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.
@article{urn:eudml:doc:44267,
title = {Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {10},
year = {1997},
pages = {291-310},
zbl = {0949.14036},
mrnumber = {MR1485306},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44267}
}
Santos, Francisco. Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.. Revista Matemática de la Universidad Complutense de Madrid, Tome 10 (1997) pp. 291-310. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44267/