < 1} O K is a principal ideal, then let β be a generator. Otherwise, let P h = (α) as in (4.9) and let β = α 1/h V K. Clearly, β has a nontriial finite aluation only at of β = p 1/e, where e is the ramification index of p. By Theorem 5 aboe, we can find η V K,S such that h 1 (βη) = log βη w w M K = log β w + log βη w w M K \S w S = log β + log βη w w S = log β + log β w, where the last equality follows from the product formula for η. That we hae equality in the third step aboe implies that either log βη w 0 for all w S or log βη w 0 for all w S. By our choice of β we hae log β < 0, and hence, by the product formula, all of the S aluations of βη must be nonnegatie. We therefore can choose α = βη and we are done. Let M K and suppose p for the rational prime p. Let G = Gal(Q/Q) be the absolute Galois group, and let H = Stab G () be the decomposition group associated to the finite place. Let α V K and take {σ 1,..., σ k } to be a set of right coset representaties for Stab H (α) in H (where k = [H : Stab H (α)]) and then define P H : V K V K to be w S P H α = (σ 1 (α) σ k (α)) 1/k. Then by Proposition 2.7, P H is a projection to V F V K for F K the fixed field of H, of operator norm 1 with respect to the Weil p-height h p for 1 p. We will now construct a system of α for each place M K \ S. Lemma 4.4. There exists a set {α V K : M K \ S} such that each α satisfies the conditions of Lemma 4.3 aboe with the following additional property: for any w S and σ G, if α w α σw then σ.

22 22 FILI AND MINER Proof. For each rational prime p which has a place in M K \ S lying aboe it, pick one particular place p lying aboe it. Choose α to be the number constructed by Lemma 4.3 aboe, and let α = P H α where H = Stab G () is the stabilizer of the place in the absolute Galois group as aboe. Notice that by the fact that P H has norm 1 and by minimality modulo V K,S of α in Lemma 4.3, h 1 (P H α ) = h 1 (α ). Since S is closed under the Galois action, and H fixes the place, α still satisfies the criteria of Lemma 4.3. For any other place w p lying aboe the same rational prime p, obsere that there exists σ G with σ = w. Define α w = σ 1 (α ), and repeat this construction for eery rational prime p whose extensions to K lie in M K \ S. This gies us the entire set of α whose existence we need to establish, and since the Galois action permutes the places lying oer p, the α thus constructed all meet the conditions of Lemma 4.3. It now remains to see that this set has the additional property claimed. This is guaranteed by the aeraging oer H done by P H in constructing the original α whose orbit we took in the aboe construction. Obsere that if σ G fixes the -adic aluation of α, then σ H. Let F K denote the fixed field of H and iew P H as the projection to V F. Then α V F is some power of an element of F / Tor(F ), so by linearity, we hae σα = α. Thus we see that for such σ G, σα w = α σ 1 w unless σ, in which case we hae the desired conclusion. Corollary 4.5. For p and α in the set as constructed in Lemma 4.4, δ(α ) is precisely the number of places of K which lie oer p. Proof. As seen in the proof, if σ(α ) α, then σ. While, if σ(α ) = α, then 1 > α = σ(α ) = α σ 1, which gies σ =. We are now ready to construct the projection P S : V K V K,S which is fundamental to the proof of Theorem 1. Proposition 4.6. There exists a linear projection P S : V K V K,S which satisfies the following properties: (1) h 1 (P S α) h 1 (α), so P S = 1 with respect to the Weil height norm, and (2) δ(p S α) δ(α), and thus P S = 1 with respect to the Mahler norm. Proof. For our gien S, let {α : M K \ S} be the set constructed by Lemma 4.4. For each M K \ S, define the map n : V K Q ia the requirement that βα n(β) = 1 for all β V K. It is easy to see that such a alue for n must exist and be unique, since the -adic aluations are discrete. 4 Further, obsere that n (βγ) = n (β) + n (γ) for all β, γ V K. 4 The reader will note that by our choice of α, the function n ( ) is essentially the linear extension of ord ( ) from K / Tor(K ) to V K.

23 EXTREMAL NORMS 23 Define the map P S : V K V K,S α α M K \S α n (α). That this is well-defined follows from the fact that n (α) = 0 for all but finitely many and from the fact that by our choice of α and n (α), P S α has support only in S and thus belongs to the Q-ector space span of the S-units V K,S. It follows easily from the definition that P S (β r γ s ) = P S (β) r P S (γ) s, hence P S is linear. We will now proe that P S satisfies the first desired property. Fix our α V K and let β = P S α V K,S. Let T denote the Galois orbit of supp(α) \ S inside M K. The claim is then that ( h 1 (β) h 1 β ) α n = h 1 (α), T where we will suppress the argument in the exponents n = n (α). Denote S = M K \ S. Then (4.10) h 1 (β) = w S log β w + w S log β w. Now, w S log β w = 0, since β V K,S. We apply the triangle inequality to the remaining term: (4.11) log β w log β w + n log α w + n log α w. w S w S T w S T Obsere that by our choice of α for T in the lemmas aboe, we hae α w 1 for all w S and α w 1 for all w S. Thus, n log α w n log α w = n ( log α w ), w S T w S T w S T where the last equality follows from the product formula. But α w = 1 for all w S \{} and α < 1, so in fact, n log α w n log α w. w S w S T On obsering that β w = 1 for all w S, we may write this same expression as: (4.12) n log α w log β w + n log α w. T w S T w S Combining equations (4.10), (4.11), and (4.12), we find that h 1 (β) log β w + ( n log α w = h 1 β ) α n, T T w M K which is the desired result. T

24 24 FILI AND MINER It now remains to proe the second claim, namely that ( δ(β) δ β ) α n = δ(α). T Suppose for some σ G that β σβ but σ(α) = α. Then for some w S, β w β σw, and so we must hae α n α n. w σw T It follows then by Lemma 4.4 that for some T we must hae σ and n n σ, else the w-adic aluation would not differ. But then it is easy to see that α nu u = α n = p n/e p nσ/e = α σ nσ σ = α nu u, σ u T where e is the ramification index of p. Thus any contribution the u T αnu u term might hae towards decreasing the orbit of α = β T αn by equating two w-adic aluations of α for w S will neertheless result in distinct -adic aluations for some T and thus the new orbit will be at least as large, proing the claim. We are now ready to proe Theorem 1. Proof of Theorem 1. Let α V K, where K is the Galois closure of the minimal field of α. Let P K : G V K be the projection to V K, S the set constructed aboe for K so that in fact α V K,S, and P S : V K V K,S the projection defined in Proposition 4.6, where V K,S is the Q-ector space span of the S-units in K modulo torsion. Notice that in fact, for some set S M Q, we hae {w M Q : w M K } = {w M Q : w p M Q } S p S by the requirement that S be closed under the Galois action. S, as a set of places on K, meets the criteria set forth in the theorem statement. Let P = P S P K : G V K,S. By Lemma 2.13 and Propositions 2.7 and 4.6, we hae that δh 1 (P β) δh 1 (β) for all β G. Since P is linear and α V K,S, note that α = P α, so if we hae a factorization of α into α i G for i = 1,..., n, then T α = α 1 α n = α = (P α 1 ) (P α n ), and P α i V K,S for all i = 1,..., n. Then by our established inequalities for P K andp S with respect to δh 1, n n δh 1 (P α i ) δh 1 (α i ). Hence we may take the infimum within V K,S. Associate to each term in the infimum its minimal subspace V F,S V K,S containing it for F K. If we hae more than one term for any gien minimal subspace V F,S, notice that the δ alues are equal and we can combine any such terms by the triangle inequality for h 1. Thus, the first part of the claim is proen. The remaining criterion easily follows from obsering that the choice of α F can be made in accord with Theorem 6. u T