On multiplicatively dependent vectors of algebraic numbers
Francesco Pappalardi, Min Sha, Igor E. Shparlinski, Cameron L. Stewart
aa r X i v : . [ m a t h . N T ] J un ON MULTIPLICATIVELY DEPENDENT VECTORS OFALGEBRAIC NUMBERS
FRANCESCO PAPPALARDI, MIN SHA, IGOR E. SHPARLINSKI,AND CAMERON L. STEWART
Abstract.
In this paper, we give several asymptotic formulas forthe number of multiplicatively dependent vectors of algebraic num-bers of fixed degree, or within a fixed number field, and boundedheight. Introduction
Background.
Let n be a positive integer, G be a multiplicativegroup and let ννν = ( ν , . . . , ν n ) be in G n . We say that ννν is multiplica-tively dependent if there is a non-zero vector k = ( k , . . . , k n ) ∈ Z n forwhich(1.1) ννν k = ν k · · · ν k n n = 1 . We denote by M n ( G ) the set of multiplicatively dependent vectors in G n .For instance, the set M n ( C ∗ ) of multiplicatively dependent vectorsin ( C ∗ ) n is of Lebesgue measure zero, since it is a countable union of setsof measure zero. Further, if we fix an exponent vector k the subvarietyof ( C ∗ ) n determined by (1.1) is an algebraic subgroup of ( C ∗ ) n .For multiplicatively dependent vectors of algebraic numbers thereare two kinds of questions which have been extensively studied. Thefirst question concerns the exponents in (1.1). Given a multiplica-tively dependent vector ννν it follows from the work of Loxton and vander Poorten [14, 21], Matveev [18], and Loher and Masser [13, Corol-lary 3.2] (attributed to K. Yu) that there is a relation of the form (1.1)with a non-zero vector k with small coordinates. The second questionis to find comparison relations among the heights of the coordinates.For example, Stewart [25, Theorem 1] has given an inequality for theheights of the coordinates of such a vector (of low multiplicative rank, Mathematics Subject Classification.
Key words and phrases.
Multiplicatively dependent vectors, divisors, smoothnumbers, naive height, Weil height. in the terminology of Section 1.2), and a lower bound for the sum ofthe heights of the coordinates is implied in [26].In this paper, we obtain severa asymptotic formulas for the numberof multiplicatively dependent n -tuples whose coordinates are algebraicnumbers of fixed degree, or within a fixed number field, and boundedheight. Aside from the results mentioned above, to the best of ourknowledge, this natural question has never been addressed in the liter-ature.We remark that the above question is interesting in its own right, butis also partially motivated by the works [20, 23], where multiplicativelyindependent vectors play an important role.1.2. Rank of multplicative independence.
The following notionplays a crucial role in our argument, and is also of independent interest.Let Q be an algebraic closure of the rational numbers Q . For each ννν in ( Q ∗ ) n , we define s , the multiplicative rank of ννν , in the following way.If ννν has a coordinate which is a root of unity, we put s = 0; otherwiselet s be the largest integer with 1 ≤ s ≤ n for which any s coordinatesof ννν form a multiplicatively independent vector. Notice that(1.2) 0 ≤ s ≤ n − , whenever ννν is multiplicatively dependent.1.3. Conventions and notation.
For any algebraic number α , let f ( x ) = a d x d + · · · + a x + a be the minimal polynomial of α over the integers Z (so with content 1and positive leading coefficient). Suppose that f is factored as f ( x ) = a d ( x − α ) · · · ( x − α d )over the complex numbers C . The naive height H ( α ) of α is given byH ( α ) = max {| a d | , . . . , | a | , | a |} , and H( α ), the height of α , also known as the absolute Weil height of α , is defined by H( α ) = a d d Y i =1 max { , | α i |} ! /d . Let K be a number field of degree d (over Q ). We use the followingstandard notation: • r and r for the number of real and non-real embeddings of K ,respectively, and put r = r + r − ULTIPLICATIVELY DEPENDENT VECTORS 3 • D, h, R and ζ K for the discriminant, class number, regulatorand Dedekind zeta function of K , respectively; • w for the number of roots of unity in K .Note that r is exactly the rank of the unit group of the ring of algebraicintegers of K . As usual, let ζ ( s ) be the Riemann zeta function.For any real number x , let ⌈ x ⌉ denote the smallest integer greaterthan or equal to x , and let ⌊ x ⌋ denote the greatest integer less than orequal to x .We always implicitly assume that H is large enough, in particular sothat the logarithmic expressions log H and log log H are well-defined.In the sequel, we use the Landau symbols O and o and the Vino-gradov symbol ≪ . We recall that the assertions U = O ( V ) and U ≪ V are both equivalent to the inequality | U | ≤ cV with some positive con-stant c , while U = o ( V ) means that U/V →
0. We also use theasymptotic notation ∼ .For a finite set S we use | S | to denote its cardinality.Throughout the paper, the implied constants in the symbols O and ≪ only depend on the given number field K , the given degree d , or thedimension n .1.4. Counting vectors within a number field.
Let K be a numberfield of degree d . Denote the set of algebraic integers of K of height atmost H by B K ( H ) and the set of algebraic numbers of K of height atmost H by B ∗ K ( H ). Set B K ( H ) = |B K ( H ) | and B ∗ K ( H ) = |B ∗ K ( H ) | . Put C ( K ) = 2 r (2 π ) r d r | D | / r ! . It follows directly from the work of Widmer [29, Theorem 1.1] (taking n = e = 1 there) that(1.3) B K ( H ) = C ( K ) H d (log H ) r + O (cid:0) H d (log H ) r − (cid:1) . If r = 0, then (1.3) can be improved to (see [2, Theorem 1.1])(1.4) B K ( H ) = C ( K ) H d + O ( H d − ) . We remark that the estimate in (1.3) is stated in [12, Chapter 3, The-orem 5.2] without the explicit constant C ( K ), and moreover Bar-roero [3] has obtained similar estimates for the number of algebraic S -integers with fixed degree and bounded height.Define C ( K ) = 2 r (2 π ) r r hR | D | wζ K (2) . F. PAPPALARDI, M. SHA, I. E. SHPARLINSKI, AND C. L. STEWART
Schanuel [22, Corollary to Theorem 3] proved in 1979 (see also [17,Equation (1.5)]) that(1.5) B ∗ K ( H ) = C ( K ) H d + O (cid:0) H d − (log H ) σ ( d ) (cid:1) , where σ (1) = 1 and σ ( d ) = 0 for d >
1. Note that the height in [22] isour height to the power d .For any positive integer n , we denote by L n,K ( H ) the number ofmultiplicatively dependent n -tuples whose coordinates are algebraicintegers of height at most H , and we denote by L ∗ n,K ( H ) the numberof multiplicatively dependent n -tuples whose coordinates are algebraicnumbers of height at most H .Put C ( n, K ) = n ( n + 1)2 wC ( K ) n − . Theorem 1.1.
Let K be a number field of degree d over Q and let n be an integer with n ≥ . We have L n,K ( H ) = C ( n, K ) H d ( n − (log H ) r ( n − + O (cid:0) H d ( n − (log H ) r ( n − − (cid:1) ;(1.6) if furthermore K = Q or is an imaginary quadratic field, we have (1.7) L n,K ( H ) = C ( n, K ) H d ( n − + O (cid:0) H d ( n − / (cid:1) . We remark that when K = Q a better error term than that givenin (1.7) is stated in Theorem 1.4 below, more precisely, see (1.16).We estimate L ∗ n,K ( H ) next. Put C ( n, K ) = n wC ( K ) n − . Theorem 1.2.
Let K be a number field of degree d , and let n be aninteger with n ≥ . Then, we have (1.8) L ∗ n,K ( H ) = C ( n, K ) H d ( n − + O (cid:0) H d ( n − − g ( H ) (cid:1) , where g ( H ) = log H if d = 1 and n = 2exp( c log H/ log log H ) if d = 1 and n > if d > and n ≥ , and c is a positive number depending only on n . We now outline the strategy of the proofs. Given a number field K ,we define L n,K,s ( H ) and L ∗ n,K,s ( H ) to be the number of multiplicatively ULTIPLICATIVELY DEPENDENT VECTORS 5 dependent n -tuples of multiplicative rank s whose coordinates are alge-braic integers in B K ( H ) and algebraic numbers in B ∗ K ( H ) respectively.It follows from (1.2) that(1.9) L n,K ( H ) = L n,K, ( H ) + · · · + L n,K,n − ( H ) L ∗ n,K ( H ) = L ∗ n,K, ( H ) + · · · + L ∗ n,K,n − ( H ) . The main term in (1.6) comes from the contributions of L n,K, ( H ) and L n,K, ( H ) in (1.9), and the main term in Theorem 1.2 comes fromthe contributions of L ∗ n,K, ( H ) and L ∗ n,K, ( H ) in (1.9). To prove Theo-rems 1.1 and 1.2, we make use of (1.9) and the following result. Proposition 1.3.
Let K be a number field of degree d . Let n and s be integers with n ≥ and ≤ s ≤ n − . Then, there exist positivenumbers c and c which depend on n and K , such that (1.10) L n,K,s ( H ) < H d ( n − − d ( ⌈ ( s +1) / ⌉− exp( c log H/ log log H ) and (1.11) L ∗ n,K,s ( H ) < H d ( n − − d ( ⌈ ( s +1) / ⌉− exp( c log H/ log log H ) . In Section 5, we show that when K = Q and s = n − c log H/ log log H ) replaced by a quantity which is o ((log H ) ( k − ), where n = 2 k .1.5. Counting vectors of fixed degree.
Let d be a positive integer,and let A d ( H ), respectively A ∗ d ( H ), be the set of algebraic integers ofdegree d (over Q ), respectively algebraic numbers of degree d , of heightat most H . We set A d ( H ) = |A d ( H ) | and A ∗ d ( H ) = |A ∗ d ( H ) | . Put C ( d ) = d d ⌊ ( d − / ⌋ Y j =1 d (2 j ) d − j − (2 j + 1) d − j and C ( d ) = d d ζ ( d + 1) ⌊ ( d − / ⌋ Y j =1 ( d + 1)(2 j ) d − j (2 j + 1) d − j +1 . It follows from the work of Barroero [2, Theorem 1.1] that (see also [2,Equation (1.2)] for a previous estimate with a weaker error term whichfollows from [6, Theorem 6])(1.12) A d ( H ) = C ( d ) H d + O (cid:0) H d ( d − (log H ) ρ ( d ) (cid:1) , F. PAPPALARDI, M. SHA, I. E. SHPARLINSKI, AND C. L. STEWART where ρ (2) = 1 and ρ ( d ) = 0 for any d = 2.Further, Masser and Vaaler [16, Equation (7)] have shown that (seealso [17, Equation (1.5)])(1.13) A ∗ d ( H ) = C ( d ) H d ( d +1) + O (cid:16) H d (log H ) ϑ ( d ) (cid:17) , where ϑ (1) = ϑ (2) = 1 and ϑ ( d ) = 0 for any d ≥ n , we denote by M n,d ( H ) the number of mul-tiplicatively dependent n -tuples whose coordinates are algebraic inte-gers in A d ( H ), and we denote by M ∗ n,d ( H ) the number of multiplica-tively dependent n -tuples whose coordinates are algebraic numbers in A ∗ d ( H ).For each positive integer d , we define w ( d ) to be the number ofroots of unity of degree d . Let ϕ denote Euler’s totient function. Since ϕ ( k ) ≫ k/ log log k for any integer k ≥
3, it follows that(1.14) w ( d ) ≪ d log log d, where d ≥ w ( d ) can be zero, such as for an odd integer d > n and d , we define C ( n, d ) and C ( n, d ) as C ( n, d ) = ( nw ( d ) + n ( n − C ( d ) n − and C ( n, d ) = ( nw ( d ) + 2 n ( n − C ( d ) n − . Theorem 1.4.
Let d and n be positive integers with n ≥ . Then, thefollowing hold. (i) We have (1.15) M n,d ( H ) = C ( n, d ) H d ( n − + O (cid:16) H d ( n − − d/ (cid:17) ; furthermore if d = 2 or d is odd, we have M n,d ( H ) = C ( n, d ) H d ( n − + O (cid:16) H d ( n − − d exp( c log H/ log log H ) (cid:17) (1.16) and (1.17) M ,d ( H ) = C (2 , d ) H d + O (cid:16) H d − d (log H ) ρ ( d ) (cid:17) , where c is a positive number which depends only on n and d ,and ρ ( d ) has been defined in (1.12) . ULTIPLICATIVELY DEPENDENT VECTORS 7 (ii)
We have (1.18) M ∗ n,d ( H ) = C ( n, d ) H d ( d +1)( n − + O (cid:0) H d ( d +1)( n − − d/ log H (cid:1) ; furthermore if d = 2 or d is odd, we have M ∗ n,d ( H ) = C ( n, d ) H d ( d +1)( n − + O (cid:0) H d ( d +1)( n − − d exp( c log H/ log log H ) (cid:1) (1.19) and (1.20) M ∗ ,d ( H ) = C (2 , d ) H d ( d +1) + O (cid:16) H d (log H ) ϑ ( d ) (cid:17) , where c is a positive number which depends only on n and d ,and ϑ ( d ) is defined in (1.13) . We remark that the case when d = 1 actually has been included inTheorems 1.1 and 1.2. However, in this case the error term in (1.16) is H n − o (1) , which is better than that in (1.7) taken with d = 1.The strategy to prove Theorem 1.4 is similar to that in proving The-orems 1.1 and 1.2. For each integer s with 0 ≤ s ≤ n −
1, we define M n,d,s ( H ) and M ∗ n,d,s ( H ) to be the number of multiplicatively depen-dent n -tuples of multiplicative rank s whose coordinates are algebraicintegers in A d ( H ) and algebraic numbers in A ∗ d ( H ) respectively. Justas in (1.9) we have(1.21) M n,d ( H ) = M n,d, ( H ) + · · · + M n,d,n − ( H ) M ∗ n,d ( H ) = M ∗ n,d, ( H ) + · · · + M ∗ n,d,n − ( H ) . For the proof of Theorem 1.4, we make use of (1.21) and the followingresult.
Proposition 1.5.
Let d , n and s be integers with d ≥ , n ≥ and ≤ s ≤ n − . Then, there exist positive numbers c and c , whichdepend on n and d , such that (1.22) M n,d,s ( H ) < H d ( n − − d ( ⌈ ( s +1) / ⌉− exp( c log H/ log log H ) and M ∗ n,d,s ( H ) < H d ( d +1)( n − − d ( ⌈ ( s +1) / ⌉− exp( c log H/ log log H ) . (1.23)We remark that the estimate (1.22) yields an improvement on theupper bound of H d ( n − and (1.23) yields an improvement of the upperbound H d ( d +1)( n − for s at least 2. F. PAPPALARDI, M. SHA, I. E. SHPARLINSKI, AND C. L. STEWART Preliminaries
Weil height.
We first record a well-known result about the ab-solute Weil height; see [12, Chapter 3].
Lemma 2.1.
Let α be a non-zero algebraic number, and let k be aninteger. Then H( α k ) = H( α ) | k | . Proof.
This follows from the product formula and the fact thatH( α ) = Y v max { , | α | v } , where the product is taken over all inequivalent valuations v appropri-ately normalized, see for example [12, Chapter 3, § (cid:3) Next we need a result that allows us to compare the naive height H and the absolute Weil height H. Lemma 2.2.
Let α be an algebraic number of degree d . Then H ( α ) ≤ (2H( α )) d . Proof.
This follows from noticing that the coefficients of the minimalpolynomial f of α can be expressed in terms of elementary symmetricpolynomials in the roots of f ; see for example [15, Equation (6)]. (cid:3) For the proofs of Theorems 1.1 and 1.2, we also need the followingresult.
Lemma 2.3.
Let α be an algebraic number of degree d , and let a bethe leading coefficient of the minimal polynomial of α over the integers.Then H( aα ) ≤ d − H( α ) d . Proof.
By definition, we haveH( α ) = a d Y i =1 max { , | α i |} ! /d , where α , . . . , α d are the roots of the minimal polynomial of α . Then, aα is an algebraic integer, andH( aα ) = d Y i =1 max { , | aα i |} ! /d . ULTIPLICATIVELY DEPENDENT VECTORS 9
Thus H( aα ) d ≤ a d d Y i =1 max { , | α i |} = a d − H( α ) d , which, together with Lemma 2.2, implies thatH( aα ) d ≤ (2H( α )) d ( d − H( α ) d = 2 d ( d − H( α ) d , and so H( aα ) ≤ d − H( α ) d as required. (cid:3) Multiplicative structure of algebraic numbers.
Let K bea number field, and let H be a positive real number. We denote by U K ( H ) the number of units in the ring of algebraic integers of K ofheight at most H . Lemma 2.4.
Let K be a number field, and let r be the rank of the unitgroup as defined in Section 1.3. Then, there exists a positive number c , depending on K , such that U K ( H ) < c (log H ) r . Proof.
This is [12, Part (ii) of Theorem 5.2 of Chapter 3]. (cid:3)
The next result shows that if algebraic numbers α , . . . , α n are mul-tiplicatively dependent, then we can find a relation as (1.1), where theexponents are not too large. Such a result has found application intranscendence theory, see for example [1, 18, 21, 24]. Lemma 2.5.
Let n ≥ , and let α , . . . , α n be multiplicatively depen-dent non-zero algebraic numbers of degree at most d and height at most H . Then, there is a positive number c , which depends only on n and d , and there are rational integers k , . . . , k n , not all zero, such that α k · · · α k n n = 1 and max ≤ i ≤ n | k i | < c (log H ) n − . Proof.
This follows from [21, Theorem 1]. For an explicit constant c ,we refer to [13, Corollary 3.2]. (cid:3) Let x and y be positive real numbers with y larger than 2, and let ψ ( x, y ) denote the number of positive integers not exceeding x whichcontain no prime factors greater than y . Put Z = (cid:18) log (cid:18) y log x (cid:19)(cid:19) log x log y + (cid:18) log (cid:18) xy (cid:19)(cid:19) y log y and u = (log x ) / (log y ) . Lemma 2.6.
For < y ≤ x , we have ψ ( x, y )= exp (cid:0) Z (cid:0) O ((log y ) − ) + O ((log log x ) − ) + O (( u + 1) − ) (cid:1)(cid:1) . Proof.
This is [4, Theorem 1]. (cid:3)
Counting special algebraic numbers.
In this section, we counttwo special kinds of algebraic numbers.
Lemma 2.7.
Let K be a number field of degree d , and let u and v be non-zero integers with u > . Then, there is a positive number c , which depends on K , such that the number of elements α in K ofheight at most H , whose minimal polynomial has leading coefficient u and constant coefficient v , is at most exp( c log H/ log log H ) . Proof.
Let c , c , . . . denote positive numbers depending on K . Let N K/ Q be the norm function from K to Q . Suppose that α is an ele-ment of K of height at most H whose minimal polynomial has leadingcoefficient u and constant coefficient v . Then, we see that uα is analgebraic integer in K , and N K/ Q ( α ) = ( − d v/u and N K/ Q ( uα ) = ( − d u d − v. By Lemma 2.3, we further have H( uα ) ≤ d − H d . Note that u isfixed, so the number of such α does not exceed the number of algebraicintegers β ∈ K of height at most 2 d − H d and satisfying(2.1) N K/ Q ( β ) = ( − d u d − v. We say that two algebraic integers β and β in K are equivalent if theprincipal integral ideals h β i and h β i are equal. We note that, using [5,Chapter 3, Equation (7.8)], the number E of equivalence classes ofsolutions of (2.1) is at most τ ( | u d − v | ) d , where, for any positive integer k , τ ( k ) denotes the number of positive integers which divide k . ByWigert’s Theorem, see [11, Theorem 317],(2.2) E < exp ( c log(3 | uv | ) / log log(3 | uv | )) . Further by Lemma 2.2 u and v are at most (2 H ) d in absolute value,hence(2.3) E < exp( c log H/ log log H ) . ULTIPLICATIVELY DEPENDENT VECTORS 11
Besides, if two solutions β and β of (2.1) are equivalent, then β /β is a unit η in the ring of algebraic integers of K . ButH( η ) ≤ H( β )H(( β ) − ) ≤ d − H d . By Lemma 2.4 the number of such units is at most(2.4) U K (2 d − H d ) ≤ c (log H ) r . Our result now follows from (2.3) and (2.4). (cid:3)
We remark that if we set u = 1, then Lemma 2.7 gives an upperbound for the number of algebraic integers in K of norm ± v and ofheight at most H .Given integer d ≥
1, let C ∗ d ( H ) be the set of algebraic numbers α ofdegree d and height at most H such that αη is also of degree d for someroot of unity η = ±
1, and let C d ( H ) be the set of algebraic integerscontained in C ∗ d ( H ). Here, we want to estimate the sizes of C d ( H ) and C ∗ d ( H ).For this we need some preparations. Given a polynomial f = a d X d + · · · + a X + a ∈ Q [ X ] of degree d , we call it degenerate if it has twodistinct roots whose quotient is a root of unity. Besides, we define its height as H( f ) = max {| a d | , . . . , | a | , | a |} , and we denote by G f the Galois group of the splitting field of f over Q . Let S d be the full symmetric group of d symbols.Define E d ( H ) = { monic f ∈ Z [ X ] of degree d : H( f ) ≤ H and G f = S d } and E ∗ d ( H ) = { f ∈ Z [ X ] of degree d : H( f ) ≤ H and G f = S d } . The study of the sizes of E d ( H ) and E ∗ d ( H ) was initiated by van derWaerden [27]. Here, we recall a recent result due to Dietmann [8,Theorem 1]:(2.5) |E d ( H ) | ≪ H d − / . Besides, by a result of Cohen [7, Theorem 1] (taking K = Q , s = n + 1and r = 1 there), we directly have(2.6) |E ∗ d ( H ) | ≪ H d +1 / log H. We also put F d ( H ) = { monic f ∈ Z [ X ] of degree d : H( f ) ≤ H , f is degenerate } and F ∗ d ( H ) = { f ∈ Z [ X ] of degree d : H( f ) ≤ H , f is degenerate } . Applying [10, Theorems 1 and 4], we have(2.7) |F d ( H ) | ≪ H d − and |F ∗ d ( H ) | ≪ H d . We are now ready to prove the following lemma.
Lemma 2.8.
We have: (i) for any integer d ≥ , |C d ( H ) | ≪ H d ( d − / and |C ∗ d ( H ) | ≪ H d ( d +1 / log H ;(ii) for d = 2 or for d odd, |C d ( H ) | ≪ H d ( d − and |C ∗ d ( H ) | ≪ H d . Proof.
Pick an arbitrary element α ∈ C d ( H ). We let f be its minimalpolynomial over Z , and let the d roots of f be α , . . . , α d with α = α .Since α is of height at most H , by Lemma 2.2 we haveH( f ) ≤ (2 H ) d . By definition, there is a root of unity η = ± αη is alsoof degree d . If η ∈ Q ( α ), then under an isomorphism sending α to α i , η is mapped to one of its conjugates η i in Q ( α i ), which impliesthat η ∈ Q ( α i ) for any 1 ≤ i ≤ d . Indeed, the image η i of η in Q ( α i )multiplicatively generates the same group as η , and thus η is a power of η i , so η ∈ Q ( α i ). Hence, T di =1 Q ( α i ) = Q , then we must have G f = S d ,that is,(2.8) f ∈ E d ((2 H ) d ) . Furthermore, since f is irreducible, in this case d = 2. We also notethat since η is of even degree ϕ ( k ), where k > η k = 1, this case does not happen when d is odd.Now, we assume that η Q ( α ). Let K = Q ( η, α , . . . , α d ), and let G be the Galois group Gal( K/ Q ), where K is indeed a Galois extensionover Q . We construct a disjoint union G = S di =1 G i , where G i = { φ ∈ G : φ ( α ) = α i } . So, for each 1 ≤ i ≤ dG i αη = { φ ( αη ) : φ ∈ G i } = { α i φ ( η ) : φ ∈ G i } . Since αη is of degree d , we have(2.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d [ i =1 G i αη (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d. ULTIPLICATIVELY DEPENDENT VECTORS 13
Note that α = α , then G = Gal( K/ Q ( α )). Since η Q ( α ), thereexist two morphisms φ , φ ∈ G such that φ ( η ) = φ ( η ). That is, | G αη | ≥
2. Trivially, | G i αη | ≥ ≤ i ≤ d . We now see from (2.9)that there are two distinct indices i, j such that G i αη ∩ G j αη = ∅ ,which implies that α i /α j is a root of unity and thus f is degenerate,that is,(2.10) f ∈ F d ((2 H ) d ) . Hence, if α ∈ C d ( H ), then combing (2.8) and (2.10) with (2.5)and (2.7), respectively, we derive the first inequality in (i). If d = 2 or d is odd, by the above discussion we always have (2.10), and thus thefirst inequality in (ii) follows from (2.7). Similar arguments also applyto estimate |C ∗ d ( H ) | by using (2.6) and (2.7). (cid:3) Proofs of Propositions 1.3 and 1.5
Proof of Proposition 1.3.
Let c , c , . . . denote positive num-bers depending on n and K . Let ννν = ( ν , . . . , ν n ) be a multiplicativelydependent vector of multiplicative rank s whose coordinates are from K and have height at most H . Set m = s + 1. Then, there are m distinct integers j , . . . , j m from { , . . . , n } for which ν j , . . . , ν j m aremultiplicatively dependent and there are non-zero integers k j , . . . , k j m for which(3.1) ν k j j · · · ν k jm j m = 1 , and further by Lemma 2.5, we can assume that(3.2) max {| k j | , . . . , | k j m |} < c (log H ) m − . Let P be the set of indices i for which k i is positive, and let N be theset of indices i for which k i is negative. Then(3.3) Y i ∈ P ν k i i = Y i ∈ N ν − k i i . Plainly, either | P | or | N | is at least ⌈ m/ ⌉ .Let I = { j , . . . , j m } , and let I be the subset of I consisting of theindices i for which k i is positive if | P | ≥ ⌈ m/ ⌉ , and otherwise let I be the subset of I consisting of the indices i for which k i is negative.Note that(3.4) | I | ≥ l m m . It follows from (3.3) that(3.5) Y i ∈ I ν | k i | i = Y i ∈ I \ I ν | k i | i . For each coordinate ν i , i ∈ I , let a i be the leading coefficient ofthe minimal polynomial of ν i over the integers. Note that a i ν i is analgebraic integer, and we can rewrite (3.5) as(3.6) Y i ∈ I ( a i ν i ) | k i | = Y i ∈ I a | k i | i Y i ∈ I \ I ν | k i | i . We first establish (1.10). Accordingly, we fix non-zero algebraic in-tegers ν i ∈ B K ( H ) for i from { , . . . , n }\ I and estimate the numberof solutions of (3.5) in algebraic integers ν i , i ∈ I , from B K ( H ). Ob-serve that the number of cases when we consider an equation of theform (3.5) is, by (3.2), at most (cid:18) nm (cid:19) (cid:0) c (log H ) ( m − (cid:1) m B K ( H ) n −| I | , and, by (1.3) and (3.4), is at most(3.7) c H d ( n −⌈ m/ ⌉ ) (log H ) c . Let q , . . . , q t be the primes which divide Y i ∈ I \ I N K/ Q ( ν i ) , where N K/ Q is the norm from K to Q . Since the height of ν i is at most H , it follows from Lemma 2.2 that(3.8) | N K/ Q ( ν i ) | ≤ (2 H ) d , i = 1 , , . . . , n, and since | I \ I | ≤ n , we see that(3.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y i ∈ I \ I N K/ Q ( ν i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 H ) dn . Let p , . . . , p k be the first k primes, where k satisfies p · · · p k ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y i ∈ I \ I N K/ Q ( ν i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < p · · · p k +1 . Let T denote the number of positive integers up to (2 H ) d which arecomposed only of primes from { q , . . . , q t } . We see that T is boundedfrom above by the number of positive integers up to (2 H ) d which arecomposed of primes from { p , . . . , p k } . By (3.9), we obtain X prime p ≤ p k log p ≪ log H, ULTIPLICATIVELY DEPENDENT VECTORS 15 which, combined with the prime number theorem, yields p k < c log H. Therefore we have T ≤ ψ (cid:0) (2 H ) d , c log H (cid:1) , and thus by Lemma 2.6,(3.10) T < exp( c log H/ log log H ) . It follows that if ( ν i , i ∈ I ) is a solution of (3.5), then | N K/ Q ( ν i ) | iscomposed only of primes from { q , . . . , q t } , and so N K/ Q ( ν i ) is one ofat most 2 T integers of absolute value at most (2 H ) d . Let a be one ofthose integers.By Lemma 2.7, the number of algebraic integers α from K of heightat most H for which(3.11) N K/ Q ( α ) = a is at most exp( c log H/ log log H ). Therefore, by (3.10), and (3.11),the number of | I | -tuples ( ν i , i ∈ I ) which give a solution of (3.5) isat most exp( c log H/ log log H ). Recalling m = s + 1, we see that ourbound (1.10) now follows from (3.7).We now establish (1.11). We first remark by Lemmas 2.2 and 2.3that(3.12) 0 < a i ≤ (2 H ) d and(3.13) H( a i ν i ) ≤ d − H d , for i = 1 , . . . , n . Moreover, without loss of generality we can assumethat I \ I is not empty. Indeed, if I \ I is empty, then we can replacean arbitrary coordinate ν i , i ∈ I , by its inverse ν − i .In view of (3.6), we proceed by fixing a i for i in I and ν i for i in { , . . . , n }\ I . Since I \ I is non-empty, say that it contains i . Wefurther fix ν i for i in I \ I with i = i , and then the correspondingleading coefficient a i is also fixed. Let β = Y i ∈ I a | k i | i Y i ∈ I \ I i = i ( a i ν i ) | k i | , which is actually a fixed non-zero algebraic integer, then N K/ Q ( β ) isa fixed non-zero integer. Note that the left-hand side of (3.6) is analgebraic integer, so βν i is an algebraic integer, and then N K/ Q ( βν i ) is also an algebraic integer. Thus, the leading coefficient a i divides N K/ Q ( β ). It follows that the prime factors of a i divide Y i ∈ I a i Y i ∈ I \ I i = i N K/ Q ( a i ν i ) . Since the heights of ν , . . . , ν n are at most H , we see, as in the proofof the estimate (3.10), that there are at most exp( c log H/ log log H )possibilities for the leading coefficient a i . Note that by Lemma 2.2there are at most 2(2 H ) d possibilities for the constant coefficient of theminimal polynomial of ν i . Thus, by Lemma 2.7, there are at most(3.14) H d exp( c log H/ log log H )possible values of ν i that we need to consider. In total we have,by (1.5), (3.12) and (3.14), at most (cid:18) nm (cid:19) (cid:0) c (log H ) ( m − (cid:1) m (2 H ) d | I | H d ( n −| I |− H d exp( c log H/ log log H )equations of the form (3.6). Since | I | ≥ ⌈ m ⌉ , the number of suchequations is at most(3.15) H dn − d ( ⌈ m ⌉ +1) exp( c log H/ log log H ) . Let us put(3.16) γ = Y i ∈ I a | k i | i Y i ∈ I \ I ( a i ν i ) | k i | and γ = Y i ∈ I \ I a | k i | i . Notice that once ν i is fixed for i in I \ I , so is a i and thus γ is fixed.Then, (3.6) can be rewritten as(3.17) γ Y i ∈ I ( a i ν i ) | k i | = γ , and we seek an estimate for the number of solutions of (3.17) in alge-braic numbers ν i from B ∗ K ( H ) with leading coefficient a i for i ∈ I .Note that γ is an algebraic integer and γ is an integer. Let q , . . . , q t be the prime factors of Y i ∈ I a i Y i ∈ I \ I N K/ Q ( a i ν i ) . ULTIPLICATIVELY DEPENDENT VECTORS 17
Then, by (3.16) and (3.17), for each index i ∈ I the prime factorsof N K/ Q ( a i ν i ) are from { q , . . . , q t } . It follows from (3.12), (3.13) andLemma 2.2 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y i ∈ I a i Y i ∈ I \ I N K/ Q ( a i ν i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 H ) d | I | (2 d H d ) d | I \ I | ≤ (2 H ) d n . We can now argue as in our proof of (1.10) that the number ofsolutions of (3.17) in algebraic integers a i ν i , i ∈ I , from K of heightat most 2 d − H d is at most exp( c log H/ log log H ). The result (1.11)now follows from (3.15).3.2. Proof of Proposition 1.5.
Let c , c , . . . denote positive num-bers depending on n and d . Notice that if ννν = ( ν , . . . , ν n ) is a mul-tiplicatively dependent vector of multiplicative rank s whose coordi-nates are from A ∗ d ( H ). Set m = s + 1. Then, there are m distinctintegers j , . . . , j m from { , . . . , n } for which ν j , . . . , ν j m are multi-plicatively dependent and there are non-zero integers k j , . . . , k j m forwhich (3.1) holds, and by Lemma 2.5, we can suppose that (3.2) holds.Let I = { j , . . . , j m } and I be defined as in the proof of Proposition 1.3,so that (3.4) and (3.5) hold.We first establish (1.22). Fixing non-zero algebraic integers ν i ∈A d ( H ) for i ∈ { , . . . , n }\ I , we want to estimate the number of solu-tions of (3.5) in algebraic integers ν i ∈ A d ( H ) for i ∈ I . The numberof cases when we consider an equation of the form (3.5) is, by (3.2), atmost (cid:18) nm (cid:19) (cid:0) c (log H ) m − (cid:1) m A d ( H ) n −| I | , which, by (1.12), is at most(3.18) c H d ( n −| I | ) (log H ) m ( m − . For each i ∈ I , by (3.5) the prime factors of N Q ( ν i ) / Q ( ν i ) divide Y j ∈ I \ I N Q ( ν j ) / Q ( ν j ) . Just as in the proof of Proposition 1.3, we can apply Lemma 2.2 andLemma 2.6 to conclude that, for i ∈ I , N Q ( ν i ) / Q ( ν i ) is one of at most T integers, where, as in (3.10), T < exp( c log H/ log log H ) . Then, estimating the number of possible choices of the minimal poly-nomial of ν i over the integers by using Lemma 2.2, we see that there are at most(3.19) d (cid:0) H ) d + 1 (cid:1) d − exp( c log H/ log log H )possible values of each ν i for i ∈ I . We now fix | I | − ν i with i in I . Let i ∈ I denote the index of the term which is notfixed. Then, ν i is a solution of(3.20) x | k i | = η , where η = Y i ∈ I i = i ν −| k i | i Y i ∈ I \ I ν | k i | i . If ν i and µ i are two solutions of (3.20) from A d ( H ), then ν i /µ i is a | k i | -th root of unity. But the degree of ν i /µ i is at most d , and sothere are at most c possibilities for ν i /µ i when d is fixed. It followsfrom (3.19) that each equation (3.5) has at most(3.21) H d ( d − | I |− exp( c log H/ log log H )solutions. Thus by (3.18) and (3.21), we have(3.22) M n,d,s ( H ) < H d ( n −| I | )+ d ( d − | I |− exp( c log H/ log log H ) . Further, by (3.4),(3.23) d ( n − | I | ) + d ( d − | I | − ≤ d ( n − − d (cid:16)l m m − (cid:17) . Now, (1.22) follows from (3.22) and (3.23).We next establish (1.23). For each i ∈ I , let a i denote the leadingcoefficient of the minimal polynomial of ν i over the integers. Withoutloss of generality, we can assume that I \ I is not empty. Indeed, if I \ I is empty, then we can replace an arbitrary coordinate ν i , i ∈ I ,by its inverse ν − i .In view of (3.6), we proceed by first fixing positive integers a i for i ∈ I . Since I \ I is non-empty, say that it contains i . We next fix ν i for i in i ∈ { , . . . , n }\ I with i = i , and then the corresponding a i isalso fixed. Let β = Y i ∈ I a | k i | i Y i ∈ I \ I i = i ( a i ν i ) | k i | , which is a fixed non-zero algebraic integer. Notice that the left-handside of (3.6) is an algebraic integer, so βν i is also an algebraic integer, ULTIPLICATIVELY DEPENDENT VECTORS 19 and thus as in the proof of (1.11) the prime factors of the leadingcoefficient a i divide Y i ∈ I a i Y i ∈ I \ I i = i N Q ( ν i ) / Q ( a i ν i ) . Since the heights of ν , . . . , ν n are at most H and their degrees areall equal to d , we see, as in the proof of (3.10), that there are atmost exp( c log H/ log log H ) possibilities for the leading coefficient a i .Then, combining this result with Lemma 2.2, we know that the numberof the possibilities for the minimal polynomial of ν i is at most H d exp( c log H/ log log H ) . Thus, there are at most(3.24) H d exp( c log H/ log log H )possible values of ν i that we need to consider.Hence, the number of cases of the equation (3.6) to be considered is,by (3.2), (3.12) and (3.24), at most (cid:18) nm (cid:19) (cid:0) c (log H ) m − (cid:1) m (2 H ) d | I | A ∗ d ( H ) n −| I |− H d exp( c log H/ log log H ) , which, by (1.13), is at most(3.25) H d ( d +1)( n −| I |− d | I | + d exp( c log H/ log log H ) . We now estimate the number of solutions of (3.6) in algebraic num-bers ν i ∈ A ∗ d ( H ) for i ∈ I with minimal polynomial having leadingcoefficient a i . It follows from (3.6) that for each i ∈ I the primefactors of N Q ( ν i ) / Q ( a i ν i ) divide Y j ∈ I a j Y j ∈ I \ I N Q ( ν j ) / Q ( a j ν j ) . Thus, by Lemma 2.2, Lemma 2.3 and Lemma 2.6, as in the proofof (3.10), there is a set of at most T integers, where T < exp( c log H/ log log H ) , and N Q ( ν i ) / Q ( a i ν i ) belongs to that set. Since a i is fixed, the norm N Q ( ν i ) / Q ( ν i ) also belongs to a set of cardinality at most T for i ∈ I .Notice that for the minimal polynomial of ν i , i ∈ I , if N Q ( ν i ) / Q ( ν i ) isfixed, then the constant coefficient is also fixed, because the leadingcoefficient a i has already been fixed. Hence, counting possible choices of the minimal polynomial of ν i by using Lemma 2.2, we see that thereare at most(3.26) H d ( d − exp( c log H/ log log H )possible values of ν i for i ∈ I . We now fix | I | − ν i with i ∈ I and argue as before to conclude from (3.26) that eachequation (3.6) has at most(3.27) H d ( d − | I |− exp( c log H/ log log H )solutions. Thus, by (3.25) and (3.27), we obtain M ∗ n,d,s ( H ) < H d ( d +1)( n −| I |− d | I | + d + d ( d − | I |− exp( c log H/ log log H ) . (3.28)Observing that d ( d + 1)( n − | I | −
1) + d | I | + d + d ( d − | I | − d ( d + 1)( n − − d ( | I | − , our result (1.23) now follows from (3.4) and (3.28).4. Proof of Main Results
Proof of Theorem 1.1.
By (1.9) and (1.10), there is a positivenumber c which depends on n and K such that L n,K ( H ) = L n,K, ( H ) + L n,K, ( H )+ O ( H d ( n − − d exp( c log H/ log log H )) . (4.1)Each such vector ννν of multiplicative rank 0 has an index i for which ν i is a root of unity. Accordingly, we have nw ( B K ( H ) − w − n − ≤ L n,K, ( H ) ≤ nwB K ( H ) n − , and thus by (1.3) L n,K, ( H ) = nwC ( K ) n − H d ( n − (log H ) r ( n − + O (cid:0) H d ( n − (log H ) r ( n − − (cid:1) . (4.2)We next estimate L n,K, ( H ). Each such vector ννν of rank 1 has a pairof indices ( i , i ), two coordinates ν i and ν i from B K ( H ) and non-zerointegers k i and k i such that ν k i i ν k i i = 1. There are n ( n − / i , i ). By Lemma 2.5, the number of such vectors associated with twodistinct such pairs ( i , i ) and ( i , i ) is(4.3) O (cid:0) B K ( H ) n − (log H ) (cid:1) . ULTIPLICATIVELY DEPENDENT VECTORS 21
We now estimate the number of n -tuples ννν whose coordinates arefrom B K ( H ) for which ν k i i ν k i i = 1with ( k i , k i ) equal to ( t, t ) or ( t, − t ) for some non-zero integer t . Wehave ( B K ( H ) − w − n − choices for the coordinates of ννν associatedwith indices different from i and i , because they are non-zero andnot roots of unity. Also there are B K ( H ) − w − i -thcoordinate, and once it is determined, say ν i , then the i -th coordinateis of the form ην i or ην − i , where η is a root of unity from K . Notethat H( ην i ) = H( ν i ) = H( ην − i ) , and that ην − i is only counted when ν i is a unit in the ring of algebraicintegers of K . Thus, we have(4.4) ( B K ( H ) − w − n − (( B K ( H ) − w − w + ( U K ( H ) − w ) w )such vectors of rank 1 associated with ( i , i ). So, by (1.3), (4.3), (4.4)and Lemma 2.4, the number of such vectors of rank 1 associated withan exponent vector k with k i = t , k i = ± t for t a non-zero integer is n ( n − wC ( K ) n − H d ( n − (log H ) r ( n − + O (cid:0) H d ( n − (log H ) r ( n − − (cid:1) . (4.5)It remains to estimate the number of such vectors of multiplicativerank 1 associated with an exponent vector k with k i = t and k i = t with t = ± t and t and t non-zero integers. Let ν , ν ∈ B K ( H ) beassociated with t , − t respectively. In this case ν t = ν t . We first consider the case when t and t are of opposite signs. Then, ν and ν are units in the ring of algebraic integers of K , and so byLemma 2.4 the number of such vectors is(4.6) O (cid:0) (log H ) r B K ( H ) n − (cid:1) . It remains to consider the case when t and t are both positive.Without loss of generality, we assume that 0 < t < t , and also t ≪ log H by Lemma 2.5.If t = 2 t , then ν is determined by ν up to a root of unity containedin K , and also we have H( ν ) ≤ H / . So, the number of such pairs( ν , ν ) is O ( H d/ (log H ) r ) by using (1.3), and thus the number of suchvectors of rank 1 is(4.7) O (cid:0) H d/ (log H ) r B K ( H ) n − (cid:1) . If t divides t and t /t ≥
3, then we have H( ν ) ≤ H / , and so asthe above the number of such vectors of rank 1 is(4.8) O (cid:0) H d/ (log H ) r +1 B K ( H ) n − (cid:1) . Now, we assume that t does not divide t . Let t be the greatestcommon divisor of t and t . Note that t /t ≥ t /t ≥
3. Put(4.9) γ = ν t = ν t , and let β be a root of x t t − γ . Observe that β t = η ν and β t = η ν for some t t -th roots of unity η and η . There exist integers u and v with ut + vt = t , and so β t = β t u β t v = η u ν u η v ν v = ηα for η a t t -th root of unity and α an algebraic integer of K . Therefore(4.10) ( ηα ) t /t = β t = η ν , and so(4.11) H( α ) t /t = H( ν ) . Since H( ν ) ≤ H , we see, from (4.10) and (4.11), that ν is determinedup to a t t -th root of unity, by an algebraic integer of K of height atmost H t/t ≤ H / . Thus, by (1.3) and Lemma 2.5, the number of suchpairs ( ν , ν ) is O ( H d/ (log H ) r +4 ), hence the number of such vectorsof rank 1 is(4.12) O (cid:0) H d/ (log H ) r +4 B K ( H ) n − (cid:1) . Thus, by (1.3), (4.5), (4.6), (4.7), (4.8) and (4.12), we get L n,K, ( H ) = n ( n − wC ( K ) n − H d ( n − (log H ) r ( n − + O (cid:0) H d ( n − (log H ) r ( n − − (cid:1) . (4.13)The estimate (1.6) now follows from (4.1), (4.2) and (4.13).Finally, assume that K is the rational number field Q or an imaginaryquadratic field. Then, r = 0, and so B K ( H ) = C ( K ) H d + O ( H d − )by (1.4). Repeating the above process, we obtain L n,K, ( H ) = nwC ( K ) n − H d ( n − + O ( H d ( n − − )and L n,K, ( H ) = n ( n − wC ( K ) n − H d ( n − + O (cid:0) H d ( n − / (cid:1) , ULTIPLICATIVELY DEPENDENT VECTORS 23 where the second error term comes from (4.7) (and also (4.4) when d = 2). Hence, noticing (4.1) and d = 1 or 2, we obtain (1.7).4.2. Proof of Theorem 1.2.
By (1.9) and (1.11), we have L ∗ n,K ( H ) = L ∗ n,K, ( H ) + L ∗ n,K, ( H )+ O (cid:0) H d ( n − − d exp( c log H/ log log H ) (cid:1) . (4.14)As in the proof of Theorem 1.1, we obtain, by using (1.5) in placeof (1.3),(4.15) L ∗ n,K, ( H ) = nwC ( K ) n − H d ( n − + O (cid:0) H d ( n − − (log H ) σ ( d ) (cid:1) , where σ (1) = 1 and σ ( d ) = 0 for d > L ∗ n,K, ( H ) = n ( n − wC ( K ) n − H d ( n − + O (cid:0) H d ( n − − (log H ) σ ( d ) (cid:1) , (4.16)where the main difference from the proof of (4.13) is that the contri-bution from the exponent vectors ( k i , k i ) equal to ( t, t ) is the same aswhen ( k i , k i ) is equal to ( t, − t ).The desired result now follows from (4.14), (4.15) and (4.16) bynoticing that L ∗ ,K ( H ) = L ∗ ,K, ( H ) + L ∗ ,K, ( H ) . Proof of Theorem 1.4.
We first establish (1.15). By (1.21)and (1.22), we have M n,d ( H ) = M n,d, ( H ) + M n,d, ( H )+ O (cid:16) H d ( n − − d exp( c log H/ log log H ) (cid:17) . (4.17)Note that each such vector ννν of multiplicative rank 0 has a coordinatewhich is a root of unity of degree d . So, in view of the definition of w ( d ) in (1.14) we have nw ( d ) ( A d ( H ) − w ( d )) n − ≤ M n,d, ( H ) ≤ nw ( d ) A d ( H ) n − , and thus by (1.12) and (1.14), M n,d, ( H ) = nw ( d ) C ( d ) n − H d ( n − + O (cid:16) H d ( n − − d (log H ) ρ ( d ) (cid:17) . (4.18)We remark that M n,d, ( H ) = 0 if w ( d ) = 0.Moreover, arguing as in the proof of Theorem 1.1, we find that themain contribution to M n,d, ( H ) comes from vectors associated with anexponent vector k which has two non-zero components one of which is t and the other of which is ± t with t a non-zero integer. Notice thatthe number U d ( H ) of algebraic integers which are units of degree d andheight at most H satisfies (by using Lemma 2.2)(4.19) U d ( H ) = O (cid:0) H d ( d − (cid:1) . We then deduce from (1.12), (1.14), (4.19) and Lemma 2.8 that(4.20) M n,d, ( H ) = n ( n − C ( d ) n − H d ( n − + O (cid:16) H d ( n − − d/ (cid:17) ;if furthermore d = 2 or d is odd, then(4.21) M n,d, ( H ) = n ( n − C ( d ) n − H d ( n − + O (cid:16) H d ( n − − d log H (cid:17) . Here, we need to note that for an algebraic integer α of degree d and aroot of unity η = ± αη might not be of degree d .The desired asymptotic formula (1.15) now follows from (4.17), (4.18)and (4.20). In order to show (1.16), we use (4.21) instead of (4.20).Besides, (1.17) follows from (4.18) and (4.21) by noticing that M ,d ( H ) = M ,d, ( H ) + M ,d, ( H ) . Finally, we prove (1.18), (1.19) and (1.20). By (1.21) and (1.23), wehave M ∗ n,d ( H ) = M ∗ n,d, ( H ) + M ∗ n,d, ( H )+ O (cid:0) H d ( d +1)( n − − d exp( c log H/ log log H ) (cid:1) . (4.22)As before, we have, by using (1.13), M ∗ n,d, ( H ) = nw ( d ) C ( d ) n − H d ( d +1)( n − + O (cid:0) H d ( d +1)( n − − d (log H ) ϑ ( d ) (cid:1) . (4.23)As in (4.20) and (4.21), we find that M ∗ n,d, ( H ) = 2 n ( n − C ( d ) n − H d ( d +1)( n − + O (cid:0) H d ( d +1)( n − − d/ log H (cid:1) ;(4.24)if furthermore d = 2 or d is odd, we have M ∗ n,d, ( H ) = 2 n ( n − C ( d ) n − H d ( d +1)( n − + O (cid:0) H d ( d +1)( n − − d (log H ) ϑ ( d ) (cid:1) . (4.25)So, (1.18) follows from (4.22), (4.23) and (4.24); then using (4.25)instead of (4.24) gives (1.19). In order to deduce (1.20), we apply (4.23)and (4.25) and notice that M ∗ ,d ( H ) = M ∗ ,d, ( H ) + M ∗ ,d, ( H ) . ULTIPLICATIVELY DEPENDENT VECTORS 25 Lower Bound
In this section, we shall prove that (1.10) is sharp, apart from afactor H o (1) , when n = s + 1 is even and K = Q .We need the following slight extension of [19, Lemma 2.3]. Lemma 5.1.
Let k and q be integers with k ≥ and q ≥ . Let γγγ = ( γ , . . . , γ k ) with γ , . . . , γ k positive real numbers. Then, thereexists a positive number Γ( q, γγγ ) such that for T → ∞ , we have X . . . X a ··· a k = b ··· b k gcd( a i b i ,q )=11 ≤ a i ,b i ≤ T γi i =1 ,...,k ∼ Γ( q, γγγ ) T γ (log T ) ( k − , where γ = γ + · · · + γ k .Proof. The proof proceeds along the same lines as in the proof of [19,Lemma 2.3]. The only difference is that the primes p which divide q are now excluded from the Euler products that appear in [19]. (cid:3) We show that apart perhaps from the factor exp( c log H/ log log H )the estimate (1.10) in Proposition 1.3 is sharp when n is even, s = n − K = Q . Theorem 5.2.
Let n = 2 k , where k is an integer with k > . Then,for sufficiently large H , there exists a positive number c depending on n such that (5.1) L n, Q ,n − ( H ) ≥ cH k (log H ) ( k − . Proof.
Fix n − p i , q i , i = 2 , . . . , k . Given positiveintegers a , . . . , a k , b , . . . , b k , we first set ν = 2 p · · · p k a and ν k +1 = 2 q · · · q k b . After this we set ν i = q i a i and ν k + i = p i b i , i = 2 , . . . , k. Clearly, if a · · · a k = b · · · b k with gcd( a i b i , p q · · · p k q k ) = 1 for any2 ≤ i ≤ k , then the integer vector ννν = ( ν , . . . , ν n ) is multiplicativelydependent of rank n − ν · · · ν k = ν k +1 · · · ν n andthat there is no non-empty subset { i , . . . , i m } of { , . . . , n } of size lessthan n for which(5.2) ν j i i · · · ν j im i m = 1 , with j i , . . . , j i m non-zero integers.For sufficiently large H , we choose such integers a i , b i ≤ c H forsome positive number c depending only on the above fixed primes such that we have | ν i | ≤ H for each 1 ≤ i ≤ n . Then, each such vector ννν contributes to L n, Q ,n − ( H ). Now applying Lemma 5.1 to count suchvectors (taking T = c H and γ i = 1 for each i = 1 , . . . , k ), we derive L n, Q ,n − ( H ) ≥ cH k (log H ) ( k − , where c is a positive number depending on n . (cid:3) Comments
It might be of interest to investigate in more detail how tight ourbounds are in Propositions 1.3 and 1.5. In Section 5 we have taken aninitial step in this direction.It would be interesting to study multiplicatively dependent vectorsof polynomials over finite fields. In this case the degree plays the role ofthe height. While we expect that most of our results can be translatedto this case many tools need to be developed and this should be ofindependent interest.
Acknowledgements
The first author was supported in part by Gruppo Nazionale per leStrutture Algebriche, Geometriche e le loro Applicazioni from IstitutoNazionale di Alta Matematica “F. Severi”. The research of the secondand third authors was supported by the Australian Research Coun-cil Grant DP130100237. The research of the fourth author was sup-ported in part by the Canada Research Chairs Program and by GrantA3528 from the Natural Sciences and Engineering Research Council ofCanada.
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Dipartimento di Matematica e Fisica, Universit`a Roma Tre, Roma,I–00146, Italy
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