# Equivalence of differentiable functions, rational functions and polynomials

Annales de l'institut Fourier (1982)

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

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topShiota, 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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- [3] M. SHIOTA, Equivalence of differentiable functions and analytic functions, Thesis, Univ. of Rennes, France, 1978.
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- [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] S. SMALE, Differentiable and combinatorial structures on manifolds, Ann. of Math., 74 (1961), 498-502. Zbl0111.18902MR24 #A2967
- [9] R. THOM, L'équivalence d'une fonction différentiable et d'un polynôme, Topology, 3., Suppl. 2 (1965), 297-307. Zbl0133.30702MR32 #4702
- [10] A. TOGNOLI, Algebraic geometry and Nash functions, Academic Press, 1978. Zbl0418.14002MR82g:14029
- [11] J. C. TOUGERON, Idéaux de fonctions différentiables I, Ann. Ins. Fourier, 18 (1968), 177-240. Zbl0188.45102MR39 #2171
- [12] J. H. C. WHITEHEAD, A certain region in Euclidean 3-space, Proc. Nat. Acad. Sci., 21 (1935), 364-366. Zbl0012.03604
- [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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