On the Hessian geometry of a real polynomial hyperbolic near infinity

Abstract Consider a polynomial f∊R[x; y] whose Hessian curve is compact and the unbounded connected component of its complement is hyperbolic. We study the fields of asymptotic directions on this component. Thus, we determine an index formula for the field of asymptotic directions involving the numb...

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Bibliographic Details
Published inAdvances in geometry Vol. 13; no. 2; pp. 277 - 292
Main Authors Hernández Martínez, Lucía Ivonne, Ortiz Rodríguez, Adriana, Sánchez-Bringas, Federico
Format Journal Article
LanguageEnglish
Published Berlin Walter de Gruyter GmbH 01.04.2013
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Summary:Abstract Consider a polynomial f∊R[x; y] whose Hessian curve is compact and the unbounded connected component of its complement is hyperbolic. We study the fields of asymptotic directions on this component. Thus, we determine an index formula for the field of asymptotic directions involving the number of connected components of the Hessian curve constituting the boundary of this component, and the number of the corresponding Gaussian cusps. As an application of this study we show an example of a polynomial of degree 4 with 10 Gaussian cusps. Moreover, we determine the parity of the Gaussian cusps on the boundary of the unbounded region of a graph of a polynomial.
ISSN:1615-715X
1615-7168
DOI:10.1515/advgeom-2012-0028