# Selected Works of Ellis Kolchin with Commentary

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*Hyman Bass; Alexandru Buium; Phyllis J. Cassidy*

The work of Joseph Fels Ritt and Ellis Kolchin in
differential algebra paved the way for exciting new applications in
constructive symbolic computation, differential Galois theory, the
model theory of fields, and Diophantine geometry. This volume
assembles Kolchin's mathematical papers, contributing solidly to the
archive on construction of modern differential algebra. This
collection of Kolchin's clear and comprehensive papers—in
themselves constituting a history of the subject—is an
invaluable aid to the student of differential algebra.

In 1910, Ritt created a theory of algebraic differential equations
modeled not on the existing transcendental methods of Lie, but rather
on the new algebra being developed by E. Noether and B. van der
Waerden. Building on Ritt's foundation, and deeply influenced by Weil
and Chevalley, Kolchin opened up Ritt theory to modern algebraic
geometry. In so doing, he led differential geometry in a new
direction. By creating differential algebraic geometry and the theory
of differential algebraic groups, Kolchin provided the foundation for
a “new geometry” that has led to both a striking and an
original approach to arithmetic algebraic geometry. Intriguing
possibilities were introduced for a new language for nonlinear
differential equations theory.

The volume includes commentary by A. Borel, M. Singer, and
B. Poizat. Also Buium and Cassidy trace the development of Kolchin's
ideas, from his important early work on the differential Galois theory
to his later groundbreaking results on the theory of differential
algebraic geometry and differential algebraic groups. Commentaries are
self-contained with numerous examples of various aspects of
differential algebra and its applications. Central topics of Kolchin's
work are discussed, presenting the history of differential algebra and
exploring how his work grew from and transformed the work of
Ritt. New directions of differential algebra are illustrated,
outlining important current advances. Prerequisite to understanding
the text is a background at the beginning graduate level in algebra,
specifically commutative algebra, the theory of field extensions, and
Galois theory.

#### Readership

Graduate students and research mathematicians working in differential algebra, symbolic computation, differential Galois theory, the model theory of fields, and arithmetic algebraic geometry.

#### Reviews & Endorsements

This book reprints all the published research papers of Ellis Kolchin, as well as some non-archival publications (such as distributed lecture notes), and publishes for the first time a short paper on Painlevé transcendents.

The volume also contains a nearly 130 page section of commentary, consisting of four expository articles. Taken together, these probably comprise the best introduction and survey of differential algebra currently available. The editors of this volume, and the authors of the articles in the Commentary section, are to be congratulated on not only making the publications of Ellis Kolchin readily available in a single volume format, but also for providing a set of excellent expository articles which will allow newcomers to the field to learn about the main themes in Kolchin's work in differential algebra and to see how those topics have developed mathematically up to the present day.

-- Zentralblatt MATH

With its relations to such varied fields as algebraic groups, transcendence, model theory, Diophantine geometry, partial differential equations and of course differential algebra itself, this book will be a welcome addition to all mathematical libraries.

-- Mathematical Reviews