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Differential Simplicity in Polynomial Rings and Algebraic Independence of Power Series
Author(s) -
Brumatti Paulo,
Lequain Yves,
Levcovitz Daniel
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610703004708
Subject(s) - mathematics , connection (principal bundle) , formal power series , polynomial ring , simplicity , polynomial , power series , zero (linguistics) , ring (chemistry) , series (stratigraphy) , algebraic number , combinatorics , simple (philosophy) , field (mathematics) , independence (probability theory) , discrete mathematics , differential (mechanical device) , pure mathematics , mathematical analysis , physics , geometry , quantum mechanics , paleontology , statistics , thermodynamics , biology , linguistics , philosophy , chemistry , organic chemistry , epistemology
Let k be a field of characteristic zero, f ( X,Y ), g ( X,Y )∈ k [ X,Y ], g ( X,Y ) ∉ ( X,Y ) and d := g ( X,Y )δ/δ X + f ( X,Y )δ/δ Y . A connection is established between the d ‐simplicity of the local ring k [ X,Y ] ( X,Y ) and the transcendency of the solution in tk [[ t ]] of the algebraic differential equation g ( t , y ( t ))·(δ/δ t ) y ( t )+ f ( t,y(t) ). This connection is used to obtain some interesting results in the theory of the formal power series and to construct new examples of differentially simple rings.