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Displaying similar documents to “How to calculate the slopes of a D -module”

Characteristic Cauchy problems and solutions of formal power series

Sunao Ouchi (1983)

Annales de l'institut Fourier

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Let L ( z , ∂ z ) = ( ∂ z 0 ) k - A ( z , ∂ z ) be a linear partial differential operator with holomorphic coefficients, where A ( z , ∂ z ) = ∑ j = 0 k - 1 A j ( z , ∂ z ' ) ( ∂ z 0 ) j , ord . A ( z , ∂ z ) = m > k and z = ( z 0 , z ' ) ∈ C n + 1 . We consider Cauchy problem with holomorphic data L ( z , ∂ z ) u ( z ) = f ( z ) , ( ∂ z 0 ) i u ( 0 , z ' ) = u ^ i ( z ' ) ( 0 ≤ i ≤ k - 1 ) . We can easily get a formal solution u ^ ( z ) = ∑ n = 0 ∞ u ^ n ( z ' ) ( z 0 ) n / n ! , bu in general it diverges. We show under some conditions that for any sector S with the opening less that a constant determined by L ( z , ∂ z ) , there is a function u S ( z ) holomorphic except on { z 0 = 0 } such that L ( z , ∂ z ) u S ( z ) = f ( z ) and u S ( z ) ∼ u ^ ( z ) as z 0 → 0 in S .