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.
Picard-Vessiot theory of partial differential fields The notion of dimension in the theory of algebraic differential equations Part I. The Papers of Ellis Kolchin: On certain ideals of differential polynomials On the basis theorem for infinite systems of differential polynomials On the exponents of differential ideals On the basis theorem for differential systems Extensions of differential fields. I Extensions of differential fields. II Algebraic matric groups The Picard-Vessiot theory of homogeneous linear ordinary differential equations Extensions of differential fields. III Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations On certain concepts in the theory of algebraic matric groups Existence theorems connected with the Picard-Vessiot theory of homogeneous linear ordinary differential equations Algebraic groups and differential equations Two proofs of a theorem on algebraic groups Picard-Vessiot theory of partial differential fields Galois theory of differential fields Differential fields and group varieties (First lecture) Differential fields and group varieties (Second lecture) On the Galois theory of differential fields Algebraic groups and the Galois theory of differential fields Rational approximation to the solutions of algebraic differential equations Existence of invariant bases Abelian extensions of differential fields Le theoreme de la base finie pour les polynomes differentiels The notion of dimension in the theory of algebraic differential equations Singular solutions of algebraic differential equations and a lemma of Arnold Shapiro Some problems in differential algebra Algebraic groups and algebraic dependence Differential polynomials and strongly normal extensions Constrained extensions of differential fields Differential equations in a projective space and linear dependence over a projective variety Differential algebraic groups Differential algebraic structures On universal extensions of differential fields Differential algebraic groups A problem on differential polynomials Painleve transcendent Part II. Commentary: Algebraic groups and Galois theory in the work of Ellis R. Kolchin by A. Borel Direct and inverse problems in differential Galois theory by M. F. Singer Les corps differentiellement clos, compagnons de route de la theorie des modeles by B. Poizat Differential algebraic geometry and differential algebraic groups: From algebraic differential equation to Diophantine geometry by A. Buium and P. J. Cassidy.