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DIFFERENTIAL SIMPLICITY IN POLYNOMIAL RINGS AND ALGEBRAIC INDEPENDENCE OF POWER SERIES

Published online by Cambridge University Press:  17 November 2003

PAULO BRUMATTI
Affiliation:
IMECC, Universidade Estadual de Campinas, 13801-970 Campinas, SP, [email protected]
YVES LEQUAIN
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, RJ, [email protected]
DANIEL LEVCOVITZ
Affiliation:
Instituto de Ciências Matemáticas e de Computação, USP-SC, Av. Dr Carlos Botelho, 1465, 13560-970 São Carlos, SP, [email protected]
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Abstract

Let $k$ be a field of characteristic zero, $f(X,Y), g(X,Y)\,{\in}\,k[X,Y]$, $g(X,Y)\,{\notin}\,(X,Y)$ and $d\,{:=} g(X,Y)\,{\partial}/ {\partial X}+f(X,Y)\,{\partial}/{\partial Y}$. 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 $g(t,y(t))\,{\cdot}\, ({\partial}/{\partial t})y(t)=f(t,y(t))$. 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.

Type
Notes and Papers
Copyright
The London Mathematical Society 2003

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Footnotes

This work was partially supported by a grant from the ALGA-PRONEX/MCT group.