Equivalence of differentiable functions, rational functions and polynomials
Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 167-204.

We show under some assumptions that a differentiable function can be transformed globally to a polynomial or a rational function by some diffeomorphism. One of the assumptions is that the function is proper, the number of critical points is finite, and the Milnor number of the germ at each critical point is finite.

Nous montrons sous certaines hypothèses qu’une fonction différentiable peut être transformée globablement en un polynôme ou une fonction rationnelle par un difféomorphisme. Une des hypothèses est que la fonction est propre, le nombre des points critiques est fini, et le nombre de Milnor du germe en chaque point critique est fini.

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     title = {Equivalence of differentiable functions, rational functions and polynomials},
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Shiota, Masahito. Equivalence of differentiable functions, rational functions and polynomials. Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 167-204. doi : 10.5802/aif.899. https://aif.centre-mersenne.org/articles/10.5802/aif.899/

[1] S. Lojasiewicz, Ensemble semi-algebraic, IHES, 1965.

[2] J. W. Milnor, Lectures on the h-cobordism theorem, Princeton U.P., 1965. | MR: 32 #8352 | Zbl: 0161.20302

[3] M. Shiota, Equivalence of differentiable functions and analytic functions, Thesis, Univ. of Rennes, France, 1978.

[4] M. Shiota, Equivalence of differentiable mappings and analytic mappings, Publ. Math. IHES, 54 (1981), 37-122. | Numdam | MR: 84k:58039 | Zbl: 0516.58012

[5] M. Shiota, On the unique factorial property of the ring of Nash functions, Publ. RIMS, Kyoto Univ., 17 (1981), 363-369. | MR: 83a:58001 | Zbl: 0503.58001

[6] M. Shiota, Sur la factorialité de l'anneau des fonctions lisses rationnelles, C.R., A.S., Paris, 282 (1981), 67-70. | MR: 82b:14009 | Zbl: 0489.14013

[7] H. Skoda, J. Briancon, Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de Cn, C.R.A.S., Paris, 278 (1974), 949-951. | MR: 49 #5394 | Zbl: 0307.32007

[8] S. Smale, Differentiable and combinatorial structures on manifolds, Ann. of Math., 74 (1961), 498-502. | MR: 24 #A2967 | Zbl: 0111.18902

[9] R. Thom, L'équivalence d'une fonction différentiable et d'un polynôme, Topology, 3., Suppl. 2 (1965), 297-307. | MR: 32 #4702 | Zbl: 0133.30702

[10] A. Tognoli, Algebraic geometry and Nash functions, Academic Press, 1978. | MR: 82g:14029 | Zbl: 0418.14002

[11] J. C. Tougeron, Idéaux de fonctions différentiables I, Ann. Ins. Fourier, 18 (1968), 177-240. | Numdam | MR: 39 #2171 | Zbl: 0188.45102

[12] J. H. C. Whitehead, A certain region in Euclidean 3-space, Proc. Nat. Acad. Sci., 21 (1935), 364-366. | Zbl: 0012.03604

[13] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math., 87 (1968), 56-88. | MR: 36 #7146 | Zbl: 0157.30603

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