The t -metric Mahler measures of surds and rational numbers
aa r X i v : . [ m a t h . N T ] A ug THE t -METRIC MAHLER MEASURES OF SURDS OFRATIONAL NUMBERS JONAS JANKAUSKAS AND CHARLES L. SAMUELS
Abstract.
A. Dubickas and C. Smyth introduced the metric Mahler measure M ( α ) = inf ( N X n =1 M ( α n ) : N ∈ N , α · · · α N = α ) , where M ( α ) denotes the usual (logarithmic) Mahler measure of α ∈ Q . Thisdefinition extends in a natural way to the t -metric Mahler measure by replacingthe sum with the usual L t norm of the vector ( M ( α ) , . . . , M ( α N )) for any t ≥
1. For α ∈ Q , we prove that the infimum in M t ( α ) may be attained usingonly rational points, establishing an earlier conjecture of the second author. Weshow that the natural analogue of this result fails for general α ∈ Q by givingan infinite family of quadratic counterexamples. As part of this construction,we provide an explicit formula to compute M t ( D /k ) for a squarefree D ∈ N . Introduction
Let f be a polynomial with complex coefficients given by f ( z ) = a · N Y n =1 ( z − α n ) . Recall that the (logarithmic) Mahler measure M of f is defined by M ( f ) = log | a | + N X n =1 log + | α n | . If α is a non-zero algebraic number, the (logarithmic) Mahler measure M ( α ) of α is defined as the Mahler measure of the minimal polynomial of α over Z .It is a consequence of a theorem of Kronecker that M ( α ) = 0 if and only if α is aroot of unity. In a famous 1933 paper, D.H. Lehmer [5] asked whether there existsa constant c > M ( α ) ≥ c in all other cases. He could find no algebraicnumber with Mahler measure smaller than that of ℓ ( x ) = x + x − x − x − x − x − x + x + 1 , which is approximately 0 . . . . . Although the best known general lower bound is M ( α ) ≫ (cid:18) log log deg α log deg α (cid:19) , Mathematics Subject Classification.
Key words and phrases.
Weil height, Mahler measure, metric Mahler measure, Lehmer’s problem.The first author was supported by the Lithuanian Research Council (student research supportproject) during his visit to the IRMACS Centre, Simon Fraser University.The second author was supported by NSERC of Canada. due to Dobrowolski [2], uniform lower bounds have been established in many specialcases (see [1, 14, 15], for instance). Furthermore, numerical evidence provided, forexample, in [6–9] suggests there exists such a constant c . Conjecture 1.1 (Lehmer’s conjecture) . There exists a real number c > such thatif α ∈ Q × is not a root of unity then M ( α ) ≥ c . For an algebraic number α , Dubickas and Smyth [3] introduced the metric Mahlermeasure M ( α ) by(1.1) M ( α ) = inf ( N X n =1 M ( α n ) : N ∈ N , α n ∈ Q × , α = N Y n =1 α n ) . Here, the infimum is taken over all ways to write α as a product of algebraicnumbers. The advantage of M over M is that it satisfies the triangle inequality M ( αβ ) ≤ M ( α ) + M ( β )for all algebraic numbers α and β . In view of this observation, M is well-definedon the quotient group G = Q × / Tor( Q × ), and the map ( α, β ) M ( αβ − ) definesa metric on G . This metric induces the discrete topology if and only if Lehmer’sconjecture is true.The metric Mahler measure M is only a special case of the t -metric Mahlermeasures , which are defined for t ≥ M t ( α ) = inf N X n =1 M ( α n ) t ! /t : N ∈ N , α n ∈ Q × , α = N Y n =1 α n . In addition, the ∞ -metric Mahler measure of α is defined by M ∞ ( α ) = inf ( max ≤ n ≤ N { M ( α n ) } : N ∈ N , α n ∈ Q × , α = N Y n =1 α n ) . The t -metric Mahler measures were introduced and studied in [12, 13]. It followsfrom the results of [12] that these functions have analogues of the triangle inequality M t ( αβ ) t ≤ M t ( α ) t + M t ( β ) t and M ∞ ( αβ ) ≤ max { M ∞ ( α ) , M ∞ ( β ) } Hence, the map ( α, β ) M t ( αβ − ) defines a metric on G that induces the discretetopology if and only if Lehmer’s conjecture is true.If t ∈ [1 , ∞ ] and α ∈ Q , we say that the infimum in M t ( α ) is attained by α , . . . , α N if we have that α = α · · · α N and M t ( α ) = (cid:16)P Nn =1 M ( α n ) t (cid:17) /t if t < ∞ max ≤ n ≤ N { M ( α n ) } if t = ∞ . If S is any subset of Q , we say the infimum in M t ( α ) is attained in S if there existpoints α , . . . , α N ∈ S that attain the infimum in M t ( α ).It is not immediately obvious that M t ( α ) is attained for all values of α and t .Dubickas and Smyth [3] conjectured that the infimum in M ( α ) is always attaineda fact later proved by the second author [11]. More specifically, if K α is the Galoisclosure of Q ( α ) over Q andRad( K α ) = { γ ∈ Q : γ n ∈ K α for some n ∈ N } , HE t -METRIC MAHLER MEASURE 3 then the infimum in M ( α ) is attained in Rad( K α ). Using the same method, thisresult was generalized for all t -metric Mahler measures in [12]. That is, for every t ≥
1, the infimum in M t ( α ) is attained in Rad( K α ).It is natural to ask if these results can be improved, having a smaller set S inplace of Rad( K α ). In particular, for each α ∈ Q , we would like to identify a set S α whose points generate a finite extension of Q and the infimum in M t ( α ) is attainedin S α for all t . This problem is of considerable importance if we hope to computeexact values of M t ( α ). For example, Conjecture 2.1 of [13] predicts that, if α isrational, then the infimum in M t ( α ) is attained in Q . With this assumption, it ispossible to graph some examples of the function t M t ( α ) where α ∈ Q .It follows from [3] and [4] that Conjecture 2.1 of [12] holds for t = 1 and t = ∞ . Unfortunately, these methods seem genuinely distinct and cannot be easilygeneralized to handle all values of t and α . As our first result, we prove thisconjecture for all t ≥ Theorem 1.2. If α is a non-zero rational number and t ∈ [1 , ∞ ] then the infimumin M t ( α ) is attained in Q . Our next question is whether Theorem 1.2 can be extended to arbitrary algebraicnumbers α . In view of Theorem 1.2, one might suspect that the infimum in M t ( α )is always attained in K α . This turns out to be false, however, as we are able toproduce an infinite family of quadratic counterexamples. More specifically, if D is a square-free positive integer, we show precisely when M t ( √ D ) is attained in K √ D = Q ( √ D ). Theorem 1.3.
Suppose that p , . . . , p L are distinct primes written in decreasingorder, D = p · · · p L , and t ∈ (1 , ∞ ] . The infimum in M t ( √ D ) is attained in Q ( √ D ) if and only if D < p . In this situation, the infimum is attained by points r p p · · · p L , p , . . . , p L ∈ Q ( √ D ) , and we have that M t ( √ D ) = (cid:16)P Lℓ =1 (log p ℓ ) t (cid:17) /t if t ∈ (1 , ∞ )log p if t = ∞ . Theorem 1.3 enables the construction of infinitely many integers D such that M t ( √ D ) is not attained in K √ D = Q ( √ D ) for any t >
1. Theorem 1.4 below givesa set of points that attain the infimum in M t ( α ) for algebraic numbers α = D /k ,where D >
Theorem 1.4. If p , . . . , p L are distinct primes, D = p · · · p L , and t ∈ [1 , ∞ ] ,then the infimum in M t ( D /k ) is attained by p /k , . . . , p /kL and M t ( D /k ) = (cid:16)P Lℓ =1 (log p ℓ ) t (cid:17) /t if t ∈ [1 , ∞ )max ≤ ℓ ≤ L { log p ℓ } if t = ∞ . As an example, for D = 30 = 2 · ·
5, Theorem 1.4 asserts that M t ( √
30) isattained by √ , √ , √
5, and M t ( √ t = (log 5) t + (log 3) t + (log 2) t . J. JANKAUSKAS AND C.L. SAMUELS
While it is obvious that √ , √ , √ Q ( √ M t ( √
30) might beattained by some distinct set of points in Q ( √ D = 42 = 2 · ·
7, Theorem 1.4 establishes that M t ( √
42) is attainedby √ , √ , √
7, and M t ( √ t = (log 7) t + (log 3) t + (log 2) t . Nonetheless, Theorem 1.3 identifies the slightly more subtle points p / , , ∈ Q ( √
42) that also attain the infimum in M t ( √ M t ( √ D ) can be attainedonly by rational numbers and their square roots. This intuition is misleading,however, as we see in the following example. Let t = ∞ and take D = 21 = 7 · M t ( √
21) is attained by the points p / , ∈ Q ( √
21) and M t ( √ t = (log 7) t + (log 3) t . Now consider(1.2) √
21 = ( − · √ ! · − √ ! , and we verify easily that M √ ! = log 7 and M − √ ! < log 7 . In other words, M ∞ ( √
21) is attained by the points on the right hand side of (1.2)and these points belong to Q ( √ M (cid:0) (3 − √ / (cid:1) > log 3, so these points cannot be used to attain the infimum in M t ( √
21) for othervalues of t . Nonetheless, this example illustrates that the infimum in M t ( √ D ) maybe attained by using distinct non-trivial sets of points contained in Q ( √ D ).We would like to conclude with the following question. Question 1.5.
Is the infimum in M t ( α ) always attained by points α , . . . , α N suchthat [ Q ( α n ) : Q ] ≤ [ Q ( α ) : Q ] for all n ? According to Theorem 1.4, the answer is ’yes’ when α is a surd, although weknow of little other evidence.2. The rational case
Recall that the (logarithmic) Weil height of an algebraic number α is given by h ( α ) = M ( α )deg α . It is well-known that if ζ is a root of unity, then h ( α ) = h ( ζα ) so that h is well-defined on our quotient group G . Furthermore, if n is an integer, then we havethat h ( α n ) = | n | · h ( α ). Also recall that a surd is an algebraic number α such that α n ∈ Q for some positive integer n . HE t -METRIC MAHLER MEASURE 5 Suppose now that F is any number field containing the algebraic number α .Further assume that K is an extension of F which is Galois over Q . We set G = Gal( K/ Q ) and H = Gal( K/F ) , and let S be a set of left coset representatives of H in G . Recall that the norm of α from F to Q is given by Norm F/ Q ( α ) = Y σ ∈ S σ ( α ) . It follows from standard Galois Theory that Norm F/ Q is a homomorphism from F to Q which does not depend on the choice of K or S . In addition, if E is anyextension of F , then it is easily verified that(2.1) Norm E/ Q ( α ) = Norm F/ Q ( α ) [ E : F ] . We begin our proof of Theorem 1.2 with a lemma that relates the Mahler measureof a surd to the Mahler measure of its norm.
Lemma 2.1. If γ is a surd then M ( γ ) = M (cid:0) Norm Q ( γ ) / Q ( γ ) (cid:1) .Proof. Since γ is a surd, its conjugates over Q are given by { ζ γ, ζ γ, . . . , ζ M γ } where M = deg γ and ζ m are roots of unity. It now follows that γ M M Y m =1 ζ m = Norm Q ( γ ) / Q ( γ ) ∈ Q . Since Norm Q ( γ ) / Q ( γ ) is clearly a rational number, we have that M (cid:0) Norm Q ( γ ) / Q ( γ ) (cid:1) = h γ M M Y m =1 ζ m ! = M · h ( γ ) = deg γ · h ( γ ) = M ( γ )completing the proof. (cid:3) In our proof of Theorem 1.2, it will be necessary to replace an arbitrary repre-sentation α = α · · · α N with another representation of α = β · · · β N that uses onlyrational numbers and satisfies N X n =1 M ( β n ) t ≤ N X n =1 M ( α n ) t . Our next lemma provides us with the necessary elementary number theoretic toolsto do this.
Lemma 2.2.
Suppose that m, r , . . . , r N are positive integers such that m | N Y n =1 r n . For ≤ n ≤ N , recursively define the points m n by (2.2) m = gcd( r , m ) and m n = gcd r n , m Q n − i =1 m i ! . J. JANKAUSKAS AND C.L. SAMUELS
Then we have that m = N Y n =1 m n . Before we provide the proof of Lemma 2.2, we make one clarification regardingthe definition of m n . Naively, it would appear that m Q n − i =1 m i is not necessarily an integer, so that taking its greatest common divisor with anotherinteger might not be well-defined. However, we note immediately that m | m ,which also implies that m is well-defined. Then clearly we have that m | m/m implying that m is also well-defined. As we can see, it follows inductively that m n | m Q n − i =1 m i for all 1 ≤ n ≤ N , meaning, in particular, that m n is well-defined for all such n .Now we may proceed with the proof of Lemma 2.2. Proof of Lemma 2.2.
We will assume that m = Q Nn =1 m n and find a contradiction.Since the product Q Nn =1 m n divides m , there must exist a prime number p for which(2.3) ν p ( m ) > N X j =1 ν p ( m j ) , where ν p ( x ) denotes the highest power of p dividing the integer x . It now followsthat ν p ( m n ) < ν p ( m ) − n − X j =1 ν p ( m j ) = ν p m Q n − j =1 m j ! for every n ∈ { , . . . , N } . Hence, the definition of m n implies that ν p ( m n ) = min ( ν p ( r n ) , ν p m Q n − j =1 m j !) = ν p ( r n )for every n ∈ { , . . . , N } . It now follows from (2.3) that ν p ( m ) > P Nn =1 ν p ( r n ),contradicting our assumption that m divides Q Nn =1 r n . (cid:3) Now that we have established our key lemmas, we may now proceed with theproof of Theorem 1.2.
Proof of Theorem 1.2.
As we have noted in the introduction, the case t = ∞ isknown [4], so we proceed immediately to the situation where 1 ≤ t < ∞ .We may assume without loss of generality that α >
0. Since α is rational,there exist positive integers m and m ′ such that gcd( m, m ′ ) = 1 and α = m/m ′ .Furthermore, by the results of [12], there exist surds α , . . . , α N such that(2.4) α = α · · · α N and M t ( α ) t = N X n =1 M ( α n ) t . HE t -METRIC MAHLER MEASURE 7 Let K be a number field containing α , . . . α N . Now we may take the norm from K to Q of both sides of the first equation in (2.4). We apply (2.1) and the fact thatthe Norm K/ Q is a homomorphism to establish that (cid:16) mm ′ (cid:17) [ K : Q ] = N Y n =1 Norm K/ Q ( α n ) = N Y n =1 (cid:0) Norm Q ( α n ) / Q ( α n ) (cid:1) [ K : Q ( α n )] . Suppose further that, for each 1 ≤ n ≤ N , r n and s n are relatively prime positiveintegers such that r n s n = ± Norm Q ( α n ) / Q ( α n ) . Therefore, we have that (cid:16) mm ′ (cid:17) [ K : Q ] = ± N Y n =1 (cid:18) r n s n (cid:19) [ K : Q ( α n )] . It is obvious that [ K : Q ( α n )] | [ K : Q ] so we obtain that m [ K : Q ] | N Y n =1 r n ! [ K : Q ] and m ′ [ K : Q ] | N Y n =1 s n ! [ K : Q ] . It follows from elementary number theory facts that(2.5) m | N Y n =1 r n and m ′ | N Y n =1 s n . Setting up the hypotheses of Lemma 2.2, we define recursive sequences corre-sponding to m and m ′ . First set m = gcd( r , m ) and m n = gcd r n , m Q n − i =1 m i ! and m ′ = gcd( s , m ′ ) and m ′ n = gcd s n , m ′ Q n − i =1 m ′ i ! so we clearly have that(2.6) | r n | ≥ | m n | and | s n | ≥ | m ′ n | . Applying Lemma 2.2, we have that m = N Y n =1 m n and m ′ = N Y n =1 m ′ n so that(2.7) α = mm ′ = N Y n =1 m n m ′ n . Now it follows from the definition of M t ( α ) that(2.8) M t ( α ) t ≤ N X n =1 M (cid:18) m n m ′ n (cid:19) t , so we must show that the right hand side of (2.8) is also a lower bound for M t ( α ) t . J. JANKAUSKAS AND C.L. SAMUELS
To see this, note that by Lemma 2.1, we have that M ( α n ) = M (cid:0) Norm Q ( α n ) / Q ( α n ) (cid:1) = M (cid:18) r n s n (cid:19) for all 1 ≤ n ≤ N . We have assumed that r n and s n are relatively prime, so itfollows from known facts about the Mahler measure that M ( α n ) = log max {| r n | , | s n |} . Then applying (2.6), we find that M ( α n ) ≥ log max {| m n | , | m ′ n |} ≥ M (cid:18) m n m ′ n (cid:19) , and consequently, M t ( α ) t = N X n =1 M ( α n ) t ≥ N X n =1 M (cid:18) m n m ′ n (cid:19) t . Combining this with (2.7) and (2.8), the result follows. (cid:3) The quadratic case
Our first lemma gives one particular set of points that attain the infimum in M t ( √ D ) for all t ∈ [1 , ∞ ]. When t >
1, we can also identify the Mahler measuresof any points α , . . . , α N attaining the infimum in M t ( D /k ). Lemma 3.1.
Suppose that p , . . . , p L are distinct primes written in decreasingorder, D = p · · · p L , t ∈ [1 , ∞ ) , and k ∈ N . The infimum in M t ( D /k ) is attainedby p /k , . . . , p /kL and M t ( D /k ) t = L X ℓ =1 (log p ℓ ) t . If t > and α , · · · , α N are algebraic numbers attaining the infimum in M t ( D /k ) then N ≥ L . Moreover, it is possible to relabel the elements α , . . . , α N so that(i) M ( α n ) = log p n for all n ≤ L , and(ii) M ( α n ) = 0 for all n > L .In particular, M ( α n ) ≤ log p for all n .Proof. We certainly have that D /k = p /k · · · p /kℓ , and by the definition of M t , weknow that M t ( D /k ) t ≤ L X ℓ =1 M ( p /kℓ ) t . For each ℓ , we know that x k − p ℓ vanishes at p /kℓ and is irreducible by Eisenstein’scriterion, so that M ( p /kℓ ) = M ( x k − p ℓ ) = log p ℓ . Hence, we find that(3.1) M t ( D /k ) t ≤ L X ℓ =1 (log p ℓ ) t . HE t -METRIC MAHLER MEASURE 9 To prove the first statement of the lemma, it is now sufficient to show that(3.2) M t ( D /k ) t ≥ L X ℓ =1 (log p ℓ ) t . Now suppose α , . . . , α N ∈ Q attain the infimum in M t ( D /k ) and select anumber field K containing D /k , α , . . . , α N . By definition, we know that D /k = α · · · α N . Using the fact that Norm K/ Q is a multiplicative homomorphism, weobtain that Norm K/ Q ( D /k ) = N Y n =1 Norm K/ Q ( α N )so that(3.3) (cid:16) Norm Q ( D /k ) / Q ( D /k ) (cid:17) [ K : Q ( D /k )] = N Y n =1 (cid:0) Norm Q ( α n ) / Q ( α N ) (cid:1) [ K : Q ( α n )] . Each of the above norms is a rational number. Hence, for each n , there exist positiverelatively prime integers r n and s n such that | Norm Q ( α n ) / Q ( α n ) | = r n s n . Again using Eisenstein’s Criterion, we know that x k − D is the minimal polynomialof D /k over Q , implying that | Norm Q ( D /k ) / Q ( D /k ) | = D . Substituting thesevalues into (3.3), we find that(3.4) D [ K : Q ( D /k )] = N Y n =1 (cid:18) r n s n (cid:19) [ K : Q ( α n )] . For each n , α n has minimal polynomial of the formˆ f n ( x ) = x d + a d − b d − x d − + · · · + a b x ± r n s n over Q for integers a , . . . , a d − , b , . . . , b d − with b i = 0 and ( a i , b i ) = 1. Hence,its minimal polynomial over Z is given by f n ( x ) = lcm( s n , b d − , . . . , b ) · x d + · · · ± r n · lcm( s n , b d − , . . . , b )and its Mahler measure satisfies M ( α n ) ≥ log (cid:18) r n s n · lcm( s n , b d − , . . . , b ) (cid:19) ≥ log r n . For each n , let P n = { p ∈ { p , . . . , p L } : p | r n } . We have assumed that α , . . . , α N attains the infimum in M t ( D /k ), so we get that(3.5) M t ( D /k ) t = N X n =1 M ( α n ) t ≥ N X n =1 (log r n ) t ≥ N X n =1 X p ∈ P n log p t . Since t ≥
1, we always have that(3.6) X p ∈ P n log p t ≥ X p ∈ P n (log p ) t , which implies that M t ( D /k ) t ≥ N X n =1 X p ∈ P n (log p ) t . However, applying (3.4), we know that for each ℓ ∈ { , . . . , L } , there exists n ∈{ , . . . , N } such that p ℓ ∈ P n , establishing (3.2) and the first statement of thelemma.Now assume that t >
1. If | P n | ≥
2, then we must have strict inequality in (3.6).Therefore, if | P n | ≥ n , then (3.5) implies that M t ( D /k ) t > N X n =1 X p ∈ P n (log p ) t ≥ L X ℓ =1 (log p ℓ ) t contradicting (3.1). Therefore, | P n | ≤ n and we have established that(a) For every ℓ , there exists n such that p ℓ | r n , and(b) If ℓ = ℓ then we can never have that p ℓ | r n and p ℓ | r n .It follows from the box principle that N ≥ L . Moreover, we may reorder α , . . . , α N such that p n | r n for all 1 ≤ n ≤ L , which shows that(3.7) M ( α n ) ≥ log r n ≥ log p n for 1 ≤ n ≤ L. If we have strict inequality in (3.7) for some n , then(3.8) M t ( D /k ) t = N X n =1 M ( α n ) t > L X ℓ =1 (log p ℓ ) t contradicting (3.1) and establishing (i). Similarly, if M ( α n ) > n > L ,then (3.8) holds as well verifying (ii). (cid:3) Now that we have proven Lemma 3.1, the proof of Theorem 1.4 is essentiallycomplete. Indeed, when t ∈ [1 , ∞ ) Theorem 1.4 is simply the first statement ofLemma 3.1, and the case t = ∞ was given already in [4]. The only task remainingis to prove Theorem 1.3, in which the second statement of Lemma 3.1 plays a keyrole.Before proceeding, we establish some conventions that will be used for the re-mainder of this article. For d ∈ Z and r ∈ Q , we say that d divides r if when r iswritten r = m/n with m ∈ N , n ∈ Z \ { } and ( m, n ) = 1, then either d | m or d | n . We say that d divides the numerator or denominator of r if d divides m or n , respectively.We say that an algebraic number α is stable if all of its conjugates lie either insidethe open unit disk, on the unit circle, or outside the closed unit disk. Otherwise,we say that α is unstable . It is clear that all rational numbers and all imaginaryquadratic numbers are stable, while real quadratic numbers can be either stable orunstable. If α is any algebraic number having minimal polynomial f ( x ) = a N x N + · · · + a x + a , then it is simple to verify that M ( α ) ≥ log max {| a N | , | a |} HE t -METRIC MAHLER MEASURE 11 with equality if and only if α is stable. We now state a simple criterion whichallows us to determine if a quadratic algebraic number is stable by considering thecoefficients of the minimal polynomial. Lemma 3.2.
Suppose that α is a quadratic algebraic number having minimal poly-nomial f ( x ) = ax + bx + c over Z . We have that α is stable if and only if | a + c | > | b | .In this situation, the following hold.(i) If | a | < | c | then both conjugates of α have modulus greater than one.(ii) If | a | = | c | then both conjugates of α have modulus one.(iii) If | a | > | c | then both conjugates of α have modulus less than one.Proof. Suppose that f ( x ) = a ( x − α )( x − β ). If f (1) and f ( −
1) have oppositesigns, then f has precisely one root in the interval ( − , − , α is unstable. If f (1) and f ( −
1) havethe same sign, then f has either zero or two roots in ( − , − , α is clearly stable. If f has zero roots in ( − , α is certainly stable, or two real roots both lyingoutside of [ − , α is stable.We have now shown that α is stable if and only if f (1) = a + b + c and f ( −
1) = a − b + c have the same sign. Clearly, f (1) and f ( −
1) are both positive if and onlyif a + c > | b | and both negative if and only if − a − c > | b | . Thus, α is stable if andonly if | a + c | > | b | .If, in addition, | a | < | c | , then | αβ | = | c | / | a | >
1, so both α and β have modulusgreather than 1. Similarly, if | a | > | c | , then | αβ | = | c | / | a | < α and β have modulus less than 1. Finally, if | a | = | c | then | αβ | = 1. Since α isstable, α and β must be complex conjugate numbers both of modulus 1. (cid:3) The following lemma shows us that certain quadratic algebraic numbers, whichwe will encounter in the proof of Theorem 1.3, have relatively simple minimalpolynomials.
Lemma 3.3.
Let D be a square-free integer, p be a prime divisor of D , and α aquadratic algebraic number in Q ( √ D ) . If M ( α ) ≤ log p and p divides the numer-ator of Norm Q ( √ D ) / Q ( α ) then α is stable. Moreover, the minimal polynomial of α satisfies f ( x ) = ax ± p or f ( x ) = ax ± px + p where a is a positive integer with a < p .Proof. Suppose that f ( x ) = ax + bx + c ∈ Z [ x ] is the minimal polynomial of α over Z , so we may assume that a >
0. Since α has degree 2, we have thatNorm Q ( √ D ) / Q ( α ) = α ¯ α = ca where ¯ α is the conjugate of α over Q . We have assumed that p divides the numeratorof Norm Q ( √ D ) / Q ( α ), which itself must divide c , implying that p | c . Since M ( α ) ≥ log max {| a | , | c |} , we have thatlog p ≤ log | c | ≤ log max {| a | , | c |} ≤ M ( α ) ≤ log p, and we conclude that(3.9) M ( α ) = log | c | = log p. It now follows that | a | ≤ | c | and, since M ( α ) is the log of an integer, we furtherobtain that α is stable. Hence, Lemma 3.2 implies that | a + c | > | b | .We cannot have | a | = | c | , sinceNorm Q ( √ D ) / Q ( α ) = ca = ± p , so it follows that | a | < | c | . In view of Lemma 3.2 (i), we havethat | α | , | ¯ α | >
1. Therefore, we find that(3.10) | b | < | a + c | ≤ | a | + | c | < | c | = 2 p. Now let ∆ = b − ac . Since Q ( √ ∆) = Q ( √ D ), and D is square-free, we have∆ = Dv for some v ∈ Z . The quadratic formula givesNorm Q ( √ D ) / Q ( α ) = α ¯ α = b − Dv a , and since p | D and the numerator of Norm Q ( √ D ) / Q ( α ), it follows that p divides b .Of course, this implies that p | b . Using (3.10), we now see that b ∈ { , p, − p } .If b = 0 then we have by (3.9) that f ( x ) = ax ± p , establishing the lemma inthis case. If b = ± p , then | a + c | > | b | holds if and only if a and c has the same sign.So in this situation, (3.9) yields that c = p which leads to f ( x ) = ax ± px + p . (cid:3) Proof of Theorem 1.3.
By Theorem 1.4, we know that M t ( √ D ) = (cid:16)P Lℓ =1 (log p ℓ ) t (cid:17) /t if t ∈ (1 , ∞ )log p if t = ∞ . We also observe that(3.11) √ D = r p p · · · p L · p · · · p L and that each term in the product on the right hand side of (3.11) belongs to Q ( √ D ). We obviously have that M ( p ℓ ) = log p ℓ for all ℓ . Furthermore, our as-sumption that D < p ensures that p · · · p L < p , so it follows that M (cid:18)r p p · · · p L (cid:19) = log p . Combining these observations, we see that M (cid:18)r p p · · · p L (cid:19) t + L X ℓ =2 M ( p ℓ ) t = L X ℓ =1 (log p ℓ ) t = M t ( √ D ) t when 1 < t < ∞ andmax (cid:26) M (cid:18)r p p · · · p L (cid:19) , M ( p ) , . . . , M ( p L ) (cid:27) = log p = M ∞ ( √ D ) . establishing one direction of the theorem as well as the second statement.To prove the other direction, we assume that there exist points α , . . . , α N ∈ Q ( √ D ) that attain the infimum in M t ( √ D ), and for simplicity, we set p = p .When t ∈ (1 , ∞ ), Lemma 3.1 establishes that that M ( α n ) ≤ log p for all n . In the HE t -METRIC MAHLER MEASURE 13 case t = ∞ , we also have M ( α n ) ≤ log p for all n as a consequence of Theorem 1.4.Since √ D = α · · · α N , we have that(3.12) − D = Norm Q ( √ D ) / Q ( √ D ) = N Y n =1 Norm Q ( √ D ) / Q ( α n ) . Defining the set Λ = n ≤ n ≤ N : p | Norm Q ( √ D ) / Q ( α n ) o we apply (3.12) to see that(3.13) X n ∈ Λ ν p (cid:16) Norm Q ( √ D ) / Q ( α n ) (cid:17) = ν p ( D ) = 1 , where the last equality follows since D is square-free. If Λ contains no irrationalpoints, then we have that p | Norm Q ( √ D ) / Q ( α n ) = α n for all n ∈ Λ. However, this implies that ν p (Norm Q ( √ D ) / Q ( α n )) is even for all n ∈ Λ. It follows that the left hand side of (3.13) is also even, a contradiction.We have shown that there must exist n such that α n is quadratic, M ( α n ) ≤ log p ,and p divides Norm Q ( √ D ) / Q ( α n ). If p divides the numerator of Norm Q ( √ D ) / Q ( α n ),then we may apply Lemma 3.3 to see that α n is stable and is a root of f ( x ) = ax ± p or f ( x ) = ax ± px + p for some positive integer a < p .Suppose now that ∆ is the discriminant of f . Since α n is quadratic over Q ,we have Q ( √ ∆) = Q ( √ D ). Furthermore, since D is a square-free, we have that∆ = Dv for some v ∈ N . If f ( x ) = ax ± p , we see that ∆ = ± ap , so that pp · · · p L v = Dv = ± ap. Since p , . . . , p L are distinct primes, we obtain that p · · · p L | a , and hence, p · · · p L ≤ a < p, establishing that D < p in this case.If f ( x ) = ax ± px + p then ∆ = p − ap = p ( p − a ). We have assume that D ispositive so that p − a >
0, and, trivially, p − a < p . Hence, D ≤ ∆ = p ( p − a ) < p completing the proof when p divides the numerator of Norm Q ( √ D ) / Q ( α n ).If p divides the denominator of Norm Q ( √ D ) / Q ( α n ) instead, the p must divide thenumerator of Norm Q ( √ D ) / Q ( α − n ). Of course, we also have that M ( α − n ) ≤ log p and α − n ∈ Q ( √ D ) is quadratic, so we may apply the above argument to α − n inplace of α n . (cid:3) References [1] P. Borwein, E. Dobrowolski and M.J. Mossinghoff,
Lehmer’s problem for polynomials with oddcoefficients , Ann. of Math. (2) (2007), no. 2, 347–366.[2] E. Dobrowolski,
On a question of Lehmer and the number of irreducible factors of a polyno-mial , Acta Arith. (1979), no. 4, 391–401.[3] A. Dubickas and C.J. Smyth, On the metric Mahler measure , J. Number Theory (2001),368–387.[4] P. Fili and C.L. Samuels, On the non-Archimedean metric Mahler measure , J. Number Theory, (2009), no. 7, 1698–1708. [5] D.H. Lehmer,
Factorization of certain cyclotomic functions , Ann. of Math. (1933), 461–479.[6] M.J. Mossinghoff, Algorithms for the determination of polynomials with small Mahler measure ,Ph.D. Thesis, University of Texas at Austin, 1995.[7] M.J. Mossinghoff, website,
Lehmer’s Problem , ∼ mjm/Lehmer .[8] M.J. Mossinghoff, C.G. Pinner and J.D. Vaaler, Perturbing polynomials with all their rootson the unit circle , Math. Comp. (1998), 1707–1726.[9] M.J. Mossinghoff, G. Rhin and Q. Wu, Minimal Mahler measures , Experiment. Math. (2008), no. 4, 451-458.[10] D.G. Northcott, An inequality on the theory of arithmetic on algebraic varieties , Proc. Cam-bridge Philos. Soc., (1949), 502–509.[11] C.L. Samuels, The infimum in the metric Mahler measure , Canad. Math. Bull., to appear.[12] C.L. Samuels,
A collection of metric Mahler measures , J. Ramanujan Math. Soc. (2010),no. 4, 433–456.[13] C.L. Samuels, The parametrized family of metric Mahler measures , J. Number Theory (2011), no. 6, 1070–1088.[14] A. Schinzel,
On the product of the conjugates outside the unit circle of an algebraic number ,Acta Arith. (1973), 385–399. Addendum, ibid. (1975), no. 3, 329–331.[15] C.J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer ,Bull. London Math. Soc. (1971), 169–175. Vilnius University, Department of Probability Theory and Number Theory, Facultyof Mathematics and Informatics, Naugarduko 24, LT-03225 Vilnius, Lithuania
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