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

[1] N. Dencker, The Propagation of Polarization in Double Refraction, J. Funct. Anal., 104 (1992), 414-468. | MR 93e:35108 | Zbl 0767.58041

[2] L. Hörmander, The Analysis of Linear Partial Differential Operators I-IV, Springer-Verlag, Berlin, 1983-1985. | Zbl 0612.35001

[3] B. Malgrange, Le théorème de préparation en géométrie différentiable, Séminaire H. Cartan, 15, 1962-1963, Exposés 11, 12, 13, 22. | Numdam | Zbl 0119.28501

[4] B. Malgrange, The preparation theorem for differentiable functions, Differential Analysis, 203-208, Oxford University Press, London, 1964. | MR 32 #178 | Zbl 0137.03601

[5] B. Malgrange, Ideals of differentiable functions, Oxford University Press, London, 1966. | Zbl 0177.17902

[6] J. Mather, Stability of C∞ mappings : I. The division theorem, Ann. of Math., 87 (1968), 89-104. | MR 38 #726 | Zbl 0159.24902

[7] J. Mather, Stability of C∞ mappings, III : Finitely determined map-germs, Publ. Math. I.H.E.S., 35 (1968), 127-156. | Numdam | MR 43 #1215a | Zbl 0159.25001

[8] L. Nirenberg, A proof of the Malgrange preparation theorem, Springer Lecture Notes in Math., 192 (1971), 97-105. | MR 54 #586 | Zbl 0212.10702

[9] B. L. Van Der Waerden, Algebra II, 5 Aufl., Springer-Verlag, Berlin, 1967. | MR 38 #1968 | Zbl 0192.33002