Analytic Elements in p-adic analysis

Includes bibliography (p. 380-384) and references.

1. Absolute values and norms 2. Infraconnected sets 3. Monotonous and circular filters 4. Ultrametric absolute values and valuation functions v(h, [mu]) on K(x) 5. Hensel Lemma 6. Ultrametric field extensions 7. Ultraproducts and spherically complete extensions 8. A study in C[subsript p], the p[superscript n]-th roots of 1 9. Algebras R(D) 10. The analytic elements 11. Composition of analytic elements 12. Mult(H(D), U[subscript D]) 13. Power series 14. Factorization of analytic elements 15. The Mittag-Leffler Theorem 16. Maximal ideals of codimension 1 17. Dual of a space H(D) 18. Algebras H(D) 19. Derivative of analytic elements 20. Valuation functions for analytic elements 21. Elements vanishing along a filter 22. Quasi-minorated elements 23. Values and zeros of power series 24. Quasi-invertible elements 25. Zeros Theorem for power series 26. Image of a disk 27. Strictly injective analytic elements.INTRODUCTION; CONTENTS; Analytic Elements in p-Adic Analysis; 1. ABSOLUTE VALUES AND NORMS; 2. INFRACONNECTED SETS; 3. MONOTONOUS AND CIRCULAR FILTERS; 4. ULTRAMETRIC ABSOLUTE VALUES AND VALUATION FUNCTIONS v(h, μ) ON K(x); 5. HENSEL LEMMA; 6. ULTRAMETRIC FIELD EXTENSIONS; 7. ULTRAPRODUCTS AND SPHERICALLY COMPLETE EXTENSIONS; 8. A STUDY IN Cp, THE pnth ROOTS OF 1; 9. ALGEBRAS R(D); 10. THE ANALYTIC ELEMENTS; 11. COMPOSITION OF ANALYTIC ELEMENTS; 12. Mult (H(D), UD); 13. POWER SERIES; 14. FACTORIZATION OF ANALYTIC ELEMENTS; 15. THE MITTAG-LEFFLER THEOREM; 16. MAXIMAL IDEALS OF CODIMENSION 1.17. DUAL OF A SPACE H(D)18. ALGEBRAS H(D); 19. DERIVATIVE OF ANALYTIC ELEMENTS; 20. VALUATION FUNCTIONS FOR ANALYTIC ELEMENTS; 21. ELEMENTS VANISHING ALONG A MONOTONOUS FILTER; 22. QUASI-MINORATED ELEMENTS; 23. VALUES AND ZEROS OF POWER SERIES; 24. QUASI-INVERTIBLE ELEMENTS; 25. ZEROS THEOREM FOR POWER SERIES; 26. IMAGE OF A DISK; 27. STRICTLY INJECTIVE ANALYTIC ELEMENTS; 28. LOGARITHM AND EXPONENTIAL; 29. A FINITE INCREASINGS PROPERTY; 30. MAXIMUM PRINCIPLE; 31. ANALYTIC ELEMENTS MEROMORPHIC IN A HOLE; 32. MOTZKIN FACTORIZATION; 33. APPLICATIONS OF THE MOTZKIN FACTORIZATION.34. MAXIMUM IN A CIRCLE WITH HOLES35. T-FILTERS AND T-SEQUENCES; 36. EXAMPLES AND COUNTER-EXAMPLES ABOUT T-FILTERS; 37. CHARACTERISTIC PROPERTY OF THE T-FILTERS; 38. APPLICATIONS OF T-FILTERS; 39. INTEGRALLY CLOSED ALGEBRAS H(D); 40. ABSOLUTE VALUES ON H(D); 41. DISTINGUISHED CIRCULAR FILTERS; 42. MAXIMAL IDEALS OF INFINITE CODIMENSION; 43. IDEMPOTENT T-SEQUENCES; 44. T-POLAR SEQUENCES; 45. ANALYTIC EXTENSION THROUGH A T-FILTER; 46. ALGEBRAS H(D)/I0(F); 47. MEROMORPHIC PRODUCTS; 48. COLLAPSING MEROMORPHIC PRODUCTS; 49. INJECTIVITY, MITTAG-LEFFLER SERIES AND MOTZKIN PRODUCTS.50. ANALYTIC FUNCTIONS AND ANALYTIC ELEMENTS51. INFINITE VAN DER MONDE MATRICES; 52. p-ADIC ANALYTIC INTERPOLATION; 53. ANALYTIC ELEMENTS WITH A ZERO DERIVATIVE; 54. GENERALITIES ON THE DIFFERENTIAL EQUATION y' = fy IN H(D); 55. THE DIFFERENTIAL EQUATION y' = fy IN ALGEBRAS H(D); 56. THE EQUATION y' = fy IN ZERO RESIDUE CHARACTERISTIC; 57. THE EQUATION y' =fy IN Cp WITH f NOT QUASI-INVERTIBLE; 58. THE EQUATION y' =fy IN Cp WITH f QUASI-INVERTIBLE; 59. RESIDUES AND EQUATION y'=fy; 60. EQUATION g' = fg WITH gn ∈ H(D); 61. THE p-ADIC FOURIER TRANSFORM; REFERENCES; DEFINITIONS; NOTATIONS.

This is probably the first book dedicated to this topic. The behaviour of the analytic elements on an infraconnected set D in K an algebraically closed complete ultrametric field is mainly explained by the circular filters and the monotonous filters on D, especially the T-filters: zeros of the elements, Mittag-Leffler series, factorization, Motzkin factorization, maximum principle, injectivity, algebraic properties of the algebra of the analytic elements on D, problems of analytic extension, factorization into meromorphic products and connections with Mittag-Leffler series.