By Kuik G.R.

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**Extra resources for Transseries in difference and differential equations**

**Example text**

M, let σi denote the ith elementary symmetric function in m variables X1 , . . , σi (X1 , . . , Xm ) is a polynomial of degree i deﬁned by m (1 + Xi ) = 1 + σ1 (X1 , . . , Xm ) + · · · + σm (X1 , . . , Xm ). i=1 Since diﬀerential forms of even degrees commute with one another with respect to the exterior product, we may treat κ as an ordinary matrix whose entries are numbers. We deﬁne a 2i-form σi (κ) on U by det(I + κ) = 1 + σ1 (κ) + · · · + σm (κ), where I denotes the identity matrix of rank k.

Let M be a C ∞ manifold of dimension m . For an open set U in M , we denote by Ap (U ) the complex vector space of complex valued C ∞ p-forms on U . , C ∞ sections p of the bundle (TRc M )∗ ⊗ E on U , where (TRc M )∗ denotes the dual of the complexiﬁcation of the real tangent bundle TR M of M . Thus A0 (U ) is the ring of C ∞ functions and A0 (U, E) is the A0 (U )-module of C ∞ sections of E on U . 1. A connection for E is a C-linear map ∇ : A0 (M, E) −→ A1 (M, E) satisfying the “Leibniz rule”: ∇(f s) = df ⊗ s + f ∇(s) for f ∈ A0 (M ) and s ∈ A0 (M, E).

Since each stratum Vα is itself a complex manifold we have its tangent bundle T Vα . 3 Radial Extension of Vector Fields 35 and the regular one Vreg = V \ Sing(V ). If V is reducible, we assume it is pure dimensional. 1. A stratiﬁed vector ﬁeld on V means a (continuous, smooth) section v of the complex tangent bundle T M |V such that for each x ∈ V the vector v(x) is contained in the tangent space of the stratum Vα that contains x. First we describe the local extension process. 1. Let vα be a vector ﬁeld in a neighborhood of a point x ∈ Vα with possibly a singularity at x.