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These provide the foundational language for later chapters.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the book treat regular singularities in the p-adic setting?",{"text":67,"@type":63},"Regular singularities are addressed in a dedicated section, discussing irregularity, exponents in the complex analytic setting, formal solutions, and the relationship between index and irregularity.",{"name":69,"@type":60,"acceptedAnswer":70},"What role do Frobenius structures and monodromy play?",{"text":71,"@type":63},"Later parts focus on difference algebra and Frobenius structures, then establish effective convergence bounds and study quasiconstant modules. The p-adic local monodromy theorem is presented with multiple approaches and applications.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},638664,1792934639,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":26,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},3985747874027,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","p-adic di􀀋erential equations (version of 7 Jan  \n08)  \nKiran S. Kedlaya  \nContents  \nPreface ix  \n1. About the book ix  \n2. Structure of the book x  \n3. Acknowledgments x  \nChapter 0 . Introductory remarks 1  \n1. Why p-adic di􀀋erential equations? 1  \n2. Zeta functions of varieties 2  \n3. Zeta functions and p-adic di􀀋erential equations 3  \n4. A word of caution 5  \nNotes 6  \nExercises 6  \nPart 1 . Tools of p-adic analysis 7  \nChapter 1 . Absolute values 9  \n1. Absolute values on abelian groups 9  \n2. Valuations and nonarchimedean absolute values 10  \n3. Norms on modules 11  \n4. Examples of nonarchimedean absolute values 12  \n5. Unrami􀀌ed extensions 14  \n6. Tamely rami􀀌ed extensions 15  \n7. Spherical completeness 16  \nNotes 17  \nExercises 17  \nChapter 2 . Newton polygons 19  \n1. Gauss norms and Newton polygons 19  \n2. Slope factorizations and a master factorization theorem 20  \n3. Applications to nonarchimedean 􀀌eld theory 22  \nExercises 23  \nChapter 3 . Matrix analysis 25  \n1. Singular values and eigenvalues (archimedean case) 25  \n2. Perturbations (archimedean case) 27  \n3. Singular values and eigenvalues (nonarchimedean case) 29  \n4. Perturbations (nonarchimedean case) 32  \n5. Horn’s inequalities 34  \nNotes 35  \nExercises 36  \nPart 2 . Di􀀋erential algebra 37  \nChapter 4 . Formalism of di􀀋erential algebra 39  \n1. Di􀀋erential rings and di􀀋erential modules 39  \n2. Di􀀋erential modules and di􀀋erential systems 39  \n3. Operations on di􀀋erential modules 40  \n4. Cyclic vectors 41  \n5. Di􀀋erential polynomials 42  \n6. Di􀀋erential equations 43  \n7. Cyclic vectors: a mixed blessing 44  \n8. Taylor series 45  \nNotes 45  \nExercises 46  \nChapter 5 . Metric properties of di􀀋erential modules 47  \n1. Spectral norms of linear operators 47  \n2. Spectral norms of di􀀋erential operators 48  \n3. A coordinate-free approach 51  \n4. Newton polygons for twisted polynomials 51  \n5. Twisted polynomials and spectral norms 52  \n6. The visible decomposition theorem 53  \n7. Matrices and the visible spectrum 54  \nNotes 57  \nExercises 57  \nChapter 6 . Regular singularities 59  \n1. Irregularity 59  \n2. Exponents in the complex analytic setting 60  \n3. Formal solutions of regular di􀀋erential equations 61  \n4. Index and irregularity 63  \nNotes 63  \nExercises 63  \nPart 3 . p-adic di􀀋erential equations on discs and annuli 65  \nChapter 7 . Rings of functions on discs and annuli 67  \n1. Power series on closed discs and annuli 67  \n2. Gauss norms and Newton polygons 67  \n3. Factorization results 69  \n4. Open discs and annuli 70  \nNotes 70  \nExercises 71  \nChapter 8 . Radius and generic radius of convergence 73  \n1. Di􀀋erential modules on rings and annuli 73  \n2. Radius of convergence on a disc 73  \n3. Generic radius of convergence 74  \n4. Some examples in rank 1 76  \n5. Transfer theorems 76  \n6. Geometric interpretation 78  \n7. Subsidiary radii 79  \nNotes 79  \nExercises 80  \nChapter 9 . Frobenius pullback and pushforward 81  \n1. Why Frobenius? 81  \n2. p-th roots 81  \n3. Moving along Frobenius 83  \n4. Frobenius antecedents 84  \n5. Frobenius descendants and subsidiary radii 85  \n6. Decomposition by spectral norm 86  \n7. Integrality, or lack thereof 87  \n8. O􀀋-centered Frobenius descendants 88  \nNotes 89  \nExercises 89  \nChapter 10 . Variation of generic and subsidiary radii 91  \n1. Harmonicity of the valuation function 91  \n2. Variation of Newton polygons 91  \n3. Variation of subsidiary radii: statements 93  \n4. Convexity for the generic radius 94  \n5. Finding lattices 95  \n6. Measuring small radii 96  \n7. Larger radii 96  \n8. Monotonicity 98  \n9. Radius versus generic radius 99  \n10. Subsidiary radii as radii of convergence 100  \nNotes 101  \nExercises 101  \nChapter 11 . Decomposition by subsidiary radii 103  \n1. Metrical detection of units 103  \n2. Decomposition over a closed disc 104  \n3. Decomposition over a closed annulus 105  \n4. Decomposition over an open disc or annulus 106  \n5. Modules solvable at a boundary 107  \nNotes 108  \nExercises 108  \nChapter 12 . p-adic expo","cbCaitaXevuoHqh6","https://ap.wps.com/l/cbCaitaXevuoHqh6","pdf",1369091,202,"English","# Preface\n# About the book\n# Structure of the book\n# Acknowledgments\n# Chapter 0 . Introductory remarks\n## Why p-adic di􀀋erential equations?\n## Zeta functions of varieties\n## Zeta functions and p-adic di􀀋erential equations\n## A word of caution\n# Notes\n# Exercises\n# Part 1 . Tools of p-adic analysis\n## Chapter 1 . Absolute values\n## Chapter 2 . Newton polygons\n## Chapter 3 . Matrix analysis\n# Part 2 . Di􀀋erential algebra\n## Chapter 4 . Formalism of di􀀋erential algebra\n## Chapter 5 . Metric properties of di􀀋erential modules\n## Chapter 6 . Regular singularities\n# Part 3 . p-adic di􀀋erential equations on discs and annuli\n## Chapter 7 . Rings of functions on discs and annuli\n## Chapter 8 . Radius and generic radius of convergence\n## Chapter 9 . Frobenius pullback and pushforward\n## Chapter 10 . Variation of generic and subsidiary radii\n## Chapter 11 . Decomposition by subsidiary radii\n## Chapter 12 . p-adic exponents\n# Part 4 . Di􀀋erence algebra and Frobenius structures\n## Chapter 13 . Formalism of di􀀋erence algebra\n## Chapter 14 . Frobenius modules\n## Chapter 15 . Frobenius structures on di􀀋erential modules\n## Chapter 16 . E􀀋ective convergence bounds\n## Chapter 17 . Quasiconstant di􀀋erential modules\n## Chapter 18 . The p-adic local monodromy theorem\n# Part 5 . Areas of application\n## Chapter 19 . Picard-Fuchs modules","[{\"question\":\"What topics does the book cover before developing p-adic differential equations?\",\"answer\":\"It begins with introductory remarks and then develops tools of p-adic analysis, including absolute values, Newton polygons, and matrix analysis. These provide the foundational language for later chapters.\"},{\"question\":\"How does the book treat regular singularities in the p-adic setting?\",\"answer\":\"Regular singularities are addressed in a dedicated section, discussing irregularity, exponents in the complex analytic setting, formal solutions, and the relationship between index and irregularity.\"},{\"question\":\"What role do Frobenius structures and monodromy play?\",\"answer\":\"Later parts focus on difference algebra and Frobenius structures, then establish effective convergence bounds and study quasiconstant modules. The p-adic local monodromy theorem is presented with multiple approaches and applications.\"}]","p-adic di􀀋erential equations - Version 7 - Jan 08 - Table of contents | PDF",1792692362,509]