aa r X i v : . [ m a t h . N T ] A ug ESTIMATING HEIGHTS USING AUXILIARY FUNCTIONS
CHARLES L. SAMUELS
Abstract.
Several recent papers construct auxiliary polynomials tobound the Weil height of certain classes of algebraic numbers from be-low. Following these techniques, the author gave a general method forintroducing auxiliary polynomials to problems involving the Weil height.The height appears as a solution to a certain extremal problem involvingpolynomials. We further generalize the above techniques to acquire boththe projective height and the height on subspaces in the same way. Wefurther obtain lower bounds on the heights of points on some subvarietiesof P N − ( Q ). Introduction
Let K be a number field and let v be a place of K dividing the place p of Q . Of course, if v is non-Archimedean then p is a rational prime while if v isArchimedean then p = ∞ . We write K v to denote the completion of K at v and Q p to denote the completion of Q at p . It is clear that these completionsdo not depend on a specific absolute value taken from the places v and p .We write d = [ K : Q ] for the global degree and d v = [ K v : Q p ] for the localdegree.We now select two absolute values on K v for each place v . The firstabsolute value, denoted k · k v , is the unique extension of the p -adic absolutevalue on Q p . The second, denoted | · | v , is defined by | x | v = k x k d v /dv for all x ∈ K v . We note the important identity d = X v | p d v as well as the product formula Y v | α | v = 1for all α ∈ K × . Furthermore, each of the above absolute values extendsuniquely to an algebraic closure K v . If v is Archimedean then K v is com-plete, however, in general, K v is not complete and we write Ω v to denote Mathematics Subject Classification.
Primary 11R04, 11R09.
Key words and phrases. projective height, Weil height, subspace height, Mahler mea-sure, Lehmer’s problem. its completion. It is well-known that Ω v is algebraically closed for all places v . Moreover, we may define the Weil Height of α ∈ K by h ( α ) = Y v max { , | α | v } where the product is taken over all places v of K . By the way we havenormalized our absolute values, this definition does not depend on K , andtherefore, is a well-defined function on Q .For f ∈ Z [ x ] having roots α , . . . , α d we define the Mahler measure of f by µ ( f ) = d Y k =1 h ( α k ) . Since h is invariant under Galois conjugation over Q , we note that if f isirreducible and α is any root of f then µ ( f ) = h ( α ) deg α .By Kronecker’s Theorem, µ ( f ) ≥ f is aproduct of cyclotomic polynomials and ± x . Further, in 1933, D.H. Lehmer[5] asked if there exists a constant c > µ ( f ) ≥ c in all othercases. It can be computed that µ ( x + x − x − x − x − x − x + x + 1) = 1 . . . . which remains the smallest known Mahler measure greater than 1.Since Lehmer’s famous 1933 paper, many special cases of his proposedproblem have been solved. In 1971, Smyth [8] showed that if α and α − arenot Galois conjugates, then the minimal polynomial of α over Q has Mahlermeasure at least µ ( x − x − f ∈ Q [ x ] has only real rootsthen µ ( f ) ≥ (1 + √ / f ∈ Z [ x ] has no cyclotomic factors and has coefficients congruent to 1 moduloan integer m , then(1.1) µ ( f ) ≥ c m ( T ) deg f f . Here, c m ( T ) > m and an auxiliary polynomial T ∈ Z [ x ]. They were able to obtain an explicit lower bound for µ ( f ) bymaking a choice of auxiliary polynomial T . Later, Dubickas and Mossinghoff[3] generalized the results of [1] so that the polynomial f in (1.1) maybe any factor of a polynomial having coefficient congruent to 1 mod m .They further constructed a sequence of auxiliary polynomials that furtherimproved the explicit bounds given in [1]. Following these methods, theauthor [6] constructed a function U ( α, T ) and showed that(1.2) 1 = h ( α ) · U ( α, T ) , STIMATING HEIGHTS USING AUXILIARY FUNCTIONS 3 for all polynomials T over Q with T ( α ) = 0. We now briefly recall thisconstruction.Define the local supremum norm of T ∈ Ω v [ x ] on the unit ball by(1.3) ν v ( T ) = sup {| T ( z ) | v : z ∈ Ω v and | z | v ≤ } . Consider the vector space J v of polynomials over Ω v of degree at most N − α ∈ Ω v and T ∈ J v define(1.4) U v ( α, T ) = inf { ν v ( T − f ) : f ∈ J v and f ( α ) = 0 } . Lemma 2.1 of [6] states that(1.5) | T ( α ) | v = max { , | α | v } N · U v ( α, T ) . Now assume that α ∈ Q and T ∈ Q [ x ] with T ( α ) = 0. In this situation, wemay define a global version of U v ( α, T ) by U ( α, T ) = Y v U v ( α, T )where the product runs over all places v of a number field containing α andthe coefficients of T . According to (1.5), this product is indeed finite andit does not depend on the number field we choose to contain α and thecoefficients of T . We may apply the product formula to | T ( α ) | v to obtain(1.2).The advantage of this identity is that we may freely select T in a way thatis convenient without changing the value of U ( α, T ). It can then be used toestimate the Weil height in certain special cases as found in [6]. Our goalfor this paper is to apply this strategy to obtain analogous results regardingthe projective height and, more generally, the height on subspaces.If a = ( a , . . . , a N ) ∈ Ω Nv define the local projective height of a by(1.6) H v ( a ) = max {| a | v , . . . , | a N | v } . That is, the local projective height is simply the maximum norm on Ω Nv withrespect to | · | v . It is worth noting that some authors define the projectiveheight using the maximum norm only at the non-Archmedean places whileusing the L norm on the components of a at the Archimedean places.However, we are motivated by generalizing the Weil height, so we will findit more relevant to use the maximum norm at all places in our definition.Indeed, we note that H ((1 , α, . . . , α N )) = h ( α ) N . It is clear that H v ( a ) = 1 for almost all places v of K so we may define the global projective height of a ∈ K N by H ( a ) = Y v H v ( a ) C.L. SAMUELS where the product is taken over all places v of K . Of course, by the waywe have chosen our absolute values, this definition does not depend on K . Furthermore, the product formula implies that H ( a ) is well defined on P N − ( Q ). In section 2, we define U ( a , T ) analogous to (1.4) and prove that(1.7) 1 = H ( a ) M · U ( a , T ) . Here T is an homogeneous polynomial of degree M in N variables over Q with T ( a ) = 0. We also give a simple application of this result to demon-strate how it might be applied.In a slightly different direction, suppose that W is a subspace of Q N withbasis { w , . . . , w M } . The height of W is defined to be the height of thevector w ∧ · · · ∧ w M in the exterior product ∧ M ( Q N ). That is,(1.8) H ( W ) = H ( w ∧ · · · ∧ w M ) . This definition does not depend on K , and it follows from the productformula that H ( W ) does not depend on our choice of basis. In section 3,we define U ( W, Ψ) for a surjective linear transformation Ψ : Q N → Q M andprove that(1.9) 1 = H ( W ) · U ( W, Ψ)whenever W ∩ ker Ψ = { } . This provides an analog of (1.2) using theheight on subspaces.2. The projective height using auxiliary homogeneouspolynomials
We begin by defining the function U ( a , T ) given in (1.7). Let L v denotethe vector space of homogeneous polynomials over Ω v of degree M in N variables along with the zero polynomial. We define an analog of the localsupremum norm on polynomials by(2.1) ν v ( T ) = sup {| T ( z ) | v : z ∈ Ω Mv , H v ( z ) ≤ } and set(2.2) U v ( a , T ) = inf { ν v ( T − f ) : f ∈ L v , f ( a ) = 0 } for T ∈ L v . This is the local version of U ( a , T ) that will appear in ourtheorem. Let Z ( a ) = { f ∈ L v : f ( a ) = 0 } . It is obvious that (2.2) descends to a norm on the one-dimensional quotient L v /Z ( a ) so that the ratio | T ( a ) | v /U v ( a , T ) does not depend on T . In fact,we are able to prove something much stronger. STIMATING HEIGHTS USING AUXILIARY FUNCTIONS 5
Lemma 2.1. If a ∈ Ω Nv then (2.3) | T ( a ) | v = H v ( a ) M · U v ( a , T ) for all T ∈ L v .Proof. We will assume that | a n | v = H v ( a ) and note that | T ( a ) | v = | a n | Mv · (cid:12)(cid:12)(cid:12)(cid:12) T (cid:18) a a n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) v ≤ H v ( a ) M · ν v ( T )for all homogeneous polynomials T of degree M in N variables. So if f ( a ) =0 then(2.4) | T ( a ) | v = | T ( a ) − f ( a ) | v ≤ H v ( a ) · ν v ( T − f ) . Taking the infimum of the right hand side (2.4) over all f having f ( a ) = 0we obtain(2.5) | T ( a ) | v ≤ H v ( a ) · U v ( a , T ) . We now attempt to establish the opposite inequality. We have that U v ( a , T ) = inf { ν v ( T − f ) : f ∈ Z ( a ) } = inf { ν v ( T ( z ) − ( T ( z ) − ( T ( a )( z n /a n ) M ) − T ( a ) f ( z )) : f ∈ Z ( a ) } = inf { ν v ( T ( a )( z n /a n ) M − T ( a ) f ( z )) : f ∈ Z ( a ) } = | T ( a ) | v · U v ( a , ( z n /a n ) M ) . It is clear that U v ( a , ( z n /a n ) M ) ≤ ν v (( z n /a n ) M )= sup {| z n /a n | Mv : | z n | ≤ } = | a n | − Mv and hence U v ( a , T ) ≤ | T ( a ) | v · H v ( a ) − M which completes the proof. (cid:3) If T is a homogeneous polynomial over K of degree M in N variables and a ∈ K N then Theorem 2.1 implies that ν v ( a , T ) = 1 for almost all places v of K . Hence, we may define the global functions ν ( T ) = Y v ν v ( T ) , and U ( a , T ) = Y v U v ( a , T ) . which do not depend on K . We now obtain the following projective gener-alization of (1.2). Theorem 2.2. If a ∈ Q N then (2.6) 1 = H ( a ) M · U ( a , T ) C.L. SAMUELS holds for all homogeneous polynomials T over Q of degree M in N variableshaving T ( α ) = 0 .Proof. Suppose that K is a number field containing the entries of a and thecoefficients of T . Hence, we may view a as an element of Ω Nv and T as anelement of L v for all places v of K . Thus, Lemma 2.1 implies that(2.7) | T ( a ) | v = H v ( a ) M · U v ( a , T )at every place v of K . The result follows by taking the product of (2.7)over all places of K and applying the product formula to T ( a ). (cid:3) We may construct applications of Theorem 2.2 that are similar to thosefound in [6]. Suppose, for example, that F is an homogeneous polynomialof degree M in N variables with coefficients in Z . Let X ( F ) denote thesubvariety of P N − ( Q ) consisting of all points a with F ( a ) = 0. Supposefurther that T is homogeneous of degree M in N variables and that m ∈ Z are such that T ≡ F mod m. That is, the coefficients of T are congruent to the coefficients of F modulo m . If a ∈ X ( T ) then Theorem 2.2 implies that1 = H ( a ) M · U ( a , T ) . Now select a number field K containing the entries of a . If v is non-Archimedean then U v ( a , T ) ≤ ν v ( T − F ) ≤ | m | v so that U ( a , T ) ≤ ν ∞ ( T ) Y v ∤ ∞ | m | v = m − · ν ∞ ( T ) . If T has coefficients c , . . . , c R ∈ Z define L ∞ ( T ) = R X r =1 k c r k v ! d v /d and note that by the triangle inequality we have that ν ∞ ( T ) ≤ L ∞ ( T ).Hence, we obtain a lower bound on the projective height of a (2.8) H ( a ) deg F ≥ mL ∞ ( T ) . for all a ∈ X ( F ) \ X ( T ). Hence, if L ∞ ( T ) is small relative to m then weobtain a uniform lower H ( a ) deg F over all a ∈ X ( F ) \ X ( T ). In particular,if T is a monomial having coefficient ± H ( a ) deg F ≥ m which is non-trivial for all m ≥ STIMATING HEIGHTS USING AUXILIARY FUNCTIONS 7 The height on subspaces using auxiliary lineartransformations
We now turn our attention to the height on subspaces and attempt toconstruct an analog Theorem 2.2. Suppose that X is an N -dimensionalvector space over Ω v and fix a basis { e , . . . , e N } for X . For ease of notation,we identify X with Ω Nv by writing x = x e + · · · + x N e N = ( x , . . . , x N ) . In this way, we obtain the projective height of x ∈ X by H v ( x ) = max {| x | v , . . . , | x M | v } . Of course, this is a norm on X , and therefore, it yields the natural dualnorm of an element φ ∈ X ∗ ν v ( φ ) = sup {| φ ( x ) | v : x ∈ X, H v ( x ) ≤ } . Now fix an element w ∈ X and let S ∗ ( w ) = { φ ∈ X ∗ : φ ( w ) = 0 } so that S ∗ ( w ) is an N − X ∗ . Finally, for ψ ∈ X ∗ we set U v ( w , ψ ) = inf { ν v ( ψ − φ ) : φ ∈ S ∗ ( w ) } . We note that this defines a norm on the one dimensional quotient X ∗ /S ∗ ( w ).Of course, this implies that the ratio | ψ ( w ) | v /U v ( w , ψ ) depends only on w and v . Analogous to the results of [6] and the results of section 2 we areable to determine this ratio precisely. Lemma 3.1. If w ∈ X then | ψ ( w ) | v = H v ( w ) · U v ( w , ψ ) holds for all ψ ∈ X ∗ .Proof. If ψ ( w ) = 0 then both sides of the deisred identity equal 0. Hence,we assume without loss of generality that ψ ( w ) = 0. Let w = ( w , . . . , w N )and we select an integer n such that H v ( w ) = | w n | v . Of course, w n = 0 and H v ( w /w n ) = 1 so that we obtain | ψ ( w ) | v = | w n | v · | ψ ( w /w n ) | v ≤ H v ( w ) · ν v ( ψ )for all ψ ∈ X ∗ . Hence, if φ ∈ S ∗ ( w ) then | ψ ( w ) | v = | ( ψ − φ )( w ) | v ≤ H v ( w ) · ν v ( ψ − φ )Taking the infimum of the right hand side over all φ ∈ S ∗ ( w ) we obtain(3.1) | ψ ( w ) | v ≤ H v ( w ) · U v ( w , ψ ) . C.L. SAMUELS
We now attempt to prove the opposite inequality. We define the map J : X ∗ → X by J ( φ ) = ( φ ( e ) , . . . , φ ( e N ))and note that J is a vector space isomorphism having the property that φ ( w ) = J ( φ ) · w where · represents the inner product. We now defineappropriate bases for X ∗ and S ∗ ( w ). Let c n = (0 , . . . , , w − n , , . . . , T andnote that c n · w = 1. For each index k = n , we define c k in the followingway. If w k = 0 then we let c k be the vector having w − k as the k th entryand − w − n as the n th entry. If w k = 0 then we let c k be the vector having 1as the k th entry and zero elsewhere. Hence, { J − ( c ) , . . . , J − ( c N ) } formsa basis for X ∗ and { J − ( c ) , . . . , J − ( c n − ) , J − ( c n +1 ) , . . . , J − ( c N ) } forms a basis for S ∗ ( w ).Now write ψ = ψ J − ( c ) + · · · + ψ N J − ( c N ) and note that ψ ( w ) = ψ n .Therefore, U v ( w , ψ ) = inf { ν v ( ψ − φ ) : φ ∈ S ∗ ( w ) } = inf { ν v ( ψ J − ( c ) + · · · + ψ N J − ( c N ) − φ ) : φ ∈ S ∗ ( w ) } = inf { ν v ( ψ n J − ( c n ) − ψ n φ ) : φ ∈ S ∗ ( w ) } = | ψ n | v · U v ( w , J − ( c n ))= | ψ ( w ) | v · U v ( w , J − ( c n ))Next, we observe that U v ( w , J − ( c n )) ≤ ν v ( J − ( c n ))= sup {| c n · z | v : H v ( z ) ≤ } = | w n | − v = H v ( w ) − . We have found that U v ( w , ψ ) ≤ | ψ ( w ) | v · H v ( w ) − and the result follows from (3.1). (cid:3) In order to generalize Lemma 3.1 to include the height on subspaces ratherthan simply the projective height, we must now consider the M th exteriorpower ∧ M (Ω Nv ). We define the index set I M = { I ⊂ { , , . . . , N } : | I | = M } . If { e , . . . , e N } is the standard basis for Ω Nv , we obtain a natural basis(3.2) (^ i ∈ I e i : I ∈ I ) STIMATING HEIGHTS USING AUXILIARY FUNCTIONS 9 for ∧ M (Ω Nv ) over Ω v . The height of an element x ∈ ∧ M (Ω Nv ) is computedusing the basis (3.2). For φ belonging to the dual ( ∧ M (Ω Nv )) ∗ , the norm of φ is given by ν v ( φ ) = sup {| φ ( x ) | v : x ∈ ∧ M (Ω Nv ) , H v ( x ) ≤ } . If w ∈ ∧ M (Ω Nv ) then U v ( w , ψ ) = inf { ν v ( ψ − φ ) : φ ∈ ( ∧ M (Ω Nv )) ∗ , φ ( w ) = 0 } . We also obtain the following lemma showing that a surjective linear trans-formation Ψ : Ω Nv → Ω Mv may be viewed as a map on ∧ M (Ω Nv ). Lemma 3.2.
Suppose that
Ψ : Ω Nv → Ω Mv is a surjective linear transforma-tion. Then there exists a unique linear transformation ∧ M (Ψ) : ∧ M (Ω Nv ) → Ω v such that ∧ M (Ψ)( w ∧ · · · ∧ w M ) = det Ψ( x ) ... Ψ( x M ) for all w , . . . , w M ∈ Ω Nv .Proof. Let M M × M (Ω v ) denote the vector space of M × M matrices withentries in Ω v . We note that Ψ induces a unique M -multilinear map Ψ ′ :(Ω Nv ) M → M M × M (Ω v ) given byΨ ′ ( w , . . . , w M ) = Ψ( x )...Ψ( x M ) . Furthermore, it is well-known (see, for example, [4], p. 437) that the deter-minant map det : M M × M (Ω v ) → Ω v defines an M -multilinear map on therows of the elements in M M × M (Ω v ). Hence, we conclude that the composi-tion det ◦ Ψ ′ is an M -multilinear map from (Ω Nv ) M to Ω v . Moreover, if thereexist i = j with w i = w j thendet ◦ Ψ ′ ( w , . . . , w M ) = 0It follows that det ◦ Ψ ′ is, in fact, an alternating M -multilinear map.By the universal property for alternating M -tensors, there exists a uniquelinear transformation T : ∧ M (Ω Nv ) → Ω v such that T ◦ ι = det ◦ Ψ ′ where ι : (Ω Nv ) M → ∧ M (Ω Nv ) is given by ι ( w , . . . , w M ) = w ∧ · · · ∧ w M . Therefore, we conlude that T ( w ∧ · · · ∧ w M ) = T ( ι ( w , . . . , w M ))= det(Ψ ′ ( w , . . . , w M ))= det Ψ( x )...Ψ( x M ) . By taking ∧ M (Ψ) = T we complete the proof. (cid:3) We now assume that W is an M -dimensional subspace of Q N and Ψ : Q N → Q M is a surjective linear transformation. Select a basis { w , . . . , w M } for W and assume that K is a number field containing the entries of eachbasis element w m as well as the entries of Ψ. We note that the height of W is given by H ( W ) = Y v H v ( w ∧ · · · ∧ w M )where the product is taken over all places v of K . As we noted in our intro-duction, the product formula implies that this definition does not dependon the choice of basis for W . By Lemma 3.2 we may define(3.3) U ( W, Ψ) = Y v U v ( w ∧ · · · ∧ w M , ∧ M (Ψ)) . Lemma 3.1 shows that this product is indeed finite and, by the way we havenormalized our absolute values, it does not depend on K . As in the heighton subspaces, the product formula implies that (3.3) is independent of thebasis for W as well. We may now state and prove our main result. Theorem 3.3. If W is an M -dimensional subspace of Q N then H ( W ) · U ( W, Ψ) holds for all surjective linear transformations Ψ : Q N → Q M with W ∩ ker Ψ = { } .Proof. Let { w , . . . , w M } be a basis for W and let K be a number fieldcontaining the entries of each basis element w m and the entries of Ψ. Hence, w m ∈ Ω Nv and Ψ : Ω Nv → Ω Mv for all places v of K . Therefore, Lemma 3.1implies that | ∧ M (Ψ)( w ∧ · · · ∧ w M ) | v = H v ( w ∧ · · · ∧ w M ) · U v ( w ∧ · · · ∧ w M , ∧ M (Ψ)) . (3.4)By Lemma 3.2 we have that ∧ M (Ψ)( w ∧ · · · ∧ w M ) = det Ψ( w )...Ψ( w M ) . STIMATING HEIGHTS USING AUXILIARY FUNCTIONS 11
Since W ∩ ker Ψ = { } we know that the rows in the above matrix arelinearly independent so that its determinant is non-zero. Hence, the lefthand side of (3.4) is non-zero and we may apply the product formula. Thedesired identity follows immediately. (cid:3) It is natural to consider the special case of Theorem 3.3 in which W is aone dimensional subspace spanned by an element w ∈ Q N . For ψ ∈ ( Q N ) ∗ we define U ( w , ψ ) = Y v U v ( w , ψ )and obtain the following corollary. Corollary 3.4. If w ∈ Q N then H ( w ) · U ( w , ψ ) for all ψ ∈ ( Q N ) ∗ with ψ ( w ) = 0 .Proof. If W is the one dimensional subspace spanned by w then it is easy tosee that H ( W ) = H ( w ). Furthermore, ψ : Q N → Q is a linear transforma-tion and U ( W, ψ ) = U ( w , ψ ). Theorem 3.3 yields that 1 = H ( W ) · U ( W, ψ )and the result follows immediately. (cid:3)
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