# Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.

Revista Matemática de la Universidad Complutense de Madrid (1997)

- Volume: 10, Issue: Supl., page 291-310
- ISSN: 1139-1138

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topSantos, 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 10.Supl. (1997): 291-310. <http://eudml.org/doc/44267>.

@article{Santos1997,

abstract = {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.},

author = {Santos, Francisco},

journal = {Revista Matemática de la Universidad Complutense de Madrid},

keywords = {Geometría algebraica; Espacio proyectivo; Propiedades topológicas; Curvas algebraicas; Diseño óptimo; Grado; real algebraic plane with prescribed singularities; orientable curves},

language = {eng},

number = {Supl.},

pages = {291-310},

title = {Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.},

url = {http://eudml.org/doc/44267},

volume = {10},

year = {1997},

}

TY - JOUR

AU - Santos, Francisco

TI - Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.

JO - Revista Matemática de la Universidad Complutense de Madrid

PY - 1997

VL - 10

IS - Supl.

SP - 291

EP - 310

AB - 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.

LA - eng

KW - Geometría algebraica; Espacio proyectivo; Propiedades topológicas; Curvas algebraicas; Diseño óptimo; Grado; real algebraic plane with prescribed singularities; orientable curves

UR - http://eudml.org/doc/44267

ER -

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