Diophantine Equations and Inequalities in Algebraic Number Fields

Inhaltsverzeichnis

• 1. The Circle Method and Waring’s Problem.
• 1.1 Introduction.
• 1.2 Farey Division.
• 1.3 Auxiliary Lemmas.
• 1.4 Major Arcs.
• 1.5 Singular Integral.
• 1.6 Singular Series.
• 1.7 Proof of Lemma 1.12.
• 1.8 Proof of Theorem 1.1.
• Notes.
• 2. Complete Exponential Sums.
• 2.1 Introduction.
• 2.2 Several Lemmas.
• 2.3 Mordell’s Lemma.
• 2.4 Fundamental Lemma.
• 2.5 Proof of Theorem 2.1.
• 2.6 Proof of Theorem 2.2.
• 3. Weyl’s Sums.
• 3.1 Introduction.
• 3.2 Proof of Theorem 3.1.
• 3.3 A Lemma on Units.
• 3.4 The Asymptotic Formula for N(a, T).
• 3.5 A Sum.
• 3.6 Mitsui’s Lemma.
• 3.7 Proof of Theorem 3.3.
• 3.8 Proof of Lemma 3.6.
• 3.9 Continuation.
• 4. Mean Value Theorems.
• 4.1 Introduction.
• 4.2 Proof of Theorem 4.1.
• 4.3 Proof of Theorem 4.2.
• 4.4 A Lemma on the Set D.
• 4.5 A Lemma on the Set D(x).
• 4.6 Fundamental Lemma.
• 4.7 Proof of Lemma 4.1.
• 5. The Circle Method in Algebraic Number Fields.
• 5.1 Introduction.
• 5.2 Lemmas.
• 5.3 Asympotic Expansion forSi (?, T).
• 5.4 Further Estimates on Basic Domains.
• 5.5 Proof of Theorem 5.1.
• 5.6 Proof of Theorem 5.2.
• 6. Singular Series and Singular Integrals.
• 6.1 Introduction.
• 6.2 Product Form for Singular Series.
• 6.3 Singular Series and Congruences.
• 6.4 p-adic Valuations.
• 6.5 k-th Power Residues.
• 6.6 Proof of Theorem 6.1.
• 6.7 Monotonic Functions.
• 6.8 Proof of Theorem 6.2.
• 7. Waring’s Problem.
• 7.1 Introduction.
• 7.2 The Ring Jk.
• 7.3 Proofs of Theorems 7.1 and 7.2.
• 7.4 Proof of Theorem 7.3.
• 7.5 Proof of Theorem 7.4.
• 8. Additive Equations.
• 8.1 Introduction.
• 8.2 Reductions.
• 8.3 Contraction.
• 8.4 Derived Variables.
• 8.5 Proof of Theorem 8.1.
• 8.6 Proof of Theorem 8.2.
• 8.7 Bounds for Solutions.
• 9. Small Nonnegative Solutions of Additive Equations.
• 9.1 Introduction.
• 9.2Hurwitz’s Lemma.
• 9.3 Reductions.
• 9.4 Continuation.
• 9.5 Farey Division.
• 9.6 Supplementary Domain.
• 9.7 Basic Domains.
• 9.8 Proof of Theorem 9.1.
• 10. Small Solutions of Additive Equations.
• 10.1 Introduction.
• 10.2 Reductions.
• 10.3 Continuation.
• 10.4 Farey Division.
• 10.5 Supplementary Domain.
• 10.6 Basic Domains.
• 10.7 Proof of Theorem 10.1.
• 11. Diophantine Inequalities for Forms.
• 11.1 Introduction.
• 11.2 A Single Additive Form.
• 11.3 A Variant Circle Method.
• 11.4 Continuation.
• 11.5 Proof of Lemma 11.1.
• 11.6 Linear Forms.
• 11.7 A Single Form.
• 11.8 Proof of Theorem 11.1.
• References I.
• References II.