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

Kushner L., Saia M. J.

Demonstratio Mathematica

2010

Vol. 43, nr 2

461-471

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/(...).