Geodesic
Jet continuation and reducibility of smooth function families
Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 2, pp. 515-532.

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For a family of smooth functions of several variables the problem of local reducibility (through a diffeomorphism) to a lower number of variables is considered. Some sufficient conditions of such reducibility, in terms of subfamily reducibility, are obtained. The argumentation is reduced to solution of a jet continuation problem, for jets defined on the union of a finite family of sets, where the jets can be continued from each of the sets. 
@article{FPM_2001_7_2_a11,
     author = {V. I. Rabover},
     title = {Jet continuation and reducibility of smooth function families},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {515--532},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2001_7_2_a11/}
}
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V. I. Rabover. Jet continuation and reducibility of smooth function families. Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 2, pp. 515-532. http://geodesic.mathdoc.fr/item/FPM_2001_7_2_a11/