/^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 14 (2018) 165-176 https://doi.org/10.26493/1855-3974.1348.d47 (Also available at http://amc-journal.eu) The size of algebraic integers with many real conjugates Artüras Dubickas Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania Received 20 March 2017, accepted 18 April 2017, published online 10 August 2017 Abstract In this paper we show that the relative normalised size with respect to a number field K of an algebraic integer a = -1,0,1 is greater than 1 provided that the number of real embeddings s of K satisfies s > 0.828n, where n = [K : Q]. This can be compared with the previous much more restrictive estimate s > n — 0.192 vV log n and shows that the minimum m(K) over the relative normalised size of nonzero algebraic integers a in such a field K is equal to 1 which is attained at a = ±1. Stronger than previous but apparently not optimal bound for m(K) is also obtained for the fields K satisfying 0.639 < s/n < 0.827469.... In the proof we use a lower bound for the Mahler measure of an algebraic number with many real conjugates. Keywords: Algebraic number field, relative size, relative normalised size, Mahler measure, Schur-Siegel-Smyth trace problem. Math. Subj. Class.: IIR04, 11R06 1 Introduction Let K be a number field with signature (s(K),t(K)) = (s,t) having s real embeddings ai : K ^ R, i = 1,..., s, and t conjugate pairs of complex embeddings ai+j, CTi+j : K ^ C, j = 1,..., t. Clearly, n = n(K) := [K : Q] = s + 2t. For any a G K we define IMk := (E CTi(a)2 + ]T |cts+j (a)|2)V2, (1.1) i=1 j=1 E-mail address: arturas.dubickas@mif.vu.lt (Arturas Dubickas) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 166 Ars Math. Contemp. 14 (2018) 117-128 and mK(a) := . (1.2) s +t Also, put m(K) := min mK(a), (1.3) a£0i\{0| where OK is the ring of integers of K. For any number field K we have ±1 G OK and || ± 1||K = s(K) + t(K), so that mK(±1) = 1 (see (1.1) and (1.2)). By (1.3), this yields m(K) < 1. The lower bound m(K) > -—1—r, v ' - 1 + s/n where n = [K : Q] and s = s(K), follows from [19, Lemma 1.1(ii)]. The stronger bound 2s/n m(K) > ———i— (1.4) 1 + s/n is given in [16, Theorem 5.11]. In particular, the inequality (1.4) implies m(K) = 1 if K is a totally complex field (s(K) = 0) or a totally real field (t(K) = 0). A motivation for introducing and studying the quantities ||a||K, mK(a) and m(K) is given in [7]; see also a subsequent paper [6]. There, we call ||a||K the relative size of a with respect to the number field K and \JmK(a) the relative normalised size of a (with respect to K again). Briefly speaking, it is related to some earlier work on certain lattices defined by number fields, when in the ring of integers OK of a number field K with signature (s, t), one considers the vectors ( -— = 0.942084 ... 1 + s/n 2 holds for any integers s < n, where s > 0 and n > 1. Hence, by (1.4), ^f2 < m(K) < 1. In particular, for any number field K we have either m(K) = 1 or m(K) < 1. A large class of fields for which m(K) = 1 was described in [7, Theorem 3.3], where we showed that for a number field K with signature (s, t) we have m(K) = 1 if t < 0.096^/s/ log s. (1.5) The following theorem relaxes the bound (1.5) on t to the bound t < 0.086n with the same conclusion and so strengthens the above result considerably. (Note that in view of n = s + 2t the bound (1.5) is essentially equivalent to t < 0.096^n/ log n.) A. Dubickas: The size of algebraic integers with many real conjugates 167 Theorem 1.1. For each number field K of degree n and signature (s,t) satisfying t < 0.086n we have m(K) = 1. Observe that t < 0.1038s implies t < 0.086n which is also equivalent to s > 0.828n. On the other hand, by [7, Theorem 3.5], for each integer s > 2 there exist infinitely many number fields K with signature (s, s) (so that t = s = n/3) for which m(K) < 1. This shows that Theorem 1.1 is best possible up to the constant. Moreover, the constant 0.086 cannot be replaced by the constant 1/3. Below, the bound (1.4) will also be improved for fields K of degree n with signature (s,t) satisfying 0.639 < s/n < 0.828 (see Corollary 2.4 in Section 2). Here, the constants 0.639 and 0.828 are just three decimal digit approximations from above of some presumably transcendental constants (see Proposition 2.1 below for the definition of A0 = 0.827469... ). In the next section we state Theorem 2.2 which is the main result of this paper. Section 3 contains some auxiliary results. The proofs of Proposition 2.1, Theorem 2.2 and Theorem 2.5 will be given in Sections 4, 5 and 6, respectively. 2 Main results Throughout, we shall use the following notation for fixed A > 0: g(A) := (2-1/a + Vl + 2-2A(2.1) F(A, x) := xg(A)l/x + 2(1 - x)g(A)-l/(l-x) - 2 + x, (2.2) where 0 < x < 1 and, by definition, F(A, 1) = g(A) — 1. Finally, the function y(A) is defined for positive A as follows y(A) := min F(A,x). (2.3) 0 1. With this notation, we will show that Proposition 2.1. The function y(A) is increasing for A > 0.581 and positive for A > A0 := 0.827469 .... Here, y(A0) = 0, ^(0.828) = 0.000389 ... and y(1) = 0.176732 .... More values of the function y>(A) are given in Table 1. Here, for each A G [0.83,1] the constant x0(A) is the point of absolute minimum of F(A, x) in the interval 0 < x < 1, so that <^(A) = F (A, x0(A)). Now, we can state the main result of this paper. Theorem 2.2. Let K be a number field with signature (s(K), t(K)) and degree n = s(K) + 2t(K) over Q satisfying s(K) > 0.581n, and let a = —1,0,1 be an algebraic integer in K. Then, mK(a) > 1 + , (2.4) 1+A where A := s(K)/n and the function y(A) is defined in (2.1)-(2.3). In particular, the inequality y(A) > 0 holds for each A G (A0,1], that is, for s(K) > A0n, where A0 = 0.827469.... 168 Ars Math. Contemp. 14 (2018) 117-128 Table 1: Values of y(A) in the range 0.83 < A < 1. A g(A) xo(A) ¥>(A) V(A) 1+A 0.83 1.418557 0.529769 0.001865 0.001019 0.84 1.429308 0.532299 0.009447 0.005134 0.85 1.440180 0.534841 0.017362 0.009385 0.88 1.473522 0.542547 0.043126 0.022939 0.90 1.496362 0.547749 0.061991 0.032627 0.93 1.531545 0.555648 0.092837 0.048102 0.95 1.555624 0.560980 0.115104 0.059027 0.98 1.592687 0.569077 0.151060 0.076292 0.99 1.605296 0.571802 0.163726 0.082274 1.00 1.618033 0.574542 0.176732 0.088366 In fact, the inequalities (5.7) and (5.8) which will be proved in Section 5 can be stronger than (2.4) under some additional assumptions. In particular, selecting (in Theorem 2.2) K := Q(a), we obtain the following: Corollary 2.3. Let a = -1,0,1 be an algebraic integer of degree d = s + 2t with s real conjugates ait i = 1,..., s, and t pairs of complex conjugates as+j, as+j, j = 1,..., t. Then, for A = s/d > A0 we have ( ) |ai|2 + ••• + |as+t|^ 1 + y(A) mQ(a)(a) = -~t- > 1 + Y+A, where y(A) > 0 is defined in (2.1)-(2.3). By (1.3), Theorem 2.2 immediately implies Theorem 1.1 stated in Section 1. In the range 0.581 < A = s/n < A0 Theorem 2.2 implies the following: Corollary 2.4. For a number field K of degree n and signature (s,t) satisfying 0.581 < A = s/n < A0 we have m(K) > 1 + 1+1. Note that for each A satisfying 0.639 < A < A0 the inequality of Corollary 2.4 strengthens the bound (1.4). In particular, for A = 0.639 we have m(K) > 1 + ^^ = 0.950175 ..., 1+A whereas the bound (1.4) yields the weaker inequality 2a m(K) > -= 0.950121.... ( )>1+A For further comparison of the functions 1 + y(A)/(1 + A) and 2A/(1 + A) see Table 2. If the number of complex conjugates 2t of an algebraic integer is very small compared to its degree d (which is large) then the constant 1.088366 corresponding to the case A =1 A. Dubickas: The size of algebraic integers with many real conjugates 169 Table 2: Values of 1 + vs for 0.64 < A < 0.82. A g(A) 2a -+A i + vM 1 + -+A 0.64 1.237064 0.950200 0.950299 0.65 1.245534 0.951011 0.951621 0.70 1.289701 0.955591 0.960509 0.75 1.336871 0.961024 0.973167 0.80 1.387027 0.967278 0.989505 0.82 1.407927 0.970003 0.997042 in Corollary 2.3 (see Table 1) can be improved, by using the results on the so-called Schur-Siegel-Smyth trace problem. The problem is named after the authors of the first three estimates of the trace of a totally positive algebraic integer [15], [17], [18]. The method of auxiliary functions introduced by Smyth in [18] was used in all subsequent papers on this subject. Specifically, we shall use the result of Liang and Wu [12] (see Lemma 3.5 below). See also some recent related papers [4] and [14]. Theorem 2.5. There exist two absolute positive constants D and S such that if d > D and t < Sd/ log d then for each algebraic integer a of degree d = s + 2t with s real conjugates -i, i = 1,..., s, and t pairs of complex conjugates as+j, as+j, j = 1,..., t, the inequality , s l-d2 +-----+ |-s+tI2 mQ(a)(a) = —- ' s+" > 1.79192 (2.5) holds. For large d Theorem 2.5 not only gives a better bound, but also the condition t < Sd/ log d is less restrictive than the corresponding condition t < 0.096^d/ log d of [7, Theorem 3.3]. 3 Auxiliary results Lemma 3.1. Let - = -1,0,1 be an algebraic number of degree d over Q with signature (s, t) , where A = s/d > 0. Then, M(-) > (V-/A + V1 + 2-2A)S/2. (3.1) In particular, for s > 0.581d we have M(-) > 1.090691d. (3.2) Proof. The inequality (3.1) was proved by Garza (it is the main result in [8]). In [10], Hohn gave an alternative proof of this result. By (2.1), (3.1), and s = Ad, we deduce that M(-) > (V-/A + V1 + 2-2A)Ad/2 = g(A)d/2. (3.3) Evidently, the function g(A) is increasing in A > 0, so its smallest value in the interval [0.581,1] is attained at A = 0.581. Thus, (3.3) implies (3.2) in view of g(0.581)-/2 = 1.090691.... □ 170 Ars Math. Contemp. 14 (2018) 117-128 We will also need the following inequality. Lemma 3.2. For any number fields L Ç K with signatures (s(L), t(L)) and (s(K), t(K)), respectively, we have s(K)t(L) < s(L)t(K). Proof. By the primitive element theorem, write L = Q(a) and K = Q(fi). Then, a = P(fi) with some P G Q[x]. Without restriction of generality we may assume that ,..., fis are the real conjugates of fi and fis+1, fis+1,..., fis+i, fis+i are the complex conjugates of fi. Here, s = s(K) and t = t(K). Note that in the list a(P(fi)), where a runs through all s + 2t automorphisms of the field K, each conjugate of a appears [K : L] times. In particular, each of the numbers P (fi;), where 1 < i < s, is real, so the number of real conjugates of a is at least s/[K : L]. This yields (L) s = s[L : Q] = s[L : Q] = s(s(L) + 2t(L)) S( ) " [K : L] [K : L][L : Q] [K : Q] s + 2t . Multiplying both sides by s + 2t we obtain the required inequality. □ Lemma 3.3. Let k < d be two positive integers and let S > 1, p and y1 > • • • > yk > 1 > yk+1 > • • • > yd be real numbers such that yi +-----+ yk + S(yfc+1 +-----+ yd) > S(d - k) + k + p. Then, for any positive numbers w1,..., wd satisfying max Wj < S min w; 1 1 + p min Wj. 1 ... zfc > 0 > zfc+1 > • • • > zd (3.4) and Z1 +-----+ zfc + S(zfc+1 +-----+ zd) > p. (3.5) Now, by (3.4), the bound 0 < max1 min z; + max z; 1 min w.; > z; + max > z; > 1 (z1 +-----+ zk + S(zk+1 +-----+ zd)) min w; 1 p min w;. 1 0, d > 2, and let a be an algebraic integer of degree d with signature (s, t) satisfying s > 0.581d whose conjugates ai,..., ad are labeled so that |ai| > ••• > |afc| > 1 > |afc+i| > ••• > |ad|. Then, |ai|2 + • • • + |afc|2 + 2(|afe+i|2 + • • • + |ad|2) > 2d - k + d<(A), (3.6) where A = s/d and <(A) defined in (2.1)-(2.3). Proof. Note that k > 1. Indeed k = 0 can only happen if all aj, i = 1,..., d, are of modulus 1. So, by Kronecker's theorem, a must be a root of unity which is not the case. If k = d then, by the arithmetic and geometric mean inequality (referred to as AM-GM below) and (3.3), the left side of (3.6) is at least d|Norm(a)|2/d = dM(a)2/d > dg(A), where g(A) is defined in (2.1). Since g(A) is increasing in A > 0 and #(0.581) = 1.189607..., we find that the left side of (3.6) is at least 1.189d. This is greater than its right side, since 2d - k + d<(A) = d + d<(A) < d + d<(1) < d +0.18d = 1.18d (see Proposition 2.1 and Table 1). In all what follows we thus assume that 0 < k < d. By AM-GM, estimating |ai|2 + ••• + |afc|2 > kM(a)2/k and |afc+i|2 + • • • + |ad|2 > (d - k) (JNMi^)2/(d-k) > (d - k)M(a)-2/(d-k) we find that the left side of (3.6) is at least kM(a)2/k + 2(d - k)M(a)-2/(d-k). Hence, it suffices to show that kM(a)2/k +2(d - k)M(a)-2/(d-k) k -d--2+ d > <(A) (3.7) Note that the function ky2/k + 2(d - k)y-2/(d-k) is increasing in y in the interval 8, to), since its derivative 2y2/k-i -4y-2/(d-k)-i is positive for y > 2k(d-k)/(2d) and the maximum of k(d - k) is attained at k = d/2. Also, by (3.2) and 2i/8 = 1.090507..., the inequality M(a) > 1.090691d > 2d/8 holds. Thus, replacing M(a) in (3.7) by its estimate from below as in (3.3) and setting x := k/d, we see that it suffices to prove the inequality xg(A)i/x + 2(1 - x)g(A)-i/(i-x) - 2 + x > <(A) (3.8) for 0 < x < 1. However, (3.8) clearly holds, by the definition of the function <(A) in Theorem 2.2 as the minimum of the left side of (3.8) in the interval (0,1]. EH 172 Ars Math. Contemp. 14 (2018) 117-128 The next result is given [12]. Lemma 3.5. There exist m (explicitly given) polynomials with integer coefficients Qi,..., Qm and m (explicitly given) positive numbers ei,..., em such that the inequality m y - ^ ei log |Qi(y)| > 1.79193 i=i holds for each y > 0 which is not a root of Qi... Qm. We remark that each improvement of the constant of this lemma leads to the corresponding improvement in Theorem 2.5. However, although the conjectural lower bound for the trace of a totally positive algebraic integer a is (2 - e)d, where e is an arbitrary positive number and the degree d of a is at least d(e), Serre has shown that the method of auxiliary functions as in the above lemma cannot give a constant greater than 1.8983021 (see the appendix in [1]). 4 Proof of Proposition 2.1 Note that y = #(A) > 1 for A> 0. Consider the function f (y) := xyi/x + 2(1 - x)y-i/(i-x) in the interval 1 < y < to (here, 0 < x < 1). Its derivative f'(y) = yi/x-i - 2y-i/(i-x) is positive if yi/x+V(i-x) > 2, that is, y > 2x(i-x). In particular, since x(1 - x) < 1/4, the function f (y) is increasing in the interval 2i/4 < y < to. Thus, by (2.3) and (2.1) (which implies that g(A) is increasing in A), for every fixed x in the range 0 < x < 1 the function xg(A)i/x + 2(1 - x)g(A)-i/(i-x) - 2 + x in increasing (in A) for A satisfying g(A) > 2i/4. In particular, y>(A) is increasing in A for A satisfying g(A) > 2i/4. Therefore, using the fact that g(A) is increasing in A for A > 0 and the actual expression (2.1), we find that #(0.581) = 1.189607 • • • > 1.189207 • • • = 2i/4. Consequently, the function y>(A) is increasing for A > 0.581. Evaluating y>(A) at A = 0.828 gives the positive value ^(0.828) = 0.000389..., so y(A) > 0 for A > 0.828. This, combined with evaluation of y(1) = 0.176732 ... and A0 satisfying y>(A0) = 0 completes the proof of the proposition. 5 Proof of Theorem 2.2 Let a e K and L = Q(a). Assume that the signature of a is (s, t) and the signature of K is (s(K),t(K)). Here, A = s(K)/n, where n = s(K) + 2t(K) = [K : Q]. Put also Ai := s(L)/d = s/d, where d = s + 2t = [L : Q]. We will show that Ai > A. (5.1) A. Dubickas: The size of algebraic integers with many real conjugates 173 Observe first that t(L) = 0 implies that s(L) = d, so that Ai = 1, which yields (5.1). Also, t(K) = 0 implies t(L) = 0, which leads to the situation we have just considered. So assume that t(K) = 0 and t = t(L) = 0. Then, in view of Lemma 3.2 we have s(K)/t(K) < s/t. Adding 2 to both sides we deduce n _s(K)+2t(K) s(K) s_s + 2t _ d - 2 + v < 2 + ~ - t(K) t(K) t(K) " t t t' Therefore, t/d < t(K)/n. This implies (5.1), since t/d = (1 - Ax)/2 and t(K)/n = (1 - A)/2. Let ai,..., as be the real conjugates of a. Put t d C(a):= £ |as+j|2 = 2 £ k|2. j=1 j=s+1 Assume that for each real a^1 < i < s, it appears u times under the s(K) real embeddings of K and 2vj times under the 2t(K) complex embeddings of K. Here, we have u + 2vj = [K : L] for each i. Also, s(K) = u1 + ... + us and t(K) = [K : L]t + v1 + ... + vs. (5.2) So, in view of (1.2) we can write s (s(K) + t(K))mK(a) = ^(u + Vj)a,2 + [K : L]C(a). (5.3) j=1 Here, C(a) = 1 £d=s+1 |aj|2. Setting Uj + Vj Wi := s(K) + t(K) for i = 1,..., s and [K : L] ' ' 2s(K) + 2t(K) for i = s + 1,..., d, in view of (5.2) and (5.3), we derive that d mK(a) = Wj|«j|2, (5.4) i=i where J2 d=1 w = 1 and i=i' [K : L] [K : L] 2s(K) + 2t(K) _ j " s(K) + t(K) for each i = 1,..., d. Hence, by Lemma 3.3 with S = 2, p = d^(A1), yj = |aj|2 for i = 1,..., d, and Lemma 3.4 (with A1 = s/d instead of A), it follows that d mK(a) = Wj|aj|2 > 1 + d^(A1) min Wj. 1 —tttt-7—r (5.5) i 1.79193s + EE64 log IQi(a2)|. (6.1) j=1 j=1 i=1 Note that there is nothing to prove if at least one conjugate of a is greater than %/2d, because then the right side of (2.5) is greater than 2d/(s + t) > 2 which is better than required. So, in all what follows without restriction of generality we may assume that |as+j | < V2d for j = 1,..., t. Clearly, |Qi(a2+ )| < (Di + 1)Hi(2d)Di, where A and H are the degree and the height of the polynomial respectively. Simi- I 1 ^ 1 n -I- 1 ^ W■ (OJ^Di thvt tho rlacraa r,f rJ2 larly, |Qi(OS+~2)| < (D, + 1)H,(2d)Di. Note that the degree of a2 is either d or d/2, so A. Dubickas: The size of algebraic integers with many real conjugates 175 it is greater than any Dj = deg Qj provided that d > D > E. Hence, Q^a2) = 0 for each i = 1,..., m and each j = 1,..., d. Consequently, d s t i < n iQi(«2)i=n iQi( j n q («2+ »(^j j=I J=I J=I < (2d)2tDiU2t n |Qi(a2)|, j=i where U := (Dj + 1)Hj, which yields ^log |Qj(a2)| > — 2tDj log(2d) - 2t log Uj. j=i Summing these inequalities with weights ej over i = 1,..., m we derive that s m m s ]T]Tej| log Qj(a2)| = EE e j | log Qj( j=1j=1 j=1j=1 > — E(2tDjej log(2d) + 2tej log Uj) j=1 > —At log(Bd), where the constants A, B > 2 depend on the constants e1,..., em and the polynomials Q1,..., Qm only. Combining this inequality with (6.1) we get ^a2 > 1.79193s — Atlog(Bd). j=1 To complete the proof of the theorem it suffices to show that 1.79193s — At log(Bd) > 1.79192(s +1), which is equivalent to 10-5s > At log(Bd) + 1.79192t. Multiplying both sides of this inequality by 105 and adding 2t we obtain the following equivalent inequality: d = s + 2t > 105At log(Bd) + 179192t + 2t = 105At log(Bd) + 179194t. We will show that the stronger inequality d > 105 (A + 2)t log(Bd) (6.2) holds with the constants S := 105(2A + 4) and D :=max{B,E + 1} depending on e1,..., em and Q1,..., Qm only. Indeed, in view of the upper bound on t, namely, t < Sd/ log d, the first lower bound on d, namely, d > D > B, and the choice of S the right side of (6.2) is less than r 7 r 7 105(A + 2)-log(Bd) < 105(A + 2)-log(d2) = S105(2A + 4)d = d. log d log d This completes the proof of (6.2) and the proof the theorem. 176 Ars Math. Contemp. 14 (2018) 117-128 References [1] J. Aguirre and J. C. Peral, The trace problem for totally positive algebraic integers, in: J. Mc-Kee and C. Smyth (eds.), Number Theory and Polynomials, Cambridge University Press, Cambridge, volume 352 of London Mathematical Society Lecture Note Series, pp. 1-19, 2008, doi:10.1017/cbo9780511721274.003, proceedings of the workshop held at Bristol University, Bristol, April 3-7, 2006. [2] J. H. Conway and N. J. A. 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