Preparation theorems for matrix valued functions (Q1313326)

We generalize the Malgrange preparation theorem to matrix valued functions \(F(t,x)\in C^ \infty(\mathbb{R} \times \mathbb{R}^ n)\) satisfying the condition that \(t \mapsto \text{det} F(t,0)\) 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 a polynomial function of \(t\) depending \(C^ \infty\) on \(x\). The preparation is (essentially) unique, up to functions vanishing of infinite order at \(x=0\), if we impose some additional conditions on \(P(t,x)\). We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation and division theorems.