A gigantic task undertaken by J. F. Ritt and his collaborators in the 1930's was to give the classical theory of nonlinear differential equations, similar to the theory created by Emmy Noether and her school for algebraic equations and algebraic varieties. The current book presents the results of 20 years of work on this problem. The book quickly became a classic, and thus far, it remains one of the most complete and valuable accounts of differential algebra and its applications.
Author(s): Joseph Fels Ritt
Publisher: Dover Publications Inc.
Year: 1950
Language: English
Pages: 189
Characteristic sets of prime ideals......Page 178
Title......Page 1
Preface......Page 2
Contents......Page 5
Differential fields......Page 7
Differential polynomials......Page 8
Chains......Page 9
Characteristic sets......Page 10
Reduction......Page 11
Ideals of differential polynomials......Page 13
Bases......Page 15
Strong and weak bases......Page 17
Decomposition of perfect ideals......Page 19
Relatively prime ideals......Page 20
The ideal [y^p]......Page 22
Adjunction of indeterminates......Page 24
Field extensions......Page 25
Fields of constants......Page 26
Manifolds and their decomposition......Page 27
Illustrations in analysis......Page 29
Prime ideals and regular zeros......Page 31
Generic zeros of a prime ideal......Page 32
The theorem of zeros......Page 33
General solutions......Page 36
Singular zeros and solutions......Page 38
Parametric indeterminates......Page 39
The resolvent......Page 40
Dimension of an irreducible manifold......Page 50
Order of the resolvent......Page 51
Embedded manifolds......Page 55
Prime ideals and field extensions......Page 56
Analogue of Luroth's theorem......Page 58
The polygon process......Page 63
Dimensions of components......Page 68
Preparation process......Page 69
The low power theorem......Page 70
Sufficiency proof......Page 72
Necessity proof......Page 75
An example......Page 76
Further theorems on low powers......Page 77
Terms of lowest degree......Page 80
Singular solutions......Page 81
III. Exponents of ideals......Page 84
Polynomials and their ideals......Page 87
Algebraic manifolds......Page 88
Resolvents......Page 89
Hilbert's theorem of zeros......Page 93
Characteristic sets of prime polynomial ideals......Page 94
Construction of resolvents......Page 96
Components of finite systems......Page 101
An approximation theorem......Page 109
Zeros and characteristic sets......Page 112
Characteristic sets of prime ideals......Page 113
Finite systems......Page 115
Construction of resolvents......Page 116
Constructive proof of theorem of zeros......Page 117
A second theory of elimination......Page 118
Theoretical process for decomposing the manifold of a finite system into its components......Page 124
The theorem of approximation......Page 128
Analytical treatment of low power theorem......Page 132
Differential polynomials in one indeterminate of first order......Page 135
Sequences of irreducible manifolds......Page 137
Operations upon manifolds......Page 138
Orders of components of an intersection......Page 139
Intersections of general solutions......Page 144
Intersections of components of a differential polynomial......Page 150
Analogue of a theorem of Kronecker......Page 152
Monomials......Page 153
Dissection of a Taylor series......Page 154
Marks......Page 157
Orthonomic systems......Page 158
Passive orthonomic systems......Page 166
Partial differential polynomials, ideals and manifolds......Page 169
Components of a partial differential polynomial......Page 173
The low power theorem......Page 176
Algorithm for decomposition......Page 181
The theorem of zeros......Page 182
Appendix. Questions for investigation......Page 183
Bibliography......Page 186
Index......Page 188
