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Diophantine Equations and Cyclotomic Fields

dc.contributor.advisorMihailescu, Preda Prof. Dr.
dc.contributor.advisorBilu, Yuri Prof. Dr.
dc.contributor.authorBartolomé, Boris
dc.titleDiophantine Equations and Cyclotomic Fieldsde
dc.contributor.refereeBugeaud, Yann Prof. Dr.
dc.description.abstractengThis thesis examines some approaches to address Diophantine equations, specifically we focus on the connection between the Diophantine analysis and the theory of cyclotomic fields. First, we propose a quick introduction to the methods of Diophantine approximation we have used in this research work. We remind the notion of height and introduce the logarithmic gcd. Then, we address a conjecture, made by Thoralf Skolem in 1937, on an exponential Diophantine equation. For this conjecture, let K be a number field, α1 ,…, αm , λ1 ,…, λm non-zero elements in K, and S a finite set of places of K (containing all the infinite places) such that the ring of S-integers OS = OK,S = {α ∈ K : |α|v ≤ 1 pour les places v ∈/ S} contains α1 , . . . , αm , λ1 , . . . , λm α1-1 , . . . , αm-1. For each n ∈ Z, let A(n)=λ_1 α_1^n+⋯+λ_m α_m^n∈O_S. Skolem suggested [SK1] : Conjecture (exponential local-global principle). Assume that for every non zero ideal a of the ring O_S, there exists n ∈ Z such that A(n) ≡0 mod a. Then, there exists n ∈ Z such that A(n)=0. Let Γ be the multiplicative group generated by α1 ,…, αm. Then Γ is the product of a finite abelian group and a free abelian group of finite rank. We prove that the conjecture is true when the rank of Γ is one. After that, we generalize a result previously published by Abouzaid ([A]). Let F(X,Y) ∈ Q[X,Y] be an irreducible Q-polynomial. In 2008, Abouzaid [A] proved the following theorem: Theorem (Abouzaid). Assume that (0,0) is a non-singular point of the plane curve F(X,Y) = 0. Let m = degX F, n = degY F, M = max{m, n}. Let ε satisfy 0 < ε < 1. Then for any solution (α,β) ∈ Q ̅2 of F(X,Y) = 0, we have either max{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M), or max{|h(α) − nlgcd(α, β)|,|h(β) − mlgcd(α, β)|} ≤ εmax{h(α), h(β)}+ + 742M7ε−1hp(F) + 5762M9ε−1log(2m + 2n) However, he imposed the condition that (0, 0) be a non-singular point of the plane curve F(X,Y) = 0. Using a somewhat different version of Siegel’s “absolute” lemma and of Eisenstein’s lemma, we could remove the condition and prove it in full generality. We prove the following theorem: Theorem. Let F(X,Y) ∈ Q ̅[X,Y] be an absolutely irreducible polynomial satisfying F(0,0)=0. Let m=degX F, n=degY F and r = min{i+j:(∂^(i+j) F)/(∂^i X∂^j Y)(0,0)≠0}. Let ε be such that 0 < ε < 1. Then, for all (α, β) ∈ Q ̅2 such that F(α,β) = 0, we have either h(α) ≤ 200ε−2mn6(hp(F) + 5) or |(lgcd(α,β))/r-h(α)/n|≤1/r (εh(α)+4000ε^(-1) n^4 (h_p (F)+log⁡(mn)+1)+30n^2 m(h_p (F)+log⁡(mn) )) Then, we give an overview of the tools we have used in cyclotomic fields. We try there to develop a systematic approach to address a certain type of Diophantine equations. We discuss on cyclotomic extensions and give some basic but useful properties, on group-ring properties and on Jacobi sums. Finally, we show a very interesting application of the approach developed in the previous chapter. There, we consider the Diophantine equation (1) Xn − 1 = BZn, where B ∈ Z is understood as a parameter. Define ϕ∗(B) := ϕ(rad (B)), where rad (B) is the radical of B, and assume that (2) (n, ϕ∗(B)) = 1. For a fixed B ∈ N_(>1)we let N(B) = {n ∈ N_(>1) | ∃ k > 0 such that n|ϕ∗(B)}. If p is an odd prime, we shall denote by CF the combined condition requiring that I The Vandiver Conjecture holds for p, so the class number h+ of the maximal real subfield of the cyclotomic field Q[ζp ] is not divisible by p. II We have ir>(p) < √p − 1, in other words, there is at most √p − 1 odd integers k < p such that the Bernoulli number Bk ≡ 0 mod p. Current results on Equation (1) are restricted to values of B which are built up from two small primes p ≤ 13 [BGMP] and complete solutions for B < 235 ([BBGP]). If expecting that the equation has no solutions, – possibly with the exception of some isolated examples – it is natural to consider the case when the exponent n is a prime. Of course, the existence of solutions (X,Z) for composite n imply the existence of some solutions with n prime, by raising X, Z to a power. The main contribution of our work has been to relate (1) in the case when n is a prime and (2) holds, to the diagonal Nagell – Ljunggren equation (X^n-1)/(X-1)=n^e Y^n, e={(O si X≢1[n]@1 sinon)┤ This way, we can apply results from [M] and prove the following: Theorem. Let n be a prime and B > 1 an integer with (ϕ∗(B), n) = 1. Suppose that equation (1) has a non-trivial integer solution different from n = 3 and (X,Z;B) = (18,7;17). Let X ≡ u mod n, 0 ≤ u < n and e = 1 if u = 1 and e = 0 otherwise. Then: 1. n > 163106. 2. X − 1 = ±B/ne and B < nn. 3. If u ∉ {−1,0,1}, then condition CF (II) fails for n and 2n−1 ≡ 3n−1 ≡ 1 mod n2 , and rn−1 ≡ 1 mod n2 for all r|X(X2 − 1). If u ∈ {−1, 0, 1}, then Condition CF (I) fails for n. Based on this theorem, we also prove the following: Theorem. If equation (1) has a solution for a fixed B verifying the conditions (2), then either n ∈ N(B) or there is a prime p coprime to ϕ∗(B) and a m ∈ N(B) such that n = p.m. Moreover, Xm, Ym are a solution of (1) for the prime exponent p and thus verify the conditions of the previous Theorem. This is a strong improvement of the currently known results. As we have made heavy use of [M], at the end of this thesis we have added an appendix to expose some new result that allows for a full justification of Theorem 3 of [M]. Keywords Diophantine Equations, Cyclotomic Fields, Nagell-Ljunggren Equation, Skolem, Abouzaid, Exponential Diophantine Equation, Baker’s Inequality, Subspace Theorem. References [A] M. Abouzaid, Heights and logarithmic gcd on algebraic curves, Int. J. Number Th. 4, pp. 177–197 (2008). [BBGP] A.Bazso, A.Bérczesc K.Györy and A.Pintér, On the resolution of equations Axn − Byn = C in integers x, y and n ≥ 3, II, Publicationes Mathematicae Debrecen 76, pp. 227 – 250 (2010). [BGMP] M. A. Bennett, K. Györy, M. Mignotte and A. Pintér, Binomial Thue equations and polynomial powers, Compositio Math. 142, pp. 1103–1121 (2006). [M] P. Mihăilescu Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren, Diophantine Approximation, Springer Verlag, Development in Mathematics 16, pp. 245–273 (2008). [S1] T. Skolem, Anwendung exponentieller Kongruenzen zum Beweis der Unlosbarkeit gewisser diophantischer Gleichungen, Avhdl. Norske Vid. Akad. Oslo I 12, pp. 1–16 (1929). [S2] T. Skolem, Lösung gewisser Gleichungssysteme in ganzen Zahlen oder ganzzahligen Polynomen mit beschränktem gemeinschaftlichen Teiler, Oslo Vid. Akar. Skr. I, 12 (1929).de
dc.contributor.coRefereeFuchs, Clemens Prof. Dr.
dc.contributor.thirdRefereeBrüdern, Jörg Prof. Dr.
dc.subject.engDiophantine Equationsde
dc.subject.engCyclotomic Fieldsde
dc.subject.engNagell-Ljunggren Equationde
dc.subject.engExponential Diophantine Equationde
dc.subject.engBaker’s Inequalityde
dc.subject.engSubspace Theoremde
dc.affiliation.instituteFakultät für Mathematik und Informatikde
dc.subject.gokfullMathematics (PPN61756535X)de

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