Equivalence of differentiable functions, rational functions and polynomials

Masahito Shiota

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 4, page 167-204
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Shiota, Masahito. "Equivalence of differentiable functions, rational functions and polynomials." Annales de l'institut Fourier 32.4 (1982): 167-204. <http://eudml.org/doc/74559>.

@article{Shiota1982,
abstract = {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.},
author = {Shiota, Masahito},
journal = {Annales de l'institut Fourier},
keywords = {equivalence to a polynomial; number of critical points; Milnor number of the germ at each critical point; locally equivalent to a germ of a rational function; real affine non-singular algebraic varieties; right equivalence of smooth functions},
language = {eng},
number = {4},
pages = {167-204},
publisher = {Association des Annales de l'Institut Fourier},
title = {Equivalence of differentiable functions, rational functions and polynomials},
url = {http://eudml.org/doc/74559},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Shiota, Masahito
TI - Equivalence of differentiable functions, rational functions and polynomials
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 4
SP - 167
EP - 204
AB - 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.
LA - eng
KW - equivalence to a polynomial; number of critical points; Milnor number of the germ at each critical point; locally equivalent to a germ of a rational function; real affine non-singular algebraic varieties; right equivalence of smooth functions
UR - http://eudml.org/doc/74559
ER -

References

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  1. [1] S. LOJASIEWICZ, Ensemble semi-algebraic, IHES, 1965. 
  2. [2] J. W. MILNOR, Lectures on the h-cobordism theorem, Princeton U.P., 1965. Zbl0161.20302MR32 #8352
  3. [3] M. SHIOTA, Equivalence of differentiable functions and analytic functions, Thesis, Univ. of Rennes, France, 1978. 
  4. [4] M. SHIOTA, Equivalence of differentiable mappings and analytic mappings, Publ. Math. IHES, 54 (1981), 37-122. Zbl0516.58012MR84k:58039
  5. [5] M. SHIOTA, On the unique factorial property of the ring of Nash functions, Publ. RIMS, Kyoto Univ., 17 (1981), 363-369. Zbl0503.58001MR83a:58001
  6. [6] M. SHIOTA, Sur la factorialité de l'anneau des fonctions lisses rationnelles, C.R., A.S., Paris, 282 (1981), 67-70. Zbl0489.14013MR82b:14009
  7. [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. Zbl0307.32007MR49 #5394
  8. [8] S. SMALE, Differentiable and combinatorial structures on manifolds, Ann. of Math., 74 (1961), 498-502. Zbl0111.18902MR24 #A2967
  9. [9] R. THOM, L'équivalence d'une fonction différentiable et d'un polynôme, Topology, 3., Suppl. 2 (1965), 297-307. Zbl0133.30702MR32 #4702
  10. [10] A. TOGNOLI, Algebraic geometry and Nash functions, Academic Press, 1978. Zbl0418.14002MR82g:14029
  11. [11] J. C. TOUGERON, Idéaux de fonctions différentiables I, Ann. Ins. Fourier, 18 (1968), 177-240. Zbl0188.45102MR39 #2171
  12. [12] J. H. C. WHITEHEAD, A certain region in Euclidean 3-space, Proc. Nat. Acad. Sci., 21 (1935), 364-366. Zbl0012.03604
  13. [13] F. WALDHAUSEN, On irreducible 3-manifolds which are sufficiently large, Ann. of Math., 87 (1968), 56-88. Zbl0157.30603MR36 #7146

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