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Equivalence Relations for Two Variable Real Analytic Function Germs
| Content Provider | Semantic Scholar |
|---|---|
| Author | Koike, Satoshi Parusinski, Adam |
| Copyright Year | 2008 |
| Abstract | For two variable real analytic function germs we compare the blowanalytic equivalence in the sense of Kuo to the other natural equivalence relations. Our main theorem states that C equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the C equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence in terms of the real tree model. We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent. The natural equivalence relations we first think of are the C coordinate changes for r = 1, 2, · · · ,∞, ω, where C stands for real analytic. Let f , g : (R, 0) → (R, 0) be real analytic function germs. We say that f and g are C (right) equivalent if there is a local C diffeomorphism σ : (R, 0) → (R, 0) such that f = g ◦ σ. If σ is a local bi-Lipschitz homeomorphism, resp. a local homeomorphism, then we say that f and g are bi-Lipschitz equivalent, resp. C equivalent. By definition, we have the following implications: (0.1) C-eq. ⇐ bi-Lipschitz eq. ⇐ C-eq. ⇐ C-eq. ⇐ · · · ⇐ C-eq. ⇐ C-eq. By Artin’s Approximation Theorem [2], C equivalence implies C equivalence. But the other converse implications of (0.1) do not hold. Let f , g : (R, 0) → (R, 0) be polynomial functions defined by f(x, y) = (x + y), g(x, y) = (x + y) + x for r = 1, 2, · · · . N. Kuiper [14] and F. Takens [21] showed that f and g are C equivalent, but not C equivalent. In the family of germs Kt(x, y) = x 4 + txy + y, the phenomenon of continuous C moduli appears: for t1, t2 ∈ I, Kt1 and Kt2 are C equivalent if and only if t1 = t2, where I = (−∞,−6], [−6,−2] or [−2,∞), see example 0.5 below. On the other hand, T.-C. Kuo proved that this family is The second named author was partially supported by the JSPS Invitation Fellowship Program. ID No. S-07026 1991 Mathematics Subject Classification. Primary: 32S15. Secondary: 14B05, 57R45. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://math.unice.fr/~parus/publis/2equivfinal.pdf |
| Alternate Webpage(s) | http://math.unice.fr/~parus/publis/2equivfinal.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/0801.2650v1.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Approximation Artin billiard Biologic Preservation Equivalent Weight JavaServer Pages Lambda calculus Mathematics Subject Classification NP-equivalent Name Neoplasm Metastasis Newton Plant Roots Polynomial Tooth Germ Turing completeness |
| Content Type | Text |
| Resource Type | Article |
