Displaying similar documents to “Small divisors and large multipliers”

Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations

Zhuangchu Luo, Hua Chen, Changgui Zhang (2012)

Annales de l’institut Fourier

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In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at ( t , x ) = ( 0 , 0 ) ∈ C 2 . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the k -summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

On Halphen’s Theorem and some generalizations

Alcides Lins Neto (2006)

Annales de l’institut Fourier

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Let M n be a germ at 0 ∈ ℂ m of an irreducible analytic set of dimension n , where n ≥ 2 and 0 is a singular point of M . We study the question: when does there exist a germ of holomorphic map φ : ( ℂ n , 0 ) → ( M , 0 ) such that φ - 1 ( 0 ) = { 0 } ? We prove essentialy three results. In Theorem 1 we consider the case where M is a quasi-homogeneous complete intersection of k polynomials F = ( F 1 , ... , F k ) , that is there exists a linear holomorphic vector field X on ℂ m , with eigenvalues λ 1 , ... , λ m ∈ ℚ + such that X ( F T ) = U · F T , where U is a k × k matrix with entries in 𝒪 m . We prove that if...

Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

Masafumi Yoshino, Todor Gramchev (2008)

Annales de l’institut Fourier

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We study the simultaneous linearizability of d –actions (and the corresponding d -dimensional Lie algebras) defined by commuting singular vector fields in ℂ n fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of d vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators...

On proper discs in complex manifolds

Barbara Drinovec Drnovšek (2007)

Annales de l’institut Fourier

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Let X be a complex manifold of dimension at least 2 which has an exhaustion function whose Levi form has at each point at least 2 strictly positive eigenvalues. We construct proper holomorphic discs in X through any given point and in any given direction.