Preparation theorems for matrix valued functions
Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 865-892.

Nous généralisons le théorème de préparation de Malgrange au cas des fonctions F(t,x)C (R×R n ) à valeurs matricielles. Nous supposons que t det F(t,0) s’annule à un ordre fini en t=0. Nous démontrons qu’on peut alors factoriser F sous la forme F(t,x)=C(t,x)P(t,x) au voisinage de (0,0), où C(t,x)C est inversible et P(t,x) est un polynôme en t, à coefficients qui sont des fonctions C de x. Si nous imposons des conditions supplémentaires sur P(t,x), nous montrons que la préparartion est (essentiellement) unique, modulo des fonctions s’annulant à l’ordre infini en x=0. Nous donnons aussi une généralisation du théorème de division de Malgrange, et des versions analytiques qui généralisent les théorèmes de préparation et division de Weierstrass.

We generalize the Malgrange preparation theorem to matrix valued functions F(t,x)C (R×R n ) satisfying the condition that t det F(t,0) vanishes to 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)C is inversible and P(t,x) is polynomial function of t depending C on x. The preparation is (essentially) unique, up to functions vanishing to 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.

@article{AIF_1993__43_3_865_0,
     author = {Dencker, Nils},
     title = {Preparation theorems for matrix valued functions},
     journal = {Annales de l'Institut Fourier},
     pages = {865--892},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {3},
     year = {1993},
     doi = {10.5802/aif.1359},
     zbl = {0783.58010},
     mrnumber = {95f:32009},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_3_865_0/}
}
Dencker, Nils. Preparation theorems for matrix valued functions. Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 865-892. doi : 10.5802/aif.1359. http://www.numdam.org/item/AIF_1993__43_3_865_0/

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