Full paper in PDF:
$%F. Santos, Optimal degree construction of real algebraic plane nodal curves with prescribed topology, I: the orientable case, Rev. Mat. Univ. Complut. Madrid 10 (1997), supplementary, 291310. %$

Optimal Degree Construction of Real Algebraic Plane Nodal Curves with Prescribed Topology, I: the Orientable Case
Francisco SANTOS
Dpto. de Matemáticas, Estadística y Computación
Universidad de Cantabria
E-39071 Santander Spain


We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T  to be realized has only double singularities and gives an algebraic curve of degree 2N + 2K  , where N  and K  are the numbers of double points and connected components of T  . This degree is optimal in the sense that for any choice of the numbers N  and K  there exist models which cannot be realized algebraically with lower degree. Moreover, we characterize precisely which models have this property. The construction is based on a preliminary topological manipulation of the topological model followed by some perturbation technique to obtain the polynomial which defines the algebraic curve. This paper considers only the case in which T  has an orientable neighborhood. The non-orientable case will appear in a separate paper.

1991 Mathematics Subject Classification: 14P25, 14Q05.