ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS. Volker Ziegler Technische Universität Graz, Austria
|
|
- Alexis Dickerson
- 5 years ago
- Views:
Transcription
1 GLASNIK MATEMATIČKI Vol. 1(61)(006), 9 30 ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS Volker Ziegler Techniche Univerität Graz, Autria Abtract. We conider the parameterized Thue equation X X 3 Y (ab + (a + b))x Y abxy 3 + a b Y = ±1, with a, b 1 Z uch that ab Z. By the hypergeometric method and a method of Tzanaki we find all olution, if i large with repect to a and b. 1. Introduction Let F Z[X, Y ] be a homogeneou, irreducible polynomial of degree d 3 and m a nonzero integer. Then the Diophantine equation (1.1) F (X, Y ) = m i called a Thue equation in honour of A. Thue [15] who proved that Diophantine equation (1.1) ha only finitely many olution (X, Y ) Z. The proof of thi theorem i baed on Thue approximation theorem. Given ε > 0 and an algebraic number α of degree n, then there are only finitely many integer p and q > 0 that atify α p q < q n/ 1 ε. Since the proof of thi approximation theorem i not effective we cannot olve Thue equation by exploiting the proof of Thue. However, Thue oberved that hi approximation theorem can be made effective, if one can find good 000 Mathematic Subject Claification. 11D59, 11D5, 11D09. Key word and phrae. Diophantine equation, parameterized Thue equation, norm form equation, imultaneou Pellian equation. The author wa partially upported by the Autrian Science Foundation, project S MAT. 9
2 10 V. ZIEGLER approximation to α. Although Thue never tated explicitly anything like thi, Thue [16] actually olved the family of Thue equation (a + 1)X n ay n = 1, where n 3 i a prime and a i uitable large with repect to n. He obtained hi good approximation by conidering uitable differential equation and their related hypergeometric function. Mahler [1] wa the firt who tated reult on effective meaurement of algebraic number. For Thue equation of degree 3 Chudnovky [8] give a detailed tudy on the Thue-Siegel method. In the 60 of the previou century, Baker [1, 3] conidered linear form of logarithm. In a further paper [], he ued hi reult on linear form in order to how how Thue equation can be olved algorithmically. Uing Baker method, Bugeaud and Győry [7] computed upper bound for the olution of a ingle Thue equation. Thee bound only depend on the regulator, the degree of the related number field and the degree of the Thue equation. Alo efficient algorithm have been developed by everal author. The mot famou are from Tzanaki and de Weger [18] and from Bilu and Hanrot [6]. In 1990 Thoma [1] conidered the family X 3 (n 1)X Y (n + )XY Y 3 = 1, where n i ome parameter running through all poitive integer. Thi wa the firt time that a family of Thue equation with poitive dicriminant ha been olved. Another practical approach to olve Thue equation i the method of Tzanaki [17] who howed how to reduce quartic Thue equation of certain type to a ytem of Pellian equation. Uing the method of Tzanaki, Dujella and Jadrijević [9] olved the parametrized Thue equation (1.) X nx 3 Y + (6n + )X Y + nxy 3 + Y = 1 by reducing it to the ytem (n + 1)U nv =1, (n )U nz = of Pellian equation. They olved thi ytem for all rational integer n by the method of Baker and Davenport (cf. []) combined with the congruence method (cf. [9]) and a reult of Bennett [5] about imultaneou approximation of quare root. By a refinement of their method, Dujella and Jadrijević (cf. [10]) olved the Thue inequality (1.3) X nx 3 Y + (6n + )X Y + nxy 3 + Y 6n +. The aim of thi paper i to olve following family of Thue equation: (1.) X X 3 Y (ab + (a + b))x Y abxy 3 + a b Y = µ,
3 ON A CERTAIN FAMILY OF THUE EQUATIONS 11 where µ {1, 1}, a, b 1 Z, with a b and 0 ab Z and Z large with repect to a and b. Oberve that for a =, b = 1/ and µ = 1 we obtain equation (1.). In particular we prove the following theorem: Theorem 1.1. Let (X, Y ) be a olution to Thue equation (1.) with Z, a, b 1 Z, a b, a b and 0 ab Z and uppoe > a Then necearily µ = 1. Furthermore, the only olution are (X, Y ) = (±1, 0), (X, Y ) = (0, ±1) if ab = ±1 or thoe lited in Table 1. Table 1. Solution to (1.), provided i large. a b X Y a b X Y 17 ± ±1 15 ± ±1 17 ± 1 15 ± 1 5 ± ±1 3 ± ±1 5 3 ± 1 ± 1 1 ±1 ±1 1 ±1 1 Oberve that there i no olution in the cae of µ = 1 and ufficiently large. Furthermore it i no retriction to aume that a b, ince equation (1.) i ymmetric in a and b. The ret of the paper i organized a follow. In Section we preent ome preliminary reult and invetigate aymptotic expanion of the relevant root. How to reduce Thue equation (1.) to a ytem of Pellian equation i demontrated in Section 3. For olution to thi ytem we will find an upper bound by the hypergeometric method (cf. Section ). In order to obtain a lower bound we will ue Padé approximation in Section 5. The proof of Theorem 1.1 will be finihed in Section 6, where we conider the remaining cae Y = 1. In the lat ection we will tate pecial cae of Theorem 1.1, where a, b 1 Z, repectively a, b Z, and give ome example.. Preliminarie We tart with the norm form equation (.1) N K Q (X + αy ) = ±1,
4 1 V. ZIEGLER where α = ( + a) + ( + b) + ( + a)( + b) +, with Z and a, b 1 Z, uch that ab Z, ( a b and a b. Obviouly, ( ) ( ( ) α i an element of the compoitum K := Q + a) Q + b). In any cae, K i Galoi, ince K i the compoitum of two field that are Galoi over Q. If > max( a, b ), then K i a real field and moreover K i quartic if and only if none of the quantitie ( + a), ( + b) and ( + a)( + b) i a perfect quare. Lemma.1. Aume > ( a +1/). Then K i Galoi, real and quartic with Galoi group G Z/Z Z/Z. Proof. From the dicuion above we know that K i Galoi ( ( and real. ) Let u aume ( + a)( + b) i not a perfect quare. Then Q + a) ( ( ) Q + b) = Q and therefore K i quartic and we know from Galoi theory that the Galoi group i of the wanted form (cf. [11, Chapter VI, Theorem 1.1]). So we are left to prove that neither ( + a), ( + b) nor ( + a)( + b) i a perfect quare. Aume ( + a)( + b) i a perfect quare. From the aumption on a and b we find that ( + a)( + b) i the quare of an integer. On the other hand, we have ( + (a + b)) =( + a)( + b) + (a b) >( + a)( + b), ( ( + (a + b) 1/) =( + a)( + b) + a + b (a b) 1 ) 16 ( + a)( + b) ( ( a + 1/) ) <( + a)( + b), a contradiction, hence ( + a)( + b) i not a perfect quare. Similarly we find and ( + a) >( + a), ( + a 1/) <( + a), ( + b) >( + b), ( + b 1/) <( + b). Therefore neither of ( + a), ( + b) nor ( + a)( + b) i a perfect quare. Becaue of Lemma.1 we aume for the ret of the paper that > ( a + 1/). Moreover we immediately obtain from Galoi theory the conjugate
5 ON A CERTAIN FAMILY OF THUE EQUATIONS 13 α 1,..., α of α: (.) α: α = α 1 = + ( + a) + ( + b) + ( + a)( + b); α = ( + a) + ( + b) ( + a)( + b); α 3 = + ( + a) ( + b) ( + a)( + b); α = ( + a) ( + b) + ( + a)( + b). Therefore we are able to compute the minimal polynomial f Q[X] of f(x) := X X 3 (ab + (a + b))x abx + a b, i.e. norm form equation (.1) i equivalent to Thue equation (.3) F (X, Y ) := X X 3 Y (ab + (a + b))x Y abxy 3 + a b Y = ±1. Furthermore we have proved that α i an algebraic integer. Next we want to invetigate the aymptotic of the α a. Becaue of the tructure of the α, we only have to conider the aymptotic of ( + a), ( + b) and ( + b)( + a). The following expanion i well known 1 + a = n=0 ( ) 1/ a n n n and it i valid for > a. Thi implie ( ) 1/ a n ( + a) = n n 1, n=0 ( ) 1/ b n ( + b) = n n 1, n=0 n ( )( ) 1/ 1/ a k b n k ( + b)( + a) = k n k n 1, n=0 k=0 where all three expanion are valid if > a. The following variant of the uual O-notation i ued. For two function g() and h() we write g() = L(h()) if g() h(). Thi notation i ued in the middle of an expreion in the ame way a it i uually done with O-notation. With thi L-notation we obtain N ( ) ( 1/ a n ( + a) = n n 1 + L ( ) ) 1/ a n n n 1 (.) = n=0 N n=0 n=n+1 ( ) ( ( 1/ a n 1/ n n 1 + L N + 1 ) a N+1 N ). a
6 1 V. ZIEGLER For an exact aymptotic of ( + b)( + a) we conider the N-th coefficient of it expanion. By elementary calculation we oberve that ( ) N ( )( ) C(N) := 1/ 1/ 1/ = if N ; N k N k k=0 1 if N = 0, 1. Since C(N) i decreaing with N, thi implie (.5) N n ( 1/ ( + b)( + a) = k n=0 k=0 ( + L n=n+1 )( 1/ n k C(n) a n n 1 ) a k b n k ) n 1 N n ( )( ) 1/ 1/ a k b n k = k n k n 1 n=0 k=0 ) + L (C(N + 1) a N+1 N. a 3. From Thue equation to Pellian equation In 1993, Tzanaki [17] conidered Thue equation of the form (3.1) F (X, Y ) := a 0 X + a 1 X 3 Y + 6a X Y + a 3 XY 3 + a Y = m uch that F (X, Y ) Z[X, Y ], m Z and a 0 > 0. Furthermore the correponding number field K ha to be Galoi and non-cyclic. If we aume K i not totally complex, i.e. there i ome real root of F (X, 1), then K i the compoitum of two real quadratic field. Furthermore the equation (3.) σ 3 g σ g 3 = 0 ha three ditinct rational root σ 1, σ and σ 3 ; here g and g 3 are invariant of the following form: g =a 0 a a 1 a 3 + 3a, g 3 = det a 0 a 1 a a 1 a a 3. a a 3 a Let H(X, Y ) and G(X, Y ) be the quartic and extic covariant of F (X, Y ) repectively (cf. [13]), i.e. H(X, Y ) = 1 f f X X Y, G(X, Y ) = 1 f f X Y 1 f 8 H. Y X f Y X H Y
7 ON A CERTAIN FAMILY OF THUE EQUATIONS 15 We have H(X, Y ) 1 8 Z[X, Y ], G(X, Y ) 1 96 Z[X, Y ] and (cf. [13, Chapter 5, Theorem 1]) (3.3) H 3 g Hf g 3 f 3 = G. Let u put now H = 1 8 H 0, G = 1 96 G 0, σ i = 1 1 r i (i = 1,, 3). Then H 0, G 0 Z[X, Y ] and r i Z. In view of equation (3.3) we have (H 0 r 1 f)(h 0 r f)(h 0 r 3 f) = 3G 0. Since H and f are relatively prime (cf. [17, Propoition 1]) there exit quarefree integer k 1, k and k 3 and quadratic form G i Z[X, Y ], i = 1,, 3 uch that (3.) H 0 r i f = k i G i i = 1,, 3. If (X, Y ) i a olution to (3.1), we obtain from identity (3.) the ytem (3.5) k G k 1G 1 = (r 1 r )m, k 3 G 3 k 1G 1 = (r 1 r 3 )m of Pellian equation. Applying thi procedure to Thue equation (.3) we obtain a 0 =1, a 1 =, a = g =a b ab (ab + (a + b)), ab + (a + b), a 3 = ab, a =a b ; 3 g 3 = (ab + (a + b))(ab + (b a))(ab + (a b)); 7 σ 1 = 1 (ab + a + b), 3 σ = 1 ( b + ab + a), 3 σ 3 = 1 ( a + ab + b); 3 G 1 = X aby, G = X +axy +aby, G 3 = X +bxy +aby ; k 1 = 8(a + )(b + ), k = 8(b + ), k 3 = 8(a + ). Thi yield the ytem (3.6) with (a + )U V = µa, (b + )U Z = µb, U = X aby, V = X + axy + aby, Z = X + bxy + aby and µ = ±1.
8 16 V. ZIEGLER. Hypergeometric method In thi ection we want to find an upper bound for U if (U, V, Z) i a olution to ytem (3.6). Let u firt oberve that if (U, V, Z) i a olution to (3.6), then alo (±U, ±V, ±Z) i a olution to (3.6). Therefore we may aume without lo of generality U, V, Z 0. Furthermore U = 0 yield V = ±a and, ince we aume > ( a + 1/), we have V < 1, hence V = 0 and a = 0, a contradiction. Similar argument apply to V and Z, therefore we may aume U, V, Z > 0. In order to prove an upper bound for U we will dicu firt ome approximation propertie of olution (U, V, Z) to (3.6). Lemma.1. Let (U, V, Z) be a olution to ytem (3.6) with U, V, Z > 0. Then V U 1 + a a 1 U ( + a) ; Z U 1 + b b 1 U ( + b). Proof. We only prove the firt inequality. The proof of the econd inequality i analogou. One jut ha to replace Z by V and b by a. Since U, V > 0 we have 1 + a U V + U 1 + a and therefore 1 V U + a V U 1 + a = V + U 1 + a V + U 1 + a a U 1 + a/ = a 1 U ( + a). Diviion with U yield the lemma. Since we aume a b the lemma above how that ( V (.1) max U 1 + a ), Z U 1 + b a 1 U ( + a). Hence we have found a good imultaneou approximation to 1 + a and 1 + b. The following dicuion will how that thi approximation i in ome ene too good. We tart with a theorem of Bennett [5, Theorem 3.]. Theorem.. If a i, p i, q and N are integer for 0 i with a 0 < a 1 < a, a j = 0 for ome 0 j, q nonzero and N > M 9, where M = max a i, i=0,1,
9 ON A CERTAIN FAMILY OF THUE EQUATIONS 17 then we have where and max i=0,1, ( 1 + a i N p ) i q > (130NΥ) 1 q λ = c 1 q λ, log(33n Υ) λ = 1 + log (1.7N ) 0 i<j (a i a j ) (a a 0 ) (a a 1 ) if a a 1 a 1 a 0, a Υ = a 0 a 1 (a a 0 ) (a 1 a 0 ) if a a 1 < a 1 a 0. a 1 + a a 0 We want to apply Theorem. to a i = a and a j = b with ome i, j {0, 1, }, i j, a = a /, b = b / and N =. Firt let u etimate Υ. Therefore we have to ditinguih between 6 cae (remind that we alway aume a b, hence a b ). Suppoe a > b > 0 and a b b 0. Then we have a b and Υ = (a 0) (a b ) a b 0 = a (a b ) a b 3 a 3. Let a > b > 0 and a b < b 0. The lat inequality i a < b and therefore Υ = (a 0) (b 0) b + a 0 = a b a + b 3 a 3. Aume a > 0 > b. Since a b we have a a 1 a 1 a 0, i.e. Υ = (a b ) (a 0) a b 0 = (a b ) a a b 3 a 3. Provided b > 0 > a we have a a 1 a 1 a 0, hence in both cae < and = we find Υ = (b a ) (0 a ) 0 + b a = (b a ) a a + b 3 a 3. In the cae of 0 > b > a and 0 b b a we have b a and obtain following etimation Υ = (0 b ) (0 a ) 0 b a = b a b a 3 a 3. At lat we conider the cae 0 > b > a and 0 b < b a. Therefore b < a and Υ = (0 a ) (b a ) b + 0 a = a (b a ) a + b 3 a 3.
10 18 V. ZIEGLER All cae together yield the etimation Υ 3 a 3 = 56 3 a 3. Hence we have c c := a 3, λ λ log(116 a 3 ) = log(7. ) log(1638 a 6 ). We want to have λ <. Therefore we conider the inequality or equivalently 1 < log(116 a 3 ) log(7. ) log(1638 a 6 ) log log + 3 log a < log 7. + log log log a. The lat inequality hold if log > log log a, i.e. > a 9. Therefore we will aume for the ret of thi ection > a 9. The aumption N > M 9, i.e. > 9 a 9 i now fulfilled and by an application of Theorem., together with Lemma.1, we obtain ( c 1 V U λ < max U 1 + a ), Z U 1 + b Taking logarithm and olving for log U yield log U < 1 (log λ c + log a 1 ) log(( + a)) (.) < 1 λ a U 1 ( + a). ( log log + log a 1 ) 3 ( log ) < 1 ( log a ). λ Let u aume > c 0 a 9+r. Then we have 1 λ = 1 1 (.3) log(116 a 3 ) log(7. ) log(1638 a 6 ) log(7. ) log(1638 a 6 ) = log(7. ) log(1638 a 6 ) log(116 a 3 ) = log 6 log a + log ( ) log 9 log a + log ( ) ( ) 6 9+r log r log c 0 + log ( ) < ( ) r log r log c 0 + log ( ) = 1 + r r 9 + r r ) ( 1 log ) 1 log c 0 r log ( r log + 9 log c 0 (9 + r) log ( )
11 ON A CERTAIN FAMILY OF THUE EQUATIONS 19 1 The inequality above hold, ince λ i a function increaing with a. Let u conider the econd fraction of the lat line in (.3). A one can eaily compute, the numerator i 0 if 31r 1+ (.) c 0 1 ( ) 1 r r 6. 5 If c 0 fulfill thi inequality another computation how that the denominator of thi fraction i poitive for 1. Hence we have proved (.5) r λ < r provided (.) hold. By combining (.) and (.5) we obtain (.6) 8 + 8r 1 + r log U < log a r r 8 + 8r 8 + 8r < log (9 + r)r (9 + r)r log c r. r Next we want to find an upper bound for Y. Firt let u aume ab < 0. Then we have U = X aby Y and therefore we find log Y 1 log U. Let now ab > 0. Then Z = X + bxy + aby = (X + by ) + (ab b )Y 3 Y. On the other hand, the econd Pellian equation of (3.6) yield ( Z = 1 + b ) U ± b ( ) < U < (U ), hence Therefore we have proved log Y < log U. Propoition.3. Let (X, Y ) be a olution to (1.) and aume > c 0 (r) a 9+r with r > 0 and ( ) 1 r 31r c 0 (r) := r 6. 5 Then log Y < + r + r log (9 + r)r (9 + r)r log c 1 + r 0(r) r
12 0 V. ZIEGLER 5. Approximation propertie of α In the previou ection we have found an upper bound for log Y if i large with repect to a and b. In thi ection we find a lower bound for Y provided Y > 1. Thi bound will be found by uing approximation propertie of the root α i, 1 i. We further aume > a 9. Firt we prove the following lemma: Lemma 5.1. Let (X, Y ) be a olution to (1.). Then at leat one of the following cae occur 8 X α 1 Y < Y 3 3 ; X α 8 Y < 3.99 Y 3 ; 8 X α 3 Y <.99 Y 3 ; X α 8 Y < 3.99 Y 3. Proof. From (.), (.) and (.5) we find (5.1) ( ) α 1 = + a + b + L 3 a ( ) a α 3 = b + L 3 a a ; α = L Let j be choen uch that X α j Y = Hence ( ; α = a + L 3 a ( 3 a a ). a ) ; min X α iy. Then we have i=1,,3, Y α i α j X α i Y + X α j Y X α i Y. 1 (5.) X α j Y = i j X α iy 8 Y 3 i j α j α i. Some elementary computation yield lower bound for i j α j α i and from thee we obtain the lemma. Propoition 5.. Let (X, Y ) be a olution to (1.) with Y > 1 and > a 9. According to the four cae in Lemma 5.1 we have Y > a 6 ; Y > a 7 ; Y > 3 ; Y > a a 8. Hence, in all cae we have Y > a 7.
13 ON A CERTAIN FAMILY OF THUE EQUATIONS 1 Proof. From (5.1) together with Lemma 5.1 we obtain X ( + a + b)y < Y a X + ay < Y a X + by < Y a X < Y a Y 3 < Y 0.75 a, Y Y Y 3 < Y a, < Y a, < Y a, according to the four poible cae. Since the left hand ide are 1 Z, we conclude that they vanih if (5.3) Y a, Y a, Y a, Y a. Auming (5.3) we obtain X = (+a+b)y, X = ay, X = by and X = 0 repectively. Inerting thee relation in F (X, Y ) = ±1 we deduce F (Y ( + a + b), Y ) = Y 16 (a ab + b ) + (a 3 + a b + ab + b 3 ) +(a + a 3 b + 3a b + ab 3 + b ) > Y a > 1, F ( Y a, Y ) = Y a (a b) Y > 1, F ( Y b, Y ) = Y b (a b) Y > 1, F (0, Y ) = Y > 1. In any cae we get a contradiction and therefore we may aume (5.) Y > a, Y > a, Y > a, Y > a. Now we plit the proof into the four cae according to Lemma 5.1. Cae 1: We ue the approximation (5.5) α 1 = + a + b a ab + b + a3 a b ab + b 3 16 ( 0.37 a + L Let u denote by ᾱ 1 the approximation (5.5) with omitted L-term. Padé approximation we find polynomial P and Q uch that ᾱ 1 P Q = (a3 a b ab + b 3 ) 16 = a6 a 5 b 3a b + 10a 3 b 3 3a b ab 5 + b 6 16, 3 ). Uing
14 V. ZIEGLER where Q :=16 (a ab + b ) + (3a 3 a b ab + 3b 3 ) + (a + 3a 3 b 5a b + 3ab 3 + b ), P :=(a ab + b ) + (a 3 a b ab + b 3 ). By elementary calculation we obtain P < 1 a + 6 a 3 < a. Uing Lemma 5.1 we obtain (P X QY ) Y (P ᾱ 1 Q) Y P (α 1 ᾱ 1 ) < Some elementary calculation together with (5.) yield (5.6) 8 P Y 3 3. P X QY < Y P ᾱ 1 Q + Y P (α 1 ᾱ 1 ) + Y P 10. a 8 ) 7 < Y ( a a a 10 6 < Y.83 a 6. Since P X QY 1 56 Z, we have P X QY = 0 provided Y a 6. Inerting thi relation in F (X, Y ) = ±1 yield (5.7) Y (56 A + R 1 ) = ±P, where A = ( a ab + b ) 3 (a 6 3a 5 b + 9a b 13a 3 b 3 + 9a b 3ab 5 + b 6 ). For a 0, a 1 > 0 we have (5.8) X a 0 XY + a 1 Y = Therefore we find the etimation ( a0 a 1 X a 1 Y ( ) X 1 a 0. a 1 ) ( ) + X 1 a 0 a 1 A = ( a ab + b ) 3 (a 6 3a 5 b + 9a b 13a 3 b 3 + 9a b 3ab 5 + b 6 ) > 7 6 a 6 (a ab + 5.8b )(a 0.183ab b )(a 1.818ab + b ) > a Furthermore, R 1 i ome expreion of lower term that one can etimate by R 1 < a a a a 16 < a 13. Therefore we receive from (5.7) and (5.) On the other hand Y 56 A + R 1 > Y a 1 > a. P < 073 a 8
15 ON A CERTAIN FAMILY OF THUE EQUATIONS 3 which yield a contradiction to (5.7) if > 18 a. Therefore we have proved Y > a, i.e. the firt cae. The proof of the other cae i imilar and 6 we will dicu them le detailed. Cae : For thi cae we ue the approximation :=ᾱ {}}{ α = a + a ab + a3 + a b + ab a a 3 b a b + ab 3 ( ) a 5 + L. For ᾱ we obtain a Padé approximation Q P ᾱ with Q :=16 a(a ab + b ) a(11a 3 + 9a b + 9ab + b 3 ) a(13a + 3a 3 b + 31a b + 17ab 3 + 7b ), P := 16 (a ab + b ) + a(10a a b + 7ab + b 3 ) + (5a + 30a 3 b + 9a b + 1ab 3 + b ). Similarly a in the firt cae we compute (5.9) P X QY < Y P ᾱ Q + Y P (α ᾱ ) + Y P a 8 ( ) 5 85 a 8 < Y a a 10 3 < Y 7.9 a 7. Therefore P X QY = 0 provided Y a 7. From F (X, Y ) = ±1 we obtain now (5.10) Y ( 5 A + R 1 ) = ±P, with A =10a (a b) (a ab + b ) 1a 3 + 9a b + 7ab + 5b a 10, R a a a a a 0 < a , P a 8. Comparing the bound from the right hand ide and left hand ide of (5.10), we find a < a 8, which i a contradiction for > a 6.
16 V. ZIEGLER Cae 3: Now we ue the following approximation :=ᾱ 3 {}}{ ab + b α 3 = b + + a b + ab b a3 b a b b 3 a + 5b ( ) a 5 + L. Applying Padé theorem to ᾱ 3, we obtain an approximation Q P ᾱ 3 with Q :=16 b(a ab + b ) b(a 3 + 9a b + 9ab + 11b 3 ) b(7a + 17a 3 b + 31a b + 3ab b ), P := 16 (a ab + b ) + a(a 3 + 7a b + 11ab + 10b 3 ) + (a + 1a 3 b + 9a b + 30ab 3 + 5b ). Similarly a in the firt cae, we obtain (5.11) P X QY < Y P ᾱ 3 Q + Y P (α 3 ᾱ 3 ) + Y P a 8 ( ) 5 85 a 8 < Y a a 10 3 < Y 7.9 a 7. Therefore P X QY = 0 provided Y a 7. If we inert thi relation in F (X, Y ) = ±1, we get (5.1) Y ( 5 A + R 1 ) = ±P, with A =10b (a b) (a ab + b ) 5a 3 + 7a b + 9ab + 1b a 10, R a a a a a 0 < a , P a 8. Thi time we deduce from (5.1) a < a 8, a contradiction, provided > a 6. Cae : In the lat cae we ue the method of Padé approximation twice. Firt we ue the following approximation α = :=ᾱ {}}{ ab a b + ab ( ) 0.35 a 16 +L 3
17 ON A CERTAIN FAMILY OF THUE EQUATIONS 5 and by Padé theorem we obtain an approximation Q P ᾱ with Q :=ab, P := + a + b. Similarly a in the firt cae we obtain (5.13) P X QY < Y P ᾱ Q + Y P (α ᾱ ) + Y P a 8 ( ) 5 a < Y a a 8 < Y 1.1 a. Therefore P X QY = 0 provided Y a. Thi relation together with F (X, Y ) = ±1 yield (5.1) Y ( A + R 1 ) = ±P, where From (5.1) we deduce A =16a b (a ab + b ) 1 a, R 1 a a 8 >.001 a 7, P < 57 a 6, a contradiction provided > 57 a 3. Therefore we may aume Y > a. For the econd application of Padé approximation we ue :=ᾱ {}}{ ab α = a b + ab 16 + a3 b + a b + ab a b + a 3 b + a b 3 + 5ab ( ) a 6 + L. 5 Therefore we find an approximation Q P ᾱ with Q :=ab(a ab + b ) + ab(a 3 a b ab + b 3 ), P :=16 (a ab + b ) + (3a 3 a b ab + 3b 3 ) + (a + 3a 3 b 5a b + 3ab 3 + b ). Similarly a in the firt cae we obtain P X QY < Y P ᾱ Q + Y P (α ᾱ ) + Y P a 16 9 < Y (1.0 a a a 18 ) (5.15) 7 < Y 6.9 a 8 3.
18 6 V. ZIEGLER From (5.15) we deduce P X QY = 0, if Y a 8. Let u inert thi relation into the original Thue equation F (X, Y ) = ±1. Then (5.16) Y ( A + R 1 ) = ±P, with A =56a b ( a ab + b ) 3 (a 6 3a 5 b + 9a b 13a 3 b 3 + 9a b 3ab 5 + b 6 ) > 17.1 a 1, R a a a a 0 < 3 a , P a 8. Thi, combined with (5.16), implie a < a 8, a contradiction, ince we aume > a 9 > 58 a 3. If a and b are given, one may obtain better reult than thoe proved in Propoition 5.. Indeed, one only ha to apply the method of Padé approximation ucceively to obtain reult of the form Y > µ c(a, b, µ) for ome µ >. In the general cae the difficulty to find optimal or even ueful etimation rapidly grow. Oberve that the technical bound c 0 = in Propoition 5. i quite large. In the cae of a, b Z the technical bound actually exceed the bound that one obtain by comparing lower and upper bound for log Y (cf. (5.17) and Theorem 7.1). Corollary 5.3. Let (X, Y ) be a olution to (1.). Then Y 1 provided > a Proof. Suppoe (X, Y ) i a olution to (1.) with Y > 1. Let u compare the bound from Propoition.3 and 5. and aume > c 1 a 9+r with c 1 > max( , c 0 (r)) (cf. Propoition.3). Then we obtain ( 7 ) log r 9 + r log c 1 log < log Y + r < (9 + r)r (log log c 1) r r and therefore (5.17) r + 7r 11r + log + r(r + 9) r + 9 log c 1 < r log r The coefficient of log i poitive if r > 7+ 1 and (5.17) fail provided i large enough. Suppoe now r = Then the coefficient of log
19 ON A CERTAIN FAMILY OF THUE EQUATIONS 7 i zero and (5.17) alo fail if c 1 i large enough, i.e. c 1 > > max( , c 0 (r)). 6. The cae of Y = 1 In view of Corollary 5.3 it remain to conider the cae of Y 1 in order to prove Theorem 1.1. The cae Y = 0 yield the trivial olution (X, Y ) = (±1, 0). Therefore we are left to the four cae Y = ±1 and µ = ±1 with mixed ign. Since with (X, Y ) alo ( X, Y ) i a olution to (1.) we only have to check the cae Y = 1 and µ = ±1. Let u conider firt the cae Y = 1 and µ = 1. Thue equation (1.) reduce to P (X) = X X 3 (ab + (a + b))x abx + a b 1 = 0. Since F (X, 1) = P (X) + 1, we deduce that the root of P (X) and F (X, 1) are cloe together. Therefore we want to prove that the root of P (X) lie in the dijoint interval I 1 :=( + a + b 1/8, + a + b + 1/8), I :=( a 1/8, a + 1/8), I 3 :=( b 1/8, b + 1/8) I :=( 1/8, 1/8). Let u conider the quantitie P ( + a + b 1 8 )P ( + a + b ) = (a + b) 5 + P ( a 1 8 )P ( a ) = ( 1+6a )( 1+6(a b) ) > a + and P ( b 1 8 )P ( b ) = ( 1+6b )( 1+6(a b) ) > a + P ( 1 8 )P ( 1 8 ) = ( 1+6a )( 1+6b ) > a + which yield that if i large enough then each root of P lie in one of the interval I 1, I, I 3 or I. A more detailed analyi yield that > 130. a 3 i adequate (alo for the cae µ = 1). Therefore the only integral olution may be X = + a + b, a, b or X = 0. Inerting in (1.) yield 16(a ab + b ) + (a 3 + a b + ab + b 3 ) +(a + a 3 b + 3a b + ab 3 + b ) = 1, a (a b) = 1, b (a b) = 1, a b = 1, repectively. The firt equation fail if i too large, i.e. >.36 a. The other equation only yield olution lited in Table 1. Similar argument apply for µ = 1. Oberve that in thi cae there are no olution. Therefore we have proved the following propoition:
20 8 V. ZIEGLER Propoition 6.1. Let (X, Y ) be a olution to (1.) with Y = 1 and let > 130. a 3. Then the olution (X, Y ) i lited in Theorem 1.1 or Table 1. Combining Corollary 5.3 and Propoition 6.1 we immediately obtain Theorem Some example Firt let u tate a theorem that one may obtain by recomputing the proof of Theorem 1.1. Theorem 7.1. If a, b 1 Z, repectively a, b Z. Then Theorem 1.1 alo hold for > a 9+ 1, repectively > a Let u now conider four example to illutrate Theorem 1.1 and 7.1: Let a = and b = 1/. Then we have the Thue equation (7.1) X X 3 Y + (6 + )X Y + XY 3 + Y = ±1. Let (X, Y ) be a olution to (7.1). Then (X, Y ) = (0, ±1) or (X, Y ) = (±1, 0) if > Note that Thue equation (7.1) ha been olved for all 0 in the cae of the +-ign by Dujella and Jadrijević [9]. Let a = 1 and b = 1. Then (±1, 0) and (0, ±1) are the only olution to the Thue equation (7.) X X 3 Y + X Y + XY 3 + Y = ±1 provided > Let a = 5/, b = and > Then the Diophantine equation (7.3) X X 3 Y ( )X Y 0XY 3 + 5Y = ±1 ha only the olution (±1, 0), (, 1) and (, 1). Let a = and b = 13. Then (X, Y ) = (±1, 0) i the only olution to Thue equation (7.) X X 3 Y + (6 3)X Y + 5XY Y = ±1 if we aume > All four example have been olved uing Theorem 1.1, repectively Theorem 7.1. In mot cae one could obtain harper bound for, if one would apply the method of Padé approximation everal time more. However Dujella and Jadrijević [9] ued the congruence method in order to obtain a harp etimate for in the cae of equation (7.1). In order to apply thi powerful method, we tart with ytem (3.6) and multiply the firt equation by b and
21 ON A CERTAIN FAMILY OF THUE EQUATIONS 9 put k = b + ab and imilarly we multiply the econd equation by a and put k = a + ab. In both cae we obtain a Pellian equation of the form ( (7.5) X k + ab ) ( Y k ab ) = µab. Note that the coefficient of (7.5) need not to be integer. The congruence method eentially depend on finding a fundamental olution to (7.5), i.e. in the cae above we have to find a fundamental olution correponding to X (k 1)Y = 1, X (k 1)Y = 1, X (k 5)Y = 1, X (16k 676)Y = 1, repectively. Note that the Pell equation above have integral coefficient. The firt two cae yield the fundamental olution k + k 1. For the other cae no parameterized fundamental olution are known and we cannot apply the powerful congruence method in thoe cae. A cloe look on (7.5) how that only in the cae of ab = 1 and a, b 1 Z, or ab = and a, b Z, we can find parameterized fundamental olution. In all other cae no parameterized fundamental olution are known. Reference [1] A. Baker, Linear form in the logarithm of algebraic number I, II, III, Mathematika 13 (1966), 0-16; ibid. 1 (1967), ; ibid. 1 (1967), 0-8. [] A. Baker, Contribution to the theory of Diophantine equation I. On the repreentation of integer by binary form, Philo. Tran. Roy. Soc. London Ser. A 63 (1967/1968), [3] A. Baker, Linear form in the logarithm of algebraic number IV, Mathematika 15 (1968), [] A. Baker and H. Davenport, The equation 3x = y and 8x 7 = z, Quart. J. Math. Oxford Ser. () 0 (1969), [5] M. A. Bennett, On the number of olution of imultaneou Pell equation, J. Reine Angew. Math. 98 (1998), [6] Yu. Bilu and G. Hanrot, Solving Thue equation of high degree, J. Number Theory 60 (1996), [7] Y. Bugeaud and K. Győry, Bound for the olution of Thue-Mahler equation and norm form equation, Acta Arith. 7 (1996), [8] G. V. Chudnovky, On the method of Thue-Siegel, Ann. of Math. () 117 (1983), [9] A. Dujella and B. Jadrijević, A parametric family of quartic Thue equation, Acta Arith. 101 (00), [10] A. Dujella and B. Jadrijević, A family of quartic Thue inequalitie, Acta Arith. 111 (00), [11] S. Lang, Algebra, Springer-Verlag, New York, 00. [1] K. Mahler, Zur Approximation algebraicher Zahlen (I), Math. Ann. 107 (1933),
22 30 V. ZIEGLER [13] L. J. Mordell, Diophantine equation, Academic Pre, London, [1] E. Thoma, Complete olution to a family of cubic Diophantine equation, J. Number Theory 3 (1990), [15] A. Thue, Über Annäherungwerte algebraicher Zahlen, J. Reine und Angew. Math. 135 (1909), [16] A. Thue, Berechnung aller Löungen gewier Gleichungen von der Form ax r by r = f, Vid. Skrivter I Mat.- Naturv. Klae (1918), 1-9. [17] N. Tzanaki, Explicit olution of a cla of quartic Thue equation, Acta Arith. 6 (1993), [18] N. Tzanaki and B. M. M. de Weger, On the practical olution of the Thue equation, J. Number Theory 31 (1989), Intitut für Mathematik A Techniche Univerität Graz Steyrergae 30 A-8010 Graz Autria ziegler@finanz.math.tugraz.at Received: Revied:
ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS
ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS VOLKER ZIEGLER Abtract We conider the parameterized Thue equation X X 3 Y (ab + (a + bx Y abxy 3 + a b Y = ±1, where a, b 1 Z uch that
More informationA parametric family of quartic Thue equations
A parametric family of quartic Thue equations Andrej Dujella and Borka Jadrijević Abstract In this paper we prove that the Diophantine equation x 4 4cx 3 y + (6c + 2)x 2 y 2 + 4cxy 3 + y 4 = 1, where c
More informationA family of quartic Thue inequalities
A family of quartic Thue inequalities Andrej Dujella and Borka Jadrijević Abstract In this paper we prove that the only primitive solutions of the Thue inequality x 4 4cx 3 y + (6c + 2)x 2 y 2 + 4cxy 3
More informationAn Inequality for Nonnegative Matrices and the Inverse Eigenvalue Problem
An Inequality for Nonnegative Matrice and the Invere Eigenvalue Problem Robert Ream Program in Mathematical Science The Univerity of Texa at Dalla Box 83688, Richardon, Texa 7583-688 Abtract We preent
More informationOne Class of Splitting Iterative Schemes
One Cla of Splitting Iterative Scheme v Ciegi and V. Pakalnytė Vilniu Gedimina Technical Univerity Saulėtekio al. 11, 2054, Vilniu, Lithuania rc@fm.vtu.lt Abtract. Thi paper deal with the tability analyi
More information7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281
72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition
More informationConvex Hulls of Curves Sam Burton
Convex Hull of Curve Sam Burton 1 Introduction Thi paper will primarily be concerned with determining the face of convex hull of curve of the form C = {(t, t a, t b ) t [ 1, 1]}, a < b N in R 3. We hall
More informationExplicit solution of a class of quartic Thue equations
ACTA ARITHMETICA LXIV.3 (1993) Explicit solution of a class of quartic Thue equations by Nikos Tzanakis (Iraklion) 1. Introduction. In this paper we deal with the efficient solution of a certain interesting
More informationLecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004
18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem
More informationTRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL
GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National
More informationParametrized Thue Equations A Survey
Parametrized Thue Equations A Survey Clemens Heuberger Institut fr Mathematik B Technische Universitt Graz 8010 Graz Austria clemens.heuberger@tugraz.at January 29, 2005 Abstract We consider families of
More informationSOME RESULTS ON INFINITE POWER TOWERS
NNTDM 16 2010) 3, 18-24 SOME RESULTS ON INFINITE POWER TOWERS Mladen Vailev - Miana 5, V. Hugo Str., Sofia 1124, Bulgaria E-mail:miana@abv.bg Abtract To my friend Kratyu Gumnerov In the paper the infinite
More informationBogoliubov Transformation in Classical Mechanics
Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How
More informationLecture 8: Period Finding: Simon s Problem over Z N
Quantum Computation (CMU 8-859BB, Fall 205) Lecture 8: Period Finding: Simon Problem over Z October 5, 205 Lecturer: John Wright Scribe: icola Rech Problem A mentioned previouly, period finding i a rephraing
More informationarxiv: v1 [math.mg] 25 Aug 2011
ABSORBING ANGLES, STEINER MINIMAL TREES, AND ANTIPODALITY HORST MARTINI, KONRAD J. SWANEPOEL, AND P. OLOFF DE WET arxiv:08.5046v [math.mg] 25 Aug 20 Abtract. We give a new proof that a tar {op i : i =,...,
More informationOn Waring s problem in finite fields
ACTA ARITHMETICA LXXXVII.2 (1998 On Waring problem in finite field by Arne Winterhof (Braunchweig 1. Introduction. Let g( p n be the mallet uch that every element of F p n i a um of th power in F p n.
More informationON TESTING THE DIVISIBILITY OF LACUNARY POLYNOMIALS BY CYCLOTOMIC POLYNOMIALS Michael Filaseta* and Andrzej Schinzel 1. Introduction and the Main Theo
ON TESTING THE DIVISIBILITY OF LACUNARY POLYNOMIALS BY CYCLOTOMIC POLYNOMIALS Michael Filaeta* and Andrzej Schinzel 1. Introduction and the Main Theorem Thi note decribe an algorithm for determining whether
More informationSOLUTIONS TO ALGEBRAIC GEOMETRY AND ARITHMETIC CURVES BY QING LIU. I will collect my solutions to some of the exercises in this book in this document.
SOLUTIONS TO ALGEBRAIC GEOMETRY AND ARITHMETIC CURVES BY QING LIU CİHAN BAHRAN I will collect my olution to ome of the exercie in thi book in thi document. Section 2.1 1. Let A = k[[t ]] be the ring of
More informationComputers and Mathematics with Applications. Sharp algebraic periodicity conditions for linear higher order
Computer and Mathematic with Application 64 (2012) 2262 2274 Content lit available at SciVere ScienceDirect Computer and Mathematic with Application journal homepage: wwweleviercom/locate/camwa Sharp algebraic
More informationChip-firing game and a partial Tutte polynomial for Eulerian digraphs
Chip-firing game and a partial Tutte polynomial for Eulerian digraph Kévin Perrot Aix Mareille Univerité, CNRS, LIF UMR 7279 3288 Mareille cedex 9, France. kevin.perrot@lif.univ-mr.fr Trung Van Pham Intitut
More informationLINEAR ALGEBRA METHOD IN COMBINATORICS. Theorem 1.1 (Oddtown theorem). In a town of n citizens, no more than n clubs can be formed under the rules
LINEAR ALGEBRA METHOD IN COMBINATORICS 1 Warming-up example Theorem 11 (Oddtown theorem) In a town of n citizen, no more tha club can be formed under the rule each club have an odd number of member each
More informationCHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS
CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3
More informationManprit Kaur and Arun Kumar
CUBIC X-SPLINE INTERPOLATORY FUNCTIONS Manprit Kaur and Arun Kumar manpreet2410@gmail.com, arun04@rediffmail.com Department of Mathematic and Computer Science, R. D. Univerity, Jabalpur, INDIA. Abtract:
More informationOnline Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat
Online Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat Thi Online Appendix contain the proof of our reult for the undicounted limit dicued in Section 2 of the paper,
More informationProblem Set 8 Solutions
Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem
More informationNew bounds for Morse clusters
New bound for More cluter Tamá Vinkó Advanced Concept Team, European Space Agency, ESTEC Keplerlaan 1, 2201 AZ Noordwijk, The Netherland Tama.Vinko@ea.int and Arnold Neumaier Fakultät für Mathematik, Univerität
More informationarxiv: v1 [math.ac] 30 Nov 2012
ON MODULAR INVARIANTS OF A VECTOR AND A COVECTOR YIN CHEN arxiv:73v [mathac 3 Nov Abtract Let S L (F q be the pecial linear group over a finite field F q, V be the -dimenional natural repreentation of
More informationList coloring hypergraphs
Lit coloring hypergraph Penny Haxell Jacque Vertraete Department of Combinatoric and Optimization Univerity of Waterloo Waterloo, Ontario, Canada pehaxell@uwaterloo.ca Department of Mathematic Univerity
More informationON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang
Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang
More informationControl Systems Analysis and Design by the Root-Locus Method
6 Control Sytem Analyi and Deign by the Root-Locu Method 6 1 INTRODUCTION The baic characteritic of the tranient repone of a cloed-loop ytem i cloely related to the location of the cloed-loop pole. If
More informationIEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation
IEOR 316: Fall 213, Profeor Whitt Topic for Dicuion: Tueday, November 19 Alternating Renewal Procee and The Renewal Equation 1 Alternating Renewal Procee An alternating renewal proce alternate between
More informationMATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.:
MATEMATIK Datum: 20-08-25 Tid: eftermiddag GU, Chalmer Hjälpmedel: inga A.Heintz Telefonvakt: Ander Martinon Tel.: 073-07926. Löningar till tenta i ODE och matematik modellering, MMG5, MVE6. Define what
More informationChapter 4. The Laplace Transform Method
Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination
More informationCodes Correcting Two Deletions
1 Code Correcting Two Deletion Ryan Gabry and Frederic Sala Spawar Sytem Center Univerity of California, Lo Angele ryan.gabry@navy.mil fredala@ucla.edu Abtract In thi work, we invetigate the problem of
More informationONLINE APPENDIX: TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES
ONLINE APPENDIX: TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES CHRISTOPHER P. CHAMBERS, FEDERICO ECHENIQUE, AND KOTA SAITO In thi online
More informationIntroduction to Laplace Transform Techniques in Circuit Analysis
Unit 6 Introduction to Laplace Tranform Technique in Circuit Analyi In thi unit we conider the application of Laplace Tranform to circuit analyi. A relevant dicuion of the one-ided Laplace tranform i found
More informationMulticolor Sunflowers
Multicolor Sunflower Dhruv Mubayi Lujia Wang October 19, 2017 Abtract A unflower i a collection of ditinct et uch that the interection of any two of them i the ame a the common interection C of all of
More informationTHE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON
Anal. Theory Appl. Vol. 28, No. (202), 27 37 THE HAUSDORFF MEASURE OF SIERPINSKI CARPETS BASING ON REGULAR PENTAGON Chaoyi Zeng, Dehui Yuan (Hanhan Normal Univerity, China) Shaoyuan Xu (Gannan Normal Univerity,
More informationarxiv: v2 [math.nt] 30 Apr 2015
A THEOREM FOR DISTINCT ZEROS OF L-FUNCTIONS École Normale Supérieure arxiv:54.6556v [math.nt] 3 Apr 5 943 Cachan November 9, 7 Abtract In thi paper, we etablih a imple criterion for two L-function L and
More informationSocial Studies 201 Notes for November 14, 2003
1 Social Studie 201 Note for November 14, 2003 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
More informationOSCILLATIONS OF A CLASS OF EQUATIONS AND INEQUALITIES OF FOURTH ORDER * Zornitza A. Petrova
МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2006 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2006 Proceeding of the Thirty Fifth Spring Conference of the Union of Bulgarian Mathematician Borovet, April 5 8,
More informationLecture 3. January 9, 2018
Lecture 3 January 9, 208 Some complex analyi Although you might have never taken a complex analyi coure, you perhap till know what a complex number i. It i a number of the form z = x + iy, where x and
More informationPreemptive scheduling on a small number of hierarchical machines
Available online at www.ciencedirect.com Information and Computation 06 (008) 60 619 www.elevier.com/locate/ic Preemptive cheduling on a mall number of hierarchical machine György Dóa a, Leah Eptein b,
More informationTHE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO
THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO J. W. NEUBERGER Abtract. A pecial cae, called the divergence-free cae, of the Jacobian Conjecture in dimenion two i proved. Thi note outline an
More informationc n b n 0. c k 0 x b n < 1 b k b n = 0. } of integers between 0 and b 1 such that x = b k. b k c k c k
1. Exitence Let x (0, 1). Define c k inductively. Suppoe c 1,..., c k 1 are already defined. We let c k be the leat integer uch that x k An eay proof by induction give that and for all k. Therefore c n
More informationAvoiding Forbidden Submatrices by Row Deletions
Avoiding Forbidden Submatrice by Row Deletion Sebatian Wernicke, Jochen Alber, Jen Gramm, Jiong Guo, and Rolf Niedermeier Wilhelm-Schickard-Intitut für Informatik, niverität Tübingen, Sand 13, D-72076
More informationA Constraint Propagation Algorithm for Determining the Stability Margin. The paper addresses the stability margin assessment for linear systems
A Contraint Propagation Algorithm for Determining the Stability Margin of Linear Parameter Circuit and Sytem Lubomir Kolev and Simona Filipova-Petrakieva Abtract The paper addree the tability margin aement
More informationStochastic Optimization with Inequality Constraints Using Simultaneous Perturbations and Penalty Functions
Stochatic Optimization with Inequality Contraint Uing Simultaneou Perturbation and Penalty Function I-Jeng Wang* and Jame C. Spall** The John Hopkin Univerity Applied Phyic Laboratory 11100 John Hopkin
More informationA PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY
Volume 8 2007, Iue 3, Article 68, 3 pp. A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ DEPARTMENT OF APPLIED MATHEMATICS NAVAL POSTGRADUATE
More informationNULL HELIX AND k-type NULL SLANT HELICES IN E 4 1
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 57, No. 1, 2016, Page 71 83 Publihed online: March 3, 2016 NULL HELIX AND k-type NULL SLANT HELICES IN E 4 1 JINHUA QIAN AND YOUNG HO KIM Abtract. We tudy
More informationA CATEGORICAL CONSTRUCTION OF MINIMAL MODEL
A ATEGORIAL ONSTRUTION OF MINIMAL MODEL A. Behera, S. B. houdhury M. Routaray Department of Mathematic National Intitute of Technology ROURKELA - 769008 (India) abehera@nitrkl.ac.in 512ma6009@nitrkl.ac.in
More informationLecture 9: Shor s Algorithm
Quantum Computation (CMU 8-859BB, Fall 05) Lecture 9: Shor Algorithm October 7, 05 Lecturer: Ryan O Donnell Scribe: Sidhanth Mohanty Overview Let u recall the period finding problem that wa et up a a function
More informationAssignment for Mathematics for Economists Fall 2016
Due date: Mon. Nov. 1. Reading: CSZ, Ch. 5, Ch. 8.1 Aignment for Mathematic for Economit Fall 016 We now turn to finihing our coverage of concavity/convexity. There are two part: Jenen inequality for concave/convex
More informationSimple Observer Based Synchronization of Lorenz System with Parametric Uncertainty
IOSR Journal of Electrical and Electronic Engineering (IOSR-JEEE) ISSN: 78-676Volume, Iue 6 (Nov. - Dec. 0), PP 4-0 Simple Oberver Baed Synchronization of Lorenz Sytem with Parametric Uncertainty Manih
More informationFlag-transitive non-symmetric 2-designs with (r, λ) = 1 and alternating socle
Flag-tranitive non-ymmetric -deign with (r, λ = 1 and alternating ocle Shenglin Zhou, Yajie Wang School of Mathematic South China Univerity of Technology Guangzhou, Guangdong 510640, P. R. China lzhou@cut.edu.cn
More informationAUTOMATIC SOLUTION OF FAMILIES OF THUE EQUATIONS AND AN EXAMPLE OF DEGREE 8
To appear in Journal of Symbolic Computation AUTOMATIC SOLUTION OF FAMILIES OF THUE EQUATIONS AND AN EXAMPLE OF DEGREE 8 CLEMENS HEUBERGER, ALAIN TOGBÉ, AND VOLKER ZIEGLER Abstract We describe a procedure
More informationUNIT 15 RELIABILITY EVALUATION OF k-out-of-n AND STANDBY SYSTEMS
UNIT 1 RELIABILITY EVALUATION OF k-out-of-n AND STANDBY SYSTEMS Structure 1.1 Introduction Objective 1.2 Redundancy 1.3 Reliability of k-out-of-n Sytem 1.4 Reliability of Standby Sytem 1. Summary 1.6 Solution/Anwer
More informationINTEGRAL POINTS AND ARITHMETIC PROGRESSIONS ON HESSIAN CURVES AND HUFF CURVES
INTEGRAL POINTS AND ARITHMETIC PROGRESSIONS ON HESSIAN CURVES AND HUFF CURVES SZ. TENGELY Abstract. In this paper we provide bounds for the size of the integral points on Hessian curves H d : x 3 + y 3
More informationLecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas)
Lecture 7: Analytic Function and Integral (See Chapter 4 in Boa) Thi i a good point to take a brief detour and expand on our previou dicuion of complex variable and complex function of complex variable.
More informationTheoretical Computer Science. Optimal algorithms for online scheduling with bounded rearrangement at the end
Theoretical Computer Science 4 (0) 669 678 Content lit available at SciVere ScienceDirect Theoretical Computer Science journal homepage: www.elevier.com/locate/tc Optimal algorithm for online cheduling
More informationClustering Methods without Given Number of Clusters
Clutering Method without Given Number of Cluter Peng Xu, Fei Liu Introduction A we now, mean method i a very effective algorithm of clutering. It mot powerful feature i the calability and implicity. However,
More informationWeber Schafheitlin-type integrals with exponent 1
Integral Tranform and Special Function Vol., No., February 9, 47 53 Weber Schafheitlin-type integral with exponent Johanne Kellendonk* and Serge Richard Univerité de Lyon, Univerité Lyon, Intitut Camille
More informationON ASYMPTOTIC FORMULA OF THE PARTITION FUNCTION p A (n)
#A2 INTEGERS 15 (2015) ON ASYMPTOTIC FORMULA OF THE PARTITION FUNCTION p A (n) A David Chritopher Department of Mathematic, The American College, Tamilnadu, India davchrame@yahoocoin M Davamani Chritober
More informationPrimitive Digraphs with the Largest Scrambling Index
Primitive Digraph with the Larget Scrambling Index Mahmud Akelbek, Steve Kirkl 1 Department of Mathematic Statitic, Univerity of Regina, Regina, Sakatchewan, Canada S4S 0A Abtract The crambling index of
More informationPhysics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014
Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion
More informationSymmetric Determinantal Representation of Formulas and Weakly Skew Circuits
Contemporary Mathematic Symmetric Determinantal Repreentation of Formula and Weakly Skew Circuit Bruno Grenet, Erich L. Kaltofen, Pacal Koiran, and Natacha Portier Abtract. We deploy algebraic complexity
More informationSome Sets of GCF ϵ Expansions Whose Parameter ϵ Fetch the Marginal Value
Journal of Mathematical Reearch with Application May, 205, Vol 35, No 3, pp 256 262 DOI:03770/jin:2095-26520503002 Http://jmredluteducn Some Set of GCF ϵ Expanion Whoe Parameter ϵ Fetch the Marginal Value
More informationLecture 7: Testing Distributions
CSE 5: Sublinear (and Streaming) Algorithm Spring 014 Lecture 7: Teting Ditribution April 1, 014 Lecturer: Paul Beame Scribe: Paul Beame 1 Teting Uniformity of Ditribution We return today to property teting
More informationSocial Studies 201 Notes for March 18, 2005
1 Social Studie 201 Note for March 18, 2005 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
More informationREPRESENTATION OF ALGEBRAIC STRUCTURES BY BOOLEAN FUNCTIONS. Logic and Applications 2015 (LAP 2015) September 21-25, 2015, Dubrovnik, Croatia
REPRESENTATION OF ALGEBRAIC STRUCTURES BY BOOLEAN FUNCTIONS SMILE MARKOVSKI Faculty of Computer Science and Engineering, S Ciryl and Methodiu Univerity in Skopje, MACEDONIA mile.markovki@gmail.com Logic
More informationUNIQUE CONTINUATION FOR A QUASILINEAR ELLIPTIC EQUATION IN THE PLANE
UNIQUE CONTINUATION FOR A QUASILINEAR ELLIPTIC EQUATION IN THE PLANE SEPPO GRANLUND AND NIKO MAROLA Abtract. We conider planar olution to certain quailinear elliptic equation ubject to the Dirichlet boundary
More informationTAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS ON TIME SCALES
TAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS R. ANDERSON Abtract. We are concerned with the repreentation of polynomial for nabla dynamic equation on time cale. Once etablihed,
More informationGeometric Measure Theory
Geometric Meaure Theory Lin, Fall 010 Scribe: Evan Chou Reference: H. Federer, Geometric meaure theory L. Simon, Lecture on geometric meaure theory P. Mittila, Geometry of et and meaure in Euclidean pace
More informationRobustness analysis for the boundary control of the string equation
Routne analyi for the oundary control of the tring equation Martin GUGAT Mario SIGALOTTI and Mariu TUCSNAK I INTRODUCTION AND MAIN RESULTS In thi paper we conider the infinite dimenional ytem determined
More informationarxiv: v4 [math.co] 21 Sep 2014
ASYMPTOTIC IMPROVEMENT OF THE SUNFLOWER BOUND arxiv:408.367v4 [math.co] 2 Sep 204 JUNICHIRO FUKUYAMA Abtract. A unflower with a core Y i a family B of et uch that U U Y for each two different element U
More informationLaplace Transformation
Univerity of Technology Electromechanical Department Energy Branch Advance Mathematic Laplace Tranformation nd Cla Lecture 6 Page of 7 Laplace Tranformation Definition Suppoe that f(t) i a piecewie continuou
More informationOn a family of relative quartic Thue inequalities
Journal of Number Theory 120 2006 303 325 www.elsevier.com/locate/jnt On a family of relative quartic Thue inequalities Volker Ziegler Department of Mathematics, Graz University of Technology, Steyrergasse
More informationOptimal Coordination of Samples in Business Surveys
Paper preented at the ICES-III, June 8-, 007, Montreal, Quebec, Canada Optimal Coordination of Sample in Buine Survey enka Mach, Ioana Şchiopu-Kratina, Philip T Rei, Jean-Marc Fillion Statitic Canada New
More informationNotes on Phase Space Fall 2007, Physics 233B, Hitoshi Murayama
Note on Phae Space Fall 007, Phyic 33B, Hitohi Murayama Two-Body Phae Space The two-body phae i the bai of computing higher body phae pace. We compute it in the ret frame of the two-body ytem, P p + p
More informationSOME MONOTONICITY PROPERTIES AND INEQUALITIES FOR
Kragujevac Journal of Mathematic Volume 4 08 Page 87 97. SOME MONOTONICITY PROPERTIES AND INEQUALITIES FOR THE p k-gamma FUNCTION KWARA NANTOMAH FATON MEROVCI AND SULEMAN NASIRU 3 Abtract. In thi paper
More informationON THE DETERMINATION OF NUMBERS BY THEIR SUMS OF A FIXED ORDER J. L. SELFRIDGE AND E. G. STRAUS
ON THE DETERMINATION OF NUMBERS BY THEIR SUMS OF A FIXED ORDER J. L. SELFRIDGE AND E. G. STRAUS 1. Introduction. We wih to treat the following problem (uggeted by a problem of L. Moer [2]): Let {x} = {x
More informationConvergence criteria and optimization techniques for beam moments
Pure Appl. Opt. 7 (1998) 1221 1230. Printed in the UK PII: S0963-9659(98)90684-5 Convergence criteria and optimization technique for beam moment G Gbur and P S Carney Department of Phyic and Atronomy and
More informationName: Solutions Exam 3
Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer
More informationMulti-dimensional Fuzzy Euler Approximation
Mathematica Aeterna, Vol 7, 2017, no 2, 163-176 Multi-dimenional Fuzzy Euler Approximation Yangyang Hao College of Mathematic and Information Science Hebei Univerity, Baoding 071002, China hdhyywa@163com
More informationMath Skills. Scientific Notation. Uncertainty in Measurements. Appendix A5 SKILLS HANDBOOK
ppendix 5 Scientific Notation It i difficult to work with very large or very mall number when they are written in common decimal notation. Uually it i poible to accommodate uch number by changing the SI
More informationTHE SPLITTING SUBSPACE CONJECTURE
THE SPLITTING SUBSPAE ONJETURE ERI HEN AND DENNIS TSENG Abtract We anwer a uetion by Niederreiter concerning the enumeration of a cla of ubpace of finite dimenional vector pace over finite field by proving
More informationMarch 18, 2014 Academic Year 2013/14
POLITONG - SHANGHAI BASIC AUTOMATIC CONTROL Exam grade March 8, 4 Academic Year 3/4 NAME (Pinyin/Italian)... STUDENT ID Ue only thee page (including the back) for anwer. Do not ue additional heet. Ue of
More informationA Simplified Methodology for the Synthesis of Adaptive Flight Control Systems
A Simplified Methodology for the Synthei of Adaptive Flight Control Sytem J.ROUSHANIAN, F.NADJAFI Department of Mechanical Engineering KNT Univerity of Technology 3Mirdamad St. Tehran IRAN Abtract- A implified
More informationDiophantine equations for second order recursive sequences of polynomials
Diophantine equations for second order recursive sequences of polynomials Andrej Dujella (Zagreb) and Robert F. Tichy (Graz) Abstract Let B be a nonzero integer. Let define the sequence of polynomials
More information1. The F-test for Equality of Two Variances
. The F-tet for Equality of Two Variance Previouly we've learned how to tet whether two population mean are equal, uing data from two independent ample. We can alo tet whether two population variance are
More informationREVERSE HÖLDER INEQUALITIES AND INTERPOLATION
REVERSE HÖLDER INEQUALITIES AND INTERPOLATION J. BASTERO, M. MILMAN, AND F. J. RUIZ Abtract. We preent new method to derive end point verion of Gehring Lemma uing interpolation theory. We connect revere
More informationA relationship between generalized Davenport-Schinzel sequences and interval chains
A relationhip between generalized Davenport-Schinzel equence and interval chain The MIT Faculty ha made thi article openly available. Pleae hare how thi acce benefit you. Your tory matter. Citation A Publihed
More informationBeta Burr XII OR Five Parameter Beta Lomax Distribution: Remarks and Characterizations
Marquette Univerity e-publication@marquette Mathematic, Statitic and Computer Science Faculty Reearch and Publication Mathematic, Statitic and Computer Science, Department of 6-1-2014 Beta Burr XII OR
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Laplace Tranform Paul Dawkin Table of Content Preface... Laplace Tranform... Introduction... The Definition... 5 Laplace Tranform... 9 Invere Laplace Tranform... Step Function...4
More informationConnectivity in large mobile ad-hoc networks
Weiertraß-Intitut für Angewandte Analyi und Stochatik Connectivity in large mobile ad-hoc network WOLFGANG KÖNIG (WIAS und U Berlin) joint work with HANNA DÖRING (Onabrück) and GABRIEL FARAUD (Pari) Mohrentraße
More informationON THE SMOOTHNESS OF SOLUTIONS TO A SPECIAL NEUMANN PROBLEM ON NONSMOOTH DOMAINS
Journal of Pure and Applied Mathematic: Advance and Application Volume, umber, 4, Page -35 O THE SMOOTHESS OF SOLUTIOS TO A SPECIAL EUMA PROBLEM O OSMOOTH DOMAIS ADREAS EUBAUER Indutrial Mathematic Intitute
More informationTechnical Appendix: Auxiliary Results and Proofs
A Technical Appendix: Auxiliary Reult and Proof Lemma A. The following propertie hold for q (j) = F r [c + ( ( )) ] de- ned in Lemma. (i) q (j) >, 8 (; ]; (ii) R q (j)d = ( ) q (j) + R q (j)d ; (iii) R
More informationExercises for lectures 19 Polynomial methods
Exercie for lecture 19 Polynomial method Michael Šebek Automatic control 016 15-4-17 Diviion of polynomial with and without remainder Polynomial form a circle, but not a body. (Circle alo form integer,
More informationOn the power-free parts of consecutive integers
ACTA ARITHMETICA XC4 (1999) On the power-free parts of consecutive integers by B M M de Weger (Krimpen aan den IJssel) and C E van de Woestijne (Leiden) 1 Introduction and main results Considering the
More informationBalanced Network Flows
revied, June, 1992 Thi paper appeared in Bulletin of the Intitute of Combinatoric and it Application 7 (1993), 17-32. Balanced Network Flow William Kocay* and Dougla tone Department of Computer cience
More information