Taylor expansion of implicit functions defined by linear equations of variables

Abstract

Let F be a monoid of countably many functions holomorphic at y⁰, and (X_f)_f∈F be a set of independent variables. We set F_*=F−{1}, x_*=(x_f)_f∈F*. Let (F₁, F₂, ...) be an increasing sequence of finite subsets of F such that ∪_i≥1 F_i=F. For i≥1, let A_i, denote the ring of all functions of (x_f)_f∈Fi, holomorphic at (x₁, (x_f)_f∈Fi−{1})=(x₁⁰, 0). Define A=proj lim A_i. Consider the implicit function y∈A defined by g(y)=Σ_f∈Fx_ff(y) (y(x₁⁰, 0)=y⁰). We have the Taylor expansion of y at x_*=0:
y=g⁻¹(x₁)+Σ_α(\frac{d^|α|−1}{dX₁^|α|−1} \frac{Π_f∈F*f^α(f)(g⁻¹(x₁))}{g'(g⁻¹(x₁))}) \frac{x_*^α}{α!},
where the sum runs over all maps α : F_*→{0, 1, 2, ...} such that |α| :=Σ_f∈F*α(f) are positive finite.