Preparation theorems for systems. (English) Zbl 0716.58006

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1990, No. 9, 10 p. (1990).
The Malgrange preparation theorem is generalized to matrix valued functions in the following way. Assume that \(F(t,x)\in C^{\infty}({\mathbb{R}}\times {\mathbb{R}}^ n)\) is \(N\times N\) matrix valued, such that \(t\mapsto \det F(t,0)\) vanishes of finite order at \(t=0\). Then we can factor \(F(t,x)=C(t,x)P(t,x)\) near (0,0), where \(C(t,x)\in C^{\infty}\) is invertible and P(t,x) is polynomial function of t depending \(C^{\infty}\) on x. By an orthogonal base change, P(t,0) may be put in an upper diagonal form. The preparation is (essentially) unique, up to functions vanishing of infinite order at \(x=0\), under some additional conditions on P(t,x). The diagonal elements are then determined by the elementary divisors of the Taylor expansion of \(t\mapsto F(t,0)\) at the origin. If F(t,x) is real (matrix) valued or analytic, one may choose C(t,x) and P(t,x) real (matrix) valued or analytic. We also have a generalization of the division theorem: if \(G(t,x)\in C^{\infty}\) is \(N\times N\) matrix valued and F(t,x) satisfies the above condition, then we can write \[ G(t,x)=Q(t,x)F(t,x)+R(t,x) \] where R(t,x) is polynomial function of t depending \(C^{\infty}\) on x. By taking adjoints, we also get the corresponding results for right preparation (division). These results can be used to obtain normal forms of systems of partial differential equations. The proofs will appear elsewhere.
Reviewer: N.Dencker


58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
26E10 \(C^\infty\)-functions, quasi-analytic functions
26B40 Representation and superposition of functions
35G05 Linear higher-order PDEs
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