Foundations of Analysis over Surreal Number Fields (Volume 141) (North-Holland Mathematics Studies, Volume 141)

In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Formal versions of such theorems as the Implicit Function Theorem hold over such fields. The Main Theorem states that every formal power series in a finite number of variables over a surreal field has a positive radius of hyper-convergence within which it may be evaluated. Analytic functions of several surreal and surcomplex variables can then be defined and studied. Some first results in the one variable case are derived. A primer on Conway's field of surreal numbers is also given. Throughout the manuscript, great efforts have been made to make the volume fairly self-contained. Much exposition is given. Many references are cited. While experts may want to turn quickly to new results, students should be able to find the explanation of many elementary points of interest. On the other hand, many new results are given, and much mathematics is brought to bear on the problems at hand. Content: Edited by Page iii Copyright page Page iv Preface Pages vii-x Norman L. Alling Chapter 0 Introduction Pages 1-11 Chapter 1 Preliminaries Pages 13-84 Chapter 2 The ξ-Topology Pages 85-108 Chapter 3 The ξ-Topology on Affine n-Space Pages 109-116 Chapter 4 Introduction to the Surreal Number Field No Pages 117-190 Chapter 5 The Surreal Fields ξNo, and Related Topics Pages 191-205 Chapter 6 The Valuation Theory of Ordered Fields, Applied to No and ξNo Pages 207-253 Chapter 7 Power Series: Formal and Hyper-Convergent Pages 255-331 Chapter 8 A Primer on Analytic Functions of a Surreal Variable Pages 333-351 Bibliography Pages 353-358 Index Pages 359-373