Foundations of Analysis over Surreal Number Fields, Volume 141

Introduction. 1. Preliminaries. 2. The &xgr;-Topology. 3. The &xgr;-Topology on Affine n-Space. 4. Introduction to the Surreal Field No. 5. The Surreal Fields &xgr;No, and Related Topics. 6. The Valuation Theory of Ordered Fields, Applied to No and &xgr;No. 7. Power Series: Formal and Hyper-Convergent. 8. A Primer on Analytic Functions of a Surreal Variable. Bibliography. Index.

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.