Geometry of pre quasi homogeneous polynomials of type (1/2, 1/4)

• Autorzy: Kushner L. , Saia M. J.
• Języki publikacji: EN

Abstrakty

EN

In this article we study the geometry of the orbits of the space V which consists of pre quasi homogeneous polynomials of type g(x, y) = a1x2 + a2xy2 + a3y4 + a4xy + a5y3 + a6x + a7y + a8y2 with ai ∈ R, for all i = 1, . . . , 8 under the action of the group G := {h(x, y) = (...), with (...) > 0}. To study these orbits we observe first that there are three subspaces of dimension 5, V1 := {g(x, y) = a1x2 + a2xy2 + a3y4 + a4xy + a5y3}, V2 := {g(x, y) = a1x2 + a2xy2 + a3y4 + a6x + a8y2} and V3 := {g(x, y) = a1x2 + a2xy2 + a3y4 + a7y + a8y2} of V which are also invariant under the action of this group. Then we describe the orbits which appear in these spaces and give the topological characterization of them by showing their stabilizers. We give a geometrical description of them inside R5. Moreover, we construct an appropriate map h : R6 - R5 and prove that the fibers given by the inverse image of the orbits by h are two dimensional surfaces diffeomorphic to R2 - (R × {0} ∪ {0} × R). We show that the points of these fibers which minimize the distance to the origin are indeed in the 3-torus (...)3 = S1 1/2 × S1 1/2 × S1 1/(...).

Słowa kluczowe

EN

pre-quasi homogeneous map; stabilizer; orbit; finite relative determination

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2010
• Tom: Vol. 43, nr 2
• Strony: 461--471
• Opis fizyczny: Bibliogr. 7 poz., rys.

Twórcy

autor

Kushner L.

autor

Saia M. J.

• Facultad de Ciencias Universidad Nacional Aut?noma De México, D. F.,

Bibliografia

• [1] E. Andrade, L. Kushner Schnur, Finite determination, Finite relative determination and the Artin-Rees Lemma. Notas ICMC-USP, n: 218. 2006.
• [2] V. I. Arnol’d, A. N. Varchenko, S. M. Goussein-Zad´e, Singularit´es des Applications Diff´erentiables, Editions Mir, Moscou, 1986.
• [3] R. Bulajich, L. Kushner Schnur, S. L´opez de Medrano, Cubics in R ans C, Real and Complex Singularities, D. Mond and M. J. Saia (eds.), Lecture Notes Series in Pure and Applied Maths, Marcel-Decker, 237–244, 2003.
• [4] R. Bulajich, L. Kushner, S. L´opez de Medrano, The space of quasi homogeneous maps in two variables of type (1/3, 1/6) and degree 1, preprint, 2006.
• [5] M. J. Saia, Pre-weighted homogeneous map germs, finite determinacy and topological triviality, Nagoya Math. J. 151 (1998), 209–220.
• [6] R. Thom, Structural Stability and Morphogenesis, An Outline of a General Theory of Models, W. A. Benjamin, Inc., 1975.
• [7] E. C. Zeeman, The Umbilic Bracelet and the Double Cusp Catastrophe, in: Catastrophe Theory, Selected Papers 1972-77, Addison-Wesley, 1977, pp. 563–601.