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Differential Algebra

American Mathematical Society Colloquium Publications Volume 33 Differential Algebra Joseph Fels Ritt HEMATIC AT A M R AME ICAN ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ Α ΓΕ Ω ΜΕ ΑΓΕΩΜΕ L SOCIETY FO UN 8 DED 1 88 American Mathematical Society Providence, Rhode Island . | American Mathematical Society Colloquium Publications Volume 33 Differential Algebra Joseph Fels Ritt American Mathematical Society Providence Rhode Island PREFACE In 1932 the author published Differential equations from the algebraic standpoint 1 a book dealing with differential polynomials and algebraic differential manifolds. In the sixteen years which have passed the work of a number of mathematicians has given fresh substance and new color to the subject. The complete edition of the book having been exhausted it has seemed proper to prepare a new exposition. The title Differential algebra was suggested by Dr. Kolchin. The body of algebra deals with the operations of addition and multiplication. We are concerned here with three operations addition multiplication and differentiation. If I am not mistaken the general nature of the subject here treated is now well enough known among mathematicians to permit me to dispense with a detailed introduction such as was given in A. D. E. My principal task is to show how much the present book owes to my associates. I am referring to H. w. Raudenbush w. c. Strodt E. R. Kolchin Howard Levi Eli Gourin and Richard M. Cohn. Cohn s constructive proof of the theorem of zeros will be found in Chapter V. The theorem on embedded manifolds due to Gourin is contained in Chapter II. Chapter VI contains a discussion of Strodt s work on sequences of manifolds. In Chapters I III and IX there are presented portions of Levi s work on ideals of differential polynomials and on the low power theorem. Of Kolchin s investigation of exponents of differential ideals I have been able to give only a bare idea. Other work of Kolchin for instance proofs for the abstract case of results previously established for the analytic case is given in Chapter II. His work on the Picard-Vessiot theory which employs the methods of differential algebra has just appeared in the Annals of Mathematics 2 and may be permitted to speak for itself. The contributions of .