MATHEMATICAL LOGIC QUARTERLY www.mlq-journal.org A Journal for Mathematical Logic, Foundations of Mathematics, and Logical Aspects of Theoretical Computer Science M L Q Editors Armin Hemmerling, Greifswald (Managing) Klaus Ambos-Spies, Heidelberg Klaus Meer, Cottbus Marat M. Arslanov, Kazan Wolfram Pohlers, Münster Günter Asser, Greifswald Pavel Pudlak, Prague John T. Baldwin, Chicago Andrzej Rosłanowski, Omaha Douglas S. Bridges, Canterbury Jörg Rothe, Düsseldorf Ramon Jansana, Barcelona Wilfried Sieg, Pittsburgh Carl G. Jockusch, Urbana Stephen G. Simpson, State College, PA Alexander Kechris, Pasadena Martin Weese, Potsdam Ulrich Kohlenbach, Darmstadt Klaus Weihrauch, Hagen H. Dugald Macpherson, Leeds W. Hugh Woodin, Berkeley Editorial Assistant: Heike Oberdörfer, Greifswald RE PR INT Math. Log. Quart. 56, No. 2, 175 – 184 (2010) / DOI 10.1002/malq.200910004 Two conjectures on the arithmetic in R and C∗ Apoloniusz Tyszka∗∗ University of Agriculture, Faculty of Production and Power Engineering, Balicka 116B, 30-149 Kraków, Poland Received 3 January 2009, revised 2 June 2009, accepted 3 June 2009 Published online 18 March 2010 Key words System of polynomial equations, system of linear equations, solution with minimal l∞ norm. MSC (2000) 03B30, 12D99, 14P05, 15A06 Let G be an additive subgroup of C, let Wn = {xi = 1, xi + xj = xk : i, j, k ∈ {1, . . . , n}}, and define En = {xi = 1, xi + xj = xk, xi * xj = xk : i, j, k ∈ {1, . . . , n}}. We discuss two conjectures. (1) If a system S ⊆ En is consistent over R (C), then S has a real (complex) solution which consists of numbers whose absolute values belong to [0, 22 n−2 ]. (2) If a system S ⊆ Wn is consistent over G, then S has a solution (x1, . . . , xn) ∈ (G ∩Q)n in which |xj | ≤ 2n−1 for each j. c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Systems of equations over R and C For a positive integer n we define the set of equations En by En = {xi = 1, xi + xj = xk, xi * xj = xk : i, j, k ∈ {1, . . . , n}}. Conjecture 1.1 ([20]). Let a system S ⊆ En be consistent over R (C). Then S has a real (complex) solution which consists of numbers whose absolute values belong to [0, 22 n−2 ]. Concerning the bound 22 n−2 in Conjecture 1.1, Vorobjov's theorem ([23]) allows us to compute a weaker estimation by a computable function of n. We present his result here. Let V ⊆ Rn be a real algebraic variety given by the system of equations f1 = . . . = fm = 0, where fi ∈ Q[x1, . . . , xn] (i = 1, . . . ,m). We denote by L the maximum of the bit-sizes of the coefficients of the system and set d = ∑m i=1 deg(fi), r = ( n+ 2d n ) . We recall ([1, p. 285]) that the bit-size of a non-zero integer is the number of bits in its binary representation. More precisely, the bit-size of k is τ if and only if 2τ−1 ≤ |k| < 2τ . The bit-size of a rational number is the sum of the bit-sizes of its numerator and denominator in reduced form. N. N. Vorobjov, Jr. proved that there exists (x1, . . . , xn) ∈ V such that |xi| < 2H(r,L) (i = 1, . . . , n), where H is some polynomial not depending on the initial system. For a simplified proof of Vorobjov's theorem, see [8, Lemma 9, p. 56]. For a more general theorem, see [1, Theorem 13.15, p. 516]. It is algorithmically decidable whether a system S ⊆ En has a real (complex) solution (x1, . . . , xn) with |x1|, . . . , |xn| ≤ 22 n−2 . It is also algorithmically decidable whether a system S ⊆ En is consistent over R (C). For the final problem, an appropriate algorithm follows from the theorem known as effective Hilbert Nullstellensatz. The expected complexity of such an algorithm is related to Steven Smale's conjecture, which we now recall. For an integer m denote by τ(m) the smallest positive integer s for which there exist integers x0, x1, . . . , xs such that x0 = 1, xs = m, and for each t ∈ {1, . . . , s} there are i, j ∈ {0, . . . , t − 1} with xi ◦ xj = xt. Here ◦ denotes addition, subtraction or multiplication. Smale's conjecture states that for every sequence {mk}∞k=3 ∗ This paper is a shortened version of [19]. ∗∗ e-mail: rttyszka@cyf-kr.edu.pl c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 176 A. Tyszka: Two conjectures on the arithmetic in R and C of non-zero integers, there is no constant c such that τ(mk * k!) ≤ (log2(k))c for all k ∈ {3, 4, 5, . . .}, see [2, p. 126]. This conjecture implies that there is no polynomial time algorithm for Hilbert Nullstellensatz over C, see [2, p. 126, Theorem 2]. Concerning Conjecture 1.1, for n = 1 estimation by 22 n−2 can be replaced by estimation by 1. For n > 1 estimation by 22 n−2 is the best estimation. Indeed, let n > 1 and x1 = 1, x2 = 22 0 , x3 = 22 1 , . . . , xn = 22 n−2 . In any ring K of characteristic 0, from the system of all equations belonging to En and which are satisfied under the substitution [x1 → x1, . . . , xn → xn], it follows that x1 = x1, . . . , xn = xn. Theorem 1.2 If n ∈ {1, 2, 3}, then Conjecture 1.1 holds true for each subring K ⊆ C. P r o o f. If a system S ⊆ E1 is consistent over K, then S has a solution x1 ∈ {0, 1}. If a system S ⊆ E2 is consistent over K and 1 2 ∈ K, then S has a solution (x1, x2) ∈ {(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 1)}. If a system S ⊆ E2 is consistent over K and 1 2 ∈ K, then (x1, x2) ∈ {(0, 0), (0, 1), (1, 0), (1 2 , 1), (1, 1 2 ), (1, 1), (1, 2), (2, 1)} is a solution for S. To reduce the number of studied systems S ⊆ E3, we may assume that the equation x1 = 1 belongs to S, as when all equations x1 = 1, x2 = 1, x3 = 1 do not belong to S, then S has the solution (0, 0, 0) ∈ K3. Let A2 = {x2 ∈ C : there exists x3 ∈ C for which (1, x2, x3) solves S}, A3 = {x3 ∈ C : there exists x2 ∈ C for which (1, x2, x3) solves S}. We may assume that A2 ⊆ {z ∈ C : |z| ≤ 4} or A3 ⊆ {z ∈ C : |z| ≤ 4}. Case 1: A2 ⊆ {z ∈ C : |z| ≤ 4} and A3 ⊆ {z ∈ C : |z| ≤ 4}. If (1, x2, x3) ∈ K3 solves S, then (1, 1, x3) ∈ K3 solves S. Case 2: A2 ⊆ {z ∈ C : |z| ≤ 4} and A3 ⊆ {z ∈ C : |z| ≤ 4}. If (1, x2, x3) ∈ K3 solves S, then (1, x2, 1) ∈ K3 solves S. Case 3: A2 ⊆ {z ∈ C : |z| ≤ 4} and A3 ⊆ {z ∈ C : |z| ≤ 4}. If (1, x2, x3) ∈ K3 solves S, then (1, 0, 1) ∈ K3 solves S or (1, 1, 0) ∈ K3 solves S or (1, 1, 1) ∈ K3 solves S. The following Observation borrows the idea from the proof of Theorem 1.2. Observation 1.3 Let n ∈ {1, 2, 3, 4}, and let a system S ⊆ En be consistent over the subring K ⊆ C. If (x1, . . . , xn) ∈ Kn solves S, then (x1, . . . , xn) solves S, where each xi is suitably chosen from the set {xi, 0, 1, 2, 1 2 } ∩ {z ∈ K : |z| ≤ 22n−2}. Theorem 1.4 Conjecture 1.1 holds true for each n ∈ {1, 2, 3, 4} and each subring K ⊆ C. P r o o f. It follows from Observation 1.3. Let W = { {1}, {0}, {1, 0}, {1, 2}, {1, 1 2 }, {1, 2, 1 2 }, {1, 0, 2}, {1, 0, 1 2 }, {1, 0,−1}, {1, 2,−1}, {1, 2, 3}, {1, 2, 4}, {1, 1 2 ,−1 2 }, {1, 1 2 , 1 4 }, {1, 1 2 , 3 2 }, {1,−1,−2}, {1, 1 3 , 2 3 }, {1, 2, √ 2}, {1, 1 2 , 1√ 2 }, {1, √ 2, 1√ 2 }, {1, √ 5− 1 2 , √ 5 + 1 2 }, {1, √ 5 + 1 2 , √ 5 + 3 2 }, {1, − √ 5− 1 2 , √ 5 + 3 2 }}. For each a, b, c ∈ R (C) we define S(a, b, c) as {E ∈ E3 : E is satisfied under the substitution [x1 → a, x2 → b, x3 → c]}. If a, b, c ∈ R and {a} ∪ {b} ∪ {c} ∈ W , then the system S(a, b, c) is consistent over R, has a finite number of real solutions, and each real solution of S(a, b, c) belongs to [−4, 4]3. The family {S(a, b, c) : a, b, c ∈ R ∧ {a} ∪ {b} ∪ {c} ∈ W} equals the family of all systems S ⊆ E3 which are consistent over R and maximal with respect to inclusion. c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org Math. Log. Quart. 56, No. 2 (2010) / www.mlq-journal.org 177 If a, b, c ∈ C and {a} ∪ {b} ∪ {c} ∈ W ∪{{1, −1 + √−3 2 , 1 + √−3 2 }, {1, 1− √−3 2 , 1 + √−3 2 }}, then the system S(a, b, c) is consistent over C, has a finite number of solutions, and each solution of S(a, b, c) belongs to {(z1, z2, z3) ∈ C3 : |z1| ≤ 4 ∧ |z2| ≤ 4 ∧ |z3| ≤ 4}. The family {S(a, b, c) : a, b, c ∈ C ∧ {a} ∪ {b} ∪ {c} ∈ W ∪ {{1, −1 + √−3 2 , 1 + √−3 2 }, {1, 1− √−3 2 , 1 + √−3 2 }}} equals the family of all systems S ⊆ E3 which are consistent over C and maximal with respect to inclusion. Let us consider the following four conjectures; analogous statements seem to be true for R. Conjecture 1.5 (a) If a system S ⊆ En is consistent over C and maximal with respect to inclusion, then each solution of S belongs to {(x1, . . . , xn) ∈ Cn : |x1| ≤ 22 n−2 ∧ * * * ∧ |xn| ≤ 22 n−2}. (b) If a system S ⊆ En is consistent over C and maximal with respect to inclusion, then S has a finite number of solutions (x1, . . . , xn). (c) If the equation x1 = 1 belongs to S ⊆ En and S has a finite number of complex solutions (x1, . . . , xn), then each such solution belongs to {(x1, . . . , xn) ∈ Cn : |x1| ≤ 22 n−2 ∧ * * * ∧ |xn| ≤ 22 n−2}. (d) If a system S ⊆ En has a finite number of complex solutions (x1, . . . , xn), then each such solution belongs to {(x1, . . . , xn) ∈ Cn : |x1| ≤ 22 n−1 ∧ * * * ∧ |xn| ≤ 22 n−1}. Conjecture 1.5(a) strengthens Conjecture 1.1 for C. The conjunction of Conjectures 1.5(b) and 1.5(c) implies Conjecture 1.5(a). Concerning Conjecture 1.5(d), for n = 1 estimation by 22 n−1 can be replaced by estimation by 1. For n > 1 estimation by 22 n−1 is the best estimation. Indeed, the system x1 + x1 = x2 x1 * x1 = x2 x2 * x2 = x3 x3 * x3 = x4 . . . xn−1 * xn−1 = xn has precisely two complex solutions, (0, . . . , 0), and (2, 4, 16, 256, . . . , 22 n−2 , 22 n−1 ). For the complex case of Conjectures 1.1 and 1.5(a), 1.5(b), 1.5(c), 1.5(d), the author prepared two MuPAD codes which confirm these conjectures probabilistically, see [19] and [21]. 2 Systems of equations over number rings Hilbert's tenth problem is to give a computing algorithm which will tell of a given polynomial equation with integer coefficients whether or not it has a solution in integers. Yu. V. Matijasevič proved ([13]) that there is no such algorithm, see also [14], [4], [5], [10]. It implies that Conjecture 1.1 is false for Z instead of R (C). Moreover, Matijasevič's theorem implies that Conjecture 1.1 for Z is false with any other computable estimation instead of 22 n−2 . As we have proved, Conjecture 1.1 for Z is false. We describe a counterexample showing that Conjecture 1.1 for Z is false with n = 21. Lemma 1 is a special case of the result presented in [18, p. 3]. Lemma 2.1 For each non-zero integer x there exist integers a, b such that ax = (2b− 1)(3b− 1). P r o o f. Write x as (2y−1) *2m, where y ∈ Z and m ∈ Z∩ [0,∞). Obviously, 2 2m+1 + 1 3 ∈ Z. By Chinese Remainder Theorem, we can find an integer b such that b ≡ y (mod 2y − 1) and b ≡ 2 2m+1 + 1 3 (mod 2m). Thus, 2b− 1 2y − 1 ∈ Z and 3b− 1 2m ∈ Z. Hence (2b− 1)(3b− 1) x = 2b− 1 2y − 1 * 3b− 1 2m ∈ Z. Lemma 2.2 ([9, Lemma 2.3, p. 451]) For each x ∈ Z ∩ [2,∞) there exists y ∈ Z ∩ [1,∞) such that 1 + x3(2 + x)y2 is a square. www.mlq-journal.org c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 178 A. Tyszka: Two conjectures on the arithmetic in R and C Lemma 2.3 ([9, Lemma 2.3, p. 451]) For each x ∈ Z∩ [2,∞), y ∈ Z∩ [1,∞), if 1+x3(2+x)y2 is a square, then y ≥ x+ xx−2. Theorem 2.4 Conjecture 1.1 for Z is false with n = 21. P r o o f. Let us consider the following system over Z. This system consists of two subsystems. (•) x1 = 1 x1 + x1 = x2 x2 * x2 = x3 x3 * x3 = x4 x4 * x4 = x5 x5 * x5 = x6 x6 * x6 = x7 x6 * x7 = x8 x2 + x6 = x9 x8 * x9 = x10 x11 * x11 = x12 x10 * x12 = x13 x1 + x13 = x14 x15 * x15 = x14 , () x16 + x16 = x17 x1 + x18 = x17 x16 + x18 = x19 x18 * x19 = x20 x12 * x21 = x20 . Since x1 = 1 and x12 = x11 * x11, the subsystem marked with () is equivalent to x21 * x211 = (2x16 − 1)(3x16 − 1). The subsystem marked with (•) is equivalent to x215 = 1 + (2 16)3 * (2 + 216) * x211. By Lemma 2.2, the last equation has a solution (x11, x15) ∈ Z2 such that x11 ≥ 1. By Lemma 2.1, we can find integers x16, x21 satisfying x21 * x211 = (2x16 − 1)(3x16 − 1). Thus, the whole system is consistent over Z. If (x1, . . . , x21) ∈ Z21 solves the whole system, then x215 = 1 + (2 16)3 * (2 + 216) * |x11|2 and x21 * |x11|2 = (2x16 − 1)(3x16 − 1). Since 2x16 − 1 = 0 and 3x16 − 1 = 0, |x11| ≥ 1. By Lemma 2.3, |x11| ≥ 216 + (216)2 16 − 2 > (216)216 − 2 = 2220 − 32 > 2221−2 . Lemma 2.5 ([22]). Each Diophantine equation D(x1, . . . , xp) = 0 can be equivalently written as a system S ⊆ En, where n ≥ p and both n and S are algorithmically determinable. If the equation D(x1, . . . , xp) = 0 has only finitely many solutions in a number ring K, then the system S has only finitely many solutions in K. Since there is a finite number of subsets of En, for any K there is a function χ : {1, 2, 3, . . .} −→ {1, 2, 3, . . .} with the property: for each positive integer n, if a system S ⊆ En is consistent over the number ring K, then S has a solution whose heights are less than or equal to χ(n). Theorem 2.6 If Z has a Diophantine definition in a number ring K, then any such χ is not computable. P r o o f. Let () (∀x ∈ K)(x ∈ Z ⇔ ∃t1 . . . ∃tm W (x, t1, . . . , tm) = 0) where W (x, t1, . . . , xm) ∈ Z[x, t1, . . . , xm]. Assume, on the contrary, that χ is computable. We show that it would imply a positive solution to Hilbert's tenth problem for Z. Let us consider an arbitrary Diophantine equation D(x1, . . . , xp) = 0. According to (), for each i ∈ {1, . . . , p} we construct the polynomial equation W (xi, t(1,i), . . . , t(m,i)) = 0. Applying Lemma 2.5, we write the system 0 = D(x1, . . . , xp) 0 = W (x1, t(1,1), . . . , t(m,1)) ... 0 = W (xp, t(1,p), . . . , t(m,p)) as an equivalent system T ⊆ En, where T and n are algorithmically determinable. Since χ is computable, we can decide whether T has a solution in K. Therefore, we can decide whether the equation D(x1, . . . , xp) = 0 has an integer solution. We get the contradiction to Matijasevič's theorem. The rings considered in Theorems 2.7 – 2.9 and 2.11 have the property that they allow Diophantine definitions for Z. The number 2 + 2732 is prime. c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org Math. Log. Quart. 56, No. 2 (2010) / www.mlq-journal.org 179 Theorem 2.7 If k ∈ Z ∩ [273,∞) and 2 + k2 is prime, then Conjecture 1.1 fails for n = 6 and the ring Z [ 1 2 + k2 ] = { x (2 + k2)m : x ∈ Z,m ∈ Z ∩ [0,∞) } . P r o o f. ( 1, 2, k, k2, 2 + k2, 1 2 + k2 ) solves the system x1 = 1 x1 + x1 = x2 x3 * x3 = x4 x2 + x4 = x5 x5 * x6 = x1. Assume that (x1, x2, x3, x4, x5, x6) ∈ ( Z [ 1 2 + k2 ])6 solves the system. Let x5 = a (2 + k2)p , x6 = b (2 + k2)q , a, b ∈ Z, p, q ∈ Z ∩ [0,∞). Since 2 + k2 is prime and 1 = |x1| = |x5 * x6| = |a| * |b| (2 + k2)p+q , we conclude that |a| = (2 + k2)p for some p ∈ Z ∩ [0,∞). Hence |x5| = (2 + k2)p−p. On the other hand, |x5| = |x2 + x4| = |x1 + x1 + x3 * x3| = |1 + 1 + x23| ≥ 2. Therefore, p − p ≥ 1. Consequently, |x5| = (2 + k2)p−p ≥ 2 + k2 > 22 6−2 . Theorem 2.8 If a prime number p is greater than 2256, then Conjecture 1.1 fails for n = 10 and the ring Z [ 1 p ] . P r o o f. Let us consider the system x1 = 1 x2 * x3 = x1 x3 + x4 = x2 x4 * x5 = x6 x7 + x7 = x8 x1 + x9 = x8 x7 + x9 = x10 x9 * x10 = x6. By Lemma 2.1, there exist integers u, s such that (p2 − 1) * u = (2s− 1)(3s− 1). Hence ( 1, p, 1 p , p− 1 p , p * u, (p2 − 1) * u, s, 2s, 2s− 1, 3s− 1 ) ∈ ( Z [ 1 p ])10 solves the system. If (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) ∈ ( Z [ 1 p ])10 solves the system, then we get (x2 − x3) * x5 = (2x7 − 1)(3x7 − 1). Since 2x7 − 1 = 0 and 3x7 − 1 = 0, we get x2 = x3. Since x2 * x3 = 1, we get: |x2| = pn for some n ∈ Z ∩ [1,∞) or |x3| = pn for some n ∈ Z ∩ [1,∞). Therefore, |x2| ≥ p > 2210−2 or |x3| ≥ p > 2210−2 . The number −232−216−1 is square-free, because −3*7*13*97*241*673 is the factorization of −232−216−1 into prime numbers. Theorem 2.9 Conjecture 1.1 fails for n = 6 and the ring Z [√ −232 − 216 − 1 ] = { x+ y * √ −232 − 216 − 1 : x, y ∈ Z } . P r o o f. (1, 216 + 1,−216,−232 − 216,√−232 − 216 − 1,−232 − 216 − 1) solves the system x1 = 1 x2 + x3 = x1 x2 * x3 = x4 x5 * x5 = x6 x1 + x6 = x4 which has no integer solutions. For each z ∈ Z[√−232 − 216 − 1], if |z| ≤ 226−2 , then z ∈ Z. Observation 2.10 If q, a, b, c, d ∈ Z, b = 0 or d = 0, q ≥ 2, q is square-free, and (a+ b√q) * (c+ d√q) = 1, then (a ≥ 1 ∧ b ≥ 1) ∨ (a ≤ −1 ∧ b ≤ −1) ∨ (c ≥ 1 ∧ d ≥ 1) ∨ (c ≤ −1 ∧ d ≤ −1). The number 4 *134−1 is square-free, because 3 *113 *337 is the factorization of 4 *134−1 into prime numbers. www.mlq-journal.org c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 180 A. Tyszka: Two conjectures on the arithmetic in R and C Theorem 2.11 If p ∈ Z∩ [13,∞) and 4p4− 1 is square-free, then Conjecture 1.1 fails for n = 5 and the ring Z[ √ 4p4 − 1] = {x+ y * √ 4p4 − 1 : x, y ∈ Z}. P r o o f. (1, 2p2 + √ 4p4 − 1, 2p2 − √ 4p4 − 1, 4p2, 2p) solves the system x1 = 1 x2 * x3 = x1 x2 + x3 = x4 x5 * x5 = x4. Assume that (x1, x2, x3, x4, x5) ∈ (Z[ √ 4p4 − 1])5 solves the system. Let x2 = a + b √ 4p4 − 1 and let x3 = c+ d √ 4p4 − 1, a, b, c, d ∈ Z. Since ¬((∃x2 ∈ Z)(∃x3 ∈ Z)(∃x5 ∈ Z[ √ 4p4 − 1]) (x2 * x3 = 1 ∧ x2 + x3 = x25)), we get b = 0 or d = 0. Since x2 * x3 = 1, Observation 2.10 implies that |x2| ≥ 1 + √ 4p4 − 1 > 225−2 or |x3| ≥ 1 + √ 4p4 − 1 > 225−2 . 3 Systems of equations over number fields Julia Robinson proved that Z is definable in Q by a first order formula in the language of rings. Bjorn Poonen proved ([15]) that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It is unknown whether Z is existentially definable in Q. If it is, Hilbert's tenth problem for Q is undecidable. The author conjectures that if a system S ⊆ En has at most finitely many integer (rational) solutions, then their heights are less than or equal to 22 n−1 , see [22]. This conjecture and Lemma 2.5 imply that Hilbert's tenth problem for Z (Q) has a positive solution for Diophantine equations which have at most finitely many integer (rational) solutions. Theorem 3.1 If Z is definable in Q by an existential formula, then Conjecture 1.1 fails for Q. P r o o f. If Z is definable in Q by an existential formula, then Z is definable in Q by a Diophantine formula. Let (∀x1 ∈ Q)(x1 ∈ Z ⇔ (∃x2 ∈ Q) . . . (∃xm ∈ Q)Φ(x1, x2, . . . , xm)) where Φ(x1, x2, . . . , xm) is a conjunction of formulae of the form xi = 1, xi + xj = xk, xi * xj = xk, where i, j, k ∈ {1, . . . ,m}. We find an integer n with 2n ≥ m + 10. Now we are ready to describe a counterexample to Conjecture 1.1 for Q, this counterexample uses n +m + 11 variables. Considering all equations over Q, we can equivalently write down the system Φ(x1, x2, . . . , xm)(1) x2m+2 = 1 + ( 22 n )3 * (2 + 22n) * x21(2) x1 * xm+1 = 1(3) as a conjunction of formulae of the form xi = 1, xi+xj = xk, xi *xj = xk, where i, j, k ∈ {1, . . . , n+m+11}. The system is consistent over Q. Assume that (x1, . . . , xn+m+11) ∈ Qn+m+11 solves the system. Formula (1) implies that x1 ∈ Z. By this and equation (2), xm+2 ∈ Z. Equation (3) implies that x1 = 0, so by Lemma 2.3 |x1| ≥ 22 n + (22 n )2 2n − 2 > 22n+ 2 n − 2n+1 ≥ 22n+2 n−1 ≥ 22n+m+11−2 . Theorem 3.2 Let f(x, y) ∈ Q[x, y] and the equation f(x, y) = 0 defines an irreducible algebraic curve of genus greater than 1. Let some r ∈ R satisfy (∗) (−∞, r) ⊆ {x ∈ R : (∃y ∈ R)f(x, y) = 0} ∨ (r,∞) ⊆ {x ∈ R : (∃y ∈ R)f(x, y) = 0} and let K denote the function field over Q defined by f(x, y) = 0. Then Conjecture 1.1 fails for some subfield of R that is isomorphic to K. c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org Math. Log. Quart. 56, No. 2 (2010) / www.mlq-journal.org 181 P r o o f. By Faltings' finiteness theorem ([7], cf. [12, p. 12]) the set {u ∈ K : ∃v ∈ K f(u, v) = 0} is finite. Let card {u ∈ K : ∃v ∈ K f(u, v) = 0} = n ≥ 1, and let U denote the following system of equations f(xi, yi) = 0 (1 ≤ i ≤ n) xi + ti,j = xj (1 ≤ i < j ≤ n) ti,j * si,j = 1 (1 ≤ i < j ≤ n) xn+1 = ∑n i=1 x 2 i . For some integer m > n there exists a set G of m variables such that {x1, . . . , xn xn+1, y1, . . . , yn} ∪ {ti,j , si,j : 1 ≤ i < j ≤ n} ⊆ G and the system U can be equivalently written down as a system V which contains only equations of the form X = 1, X + Y = Z, X * Y = Z, where X,Y, Z ∈ G. By (∗), we find x, ỹ ∈ R such that f(x, ỹ) = 0, x is transcendental over Q, and |x| > 22m−3 . If (x1, . . . , xm) ∈ (Q(x, ỹ))m solves V , then xn+1 = ∑n i=1 xi 2 ≥ x2 > (22m−3)2 = 22m−2 . Obviously, K is isomorphic to Q(x, ỹ). Theorem 3.3 Conjecture 1.1 fails for some subfield of R and n = 7. P r o o f. (sketch) We find α, β ∈ R such that α2 * β * (1 − α2 − β) = 1, α is transcendental over Q, and |α| > 227−2 . It is known ([16]) that the equation x+ y + z = xyz = 1 has no rational solution. Applying this, we prove: if (x1, x2, x3, x4, x5, x6, x7) ∈ Q(α, β)7 solves the system x1 = 1 x2 * x2 = x3 x3 + x4 = x5 x5 + x6 = x1 x3 * x4 = x7 x6 * x7 = x1, then |x2| = |α| > 227−2 . 4 Systems of linear equations For a positive integer n we define the set of equations Wn by Wn = {xi = 1, xi + xj = xk : i, j, k ∈ {1, . . . , n}}. Conjecture 4.1 Let G be an additive subgroup of C. If a system S ⊆ Wn is consistent over G, then S has a solution (x1, . . . , xn) ∈ (G ∩Q)n in which |xj | ≤ 2n−1 for each j. Concerning Conjecture 4.1, estimation by 2n−1 is the best estimation. Indeed, if 1 ∈ G, then the system x1 = 1 x1 + x1 = x2 x2 + x2 = x3 x3 + x3 = x4 . . . xn−1 + xn−1 = xn has a unique solution (1, 2, 4, 8, . . . , 2n−2, 2n−1) ∈ Gn. Observation 4.2 Let n ∈ {1, 2, 3, 4}, and let a system S ⊆ Wn be consistent over the additive subgroup G ⊆ C. If (x1, . . . , xn) ∈ Gn solves S, then (x1, . . . , xn) solves S, where each xi is suitably chosen from {xi, 0, 1, 2, 12} ∩ {z ∈ G : |z| ≤ 2n−1}. www.mlq-journal.org c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 182 A. Tyszka: Two conjectures on the arithmetic in R and C Theorem 4.3 Conjecture 4.1 holds true for each n ∈ {1, 2, 3, 4} and each additive subgroup G ⊆ C. P r o o f. It follows from Observation 4.2. Conjecture 4.1 restricted to the case when G ⊇ Q was probabilistically confirmed by various algorithms written in MuPAD, see [19] and [21]. In [11], a code in Mathematica illustrates the validity of Conjecture 4.1 restricted to the case when G ⊇ Q. In the case when G ⊇ Q, we will prove a weaker version of Conjecture 4.1 with the estimation given by ( √ 5)n−1. Observation 4.4 If A ⊆ Ck is an affine subspace and card A > 1, then there exists m ∈ {1, . . . , k} with ∅ = A ∩ {(x1, . . . , xk) ∈ Ck : xm + xm = xm}  A. Theorem 4.5 Let a system S ⊆ Wn be consistent over C. Then S has a rational solution (x1, . . . , xn) in which |xj | ≤ ( √ 5)n−1 for each j. P r o o f. We shall describe how to find a solution (x1, . . . , xn) ∈ Qn in which |xj | ≤ ( √ 5)n−1 for each j. We can assume that for a certain i ∈ {1, . . . , n} the equation xi = 1 belongs to S, as otherwise (0, . . . , 0) is a solution. Without loss of generality we can assume that the equation x1 = 1 belongs to S. Each equation belonging to S has the form a1x1 + . . .+ anxn = b, where a1, . . . , an, b ∈ Z. Since x1 = 1, we can equivalently write this equation as a2x2 + . . .+ anxn = b− a1. We receive a system of equations whose set of solutions is a non-empty affine subspace A ⊆ Cn−1. If card A > 1, then by Observation 4.4 we find m ∈ {2, . . . , n} for which ∅ = A ∩ {(x2, . . . , xn) ∈ Cn−1 : xm + xm = xm}  A. The procedure described in the last sentence is applied to the affine subspace A ∩ {(x2, . . . , xn) ∈ Cn−1 : xm + xm = xm} and repeated until one point is achieved. The maximum number of procedure executions is n − 1. The received one-point affine subspace is described by equations belonging to a certain set U ⊆ {xi = 1 : i ∈ {2, . . . , n}} ∪ {xi + xj = xk : i, j, k ∈ {1, . . . , n}, i+ j + k > 3}. Each equation belonging to U has the form a2x2 + . . .+ anxn = c, where a2, . . . , an, c ∈ Z. Among these equations, we choose n − 1 linearly independent equations. We can do this because the equations belonging to U describe one-point affine subspace. Let A be the matrix of the system, and the system of equations has the following form A * ⎡ ⎢⎢⎣ x2 ... xn ⎤ ⎥⎥⎦ = ⎡ ⎢⎢⎣ c2 ... cn ⎤ ⎥⎥⎦ . Let Aj be the matrix formed by replacing the j-th column of A by the column c2, . . . , cn. Clearly, det(A) ∈ Z, and det(Aj) ∈ Z for each j ∈ {1, . . . , n− 1}. By Cramer's rule xj = det(Aj−1) det(A) ∈ Q for each j ∈ {2, . . . , n}. c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org Math. Log. Quart. 56, No. 2 (2010) / www.mlq-journal.org 183 When the row of matrix A corresponds to the equation xi = 1 (i > 1), then the entries in the row are 1, 0 (n− 2 times), while the right side of the equation is 1. When the row of matrix A corresponds to the equation x1 + x1 = xi (i > 1), then the entries in the row are 1, 0 (n− 2 times), while the right side of the equation is 2. When the row of matrix A corresponds to one of the equations: x1 + xi = x1 or xi + x1 = x1 (i > 1), then the entries in the row are 1, 0 (n− 2 times), while the right side of the equation is 0. When the row of matrix A corresponds to one of the equations: x1 + xi = xj or xi + x1 = xj (i > 1, j > 1, i = j), then the entries in the row are 1, −1, 0 (n− 3 times), while the right side of the equation is 1. When the row of matrix A corresponds to the equation xi + xi = x1 (i > 1), then the entries in the row are 2, 0 (n− 2 times), while the right side of the equation is 1. When the row of matrix A corresponds to the equation xi + xj = x1 (i > 1, j > 1, i = j), then the entries in the row are 1, 1, 0 (n− 3 times), while the right side of the equation is 1. From now on we assume that i, j, k ∈ {2, . . . , n}. When the row of matrix A corresponds to the equation xi + xj = xk (i = j, i = k, j = k), then the entries in the row are 1, 1, −1, 0 (n− 4 times), while the right side of the equation is 0. When the row of matrix A corresponds to the equation xi + xi = xk (i = k), then the entries in the row are 2, −1, 0 (n− 3 times), while the right side of the equation is 0. When the row of matrix A corresponds to the equation xi + xj = xk (k = i or k = j), then the entries in the row are 1, 0 (n− 2 times), while the right side of the equation is 0. Contradictory equations, e.g. x1 + xi = xi do not belong to U , and therefore their description has been neglected. The description presented shows that each row of matrix Aj (j ∈ {1, . . . , n − 1}) has the Euclidean length less than or equal to √ 5. Hadamard's inequality states that a determinant of a real matrix is majorized by the product of the Euclidean lengths of its rows. By Hadamard's inequality | det(Aj)| ≤ ( √ 5)n−1 for each j ∈ {1, . . . , n− 1}. Hence, |xj | = | det(Aj−1)|| det(A)| ≤ |det(Aj−1)| ≤ ( √ 5)n−1 for each j ∈ {2, . . . , n}. In the case where G = Z, we will prove a weaker version of Conjecture 4.1 with the estimation given by ( √ 5)n−1. Lemma 4.6 ([3]). Let A be a matrix with m rows, n columns, and integer entries. Let b1, . . . , bm ∈ Z, and the matrix equation A * ⎡ ⎢⎢⎣ x1 ... xn ⎤ ⎥⎥⎦ = ⎡ ⎢⎢⎣ b1 ... bm ⎤ ⎥⎥⎦ defines the system of linear equations with rank m. Denote by δ the maximum of the absolute values of the m×m minors of the augmented matrix (A, b). We claim that if the system is consistent over Z, then it has a solution in (Z ∩ [−δ, δ])n. Theorem 4.7 Let a system S ⊆ Wn be consistent over Z. Then S has an integer solution (x1, . . . , xn) in which |xj | ≤ ( √ 5)n−1 for each j. P r o o f. We shall describe how to find a solution (x1, . . . , xn) ∈ Zn in which |xj | ≤ ( √ 5)n−1 for each j. We can assume that for a certain i ∈ {1, . . . , n} the equation xi = 1 belongs to S, as otherwise (0, . . . , 0) is a solution. Without loss of generality we can assume that the equation x1 = 1 belongs to S. Analogously as in the proof of Theorem 4.5, we construct a system of linear equations with variables x2, . . . , xn. For the augmented matrix of this system, the Euclidean length of each row is not greater than √ 5. We finish the proof by applying Hadamard's inequality and Lemma 4.6. Theorems 4.5 and 4.7 have similar forms, although linear systems over C and linear systems over Z have different criteria of consistency. Georg Frobenius proved that a system of linear Diophantine equations has an integer solution if and only if the rank r of the unaugmented matrix of coefficients and the greatest common divisor of the r × r minors of this matrix do not change when the augmented matrix is taken instead, see [6, p. 84]. 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