1. The Circle Method and Waringโ{128}{153}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 Mordellโ{128}{153}s Lemma -- 2.4 Fundamental Lemma -- 2.5 Proof of Theorem 2.1 -- 2.6 Proof of Theorem 2.2 -- Notes -- 3. Weylโ{128}{153}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 Mitsuiโ{128}{153}s Lemma -- 3.7 Proof of Theorem 3.3 -- 3.8 Proof of Lemma 3.6 -- 3.9 Continuation -- Notes -- 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 -- Notes -- 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 -- Notes -- 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 -- Notes -- 7. Waringโ{128}{153}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 -- Notes -- 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 -- Notes -- 9. Small Nonnegative Solutions of Additive Equations -- 9.1 Introduction -- 9.2 Hurwitzโ{128}{153}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 -- Notes -- 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 -- Notes -- 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 -- Notes -- References I -- References II