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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| Published in | Advances in geometry Vol. 13; no. 2; pp. 277 - 292 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin
Walter de Gruyter GmbH
01.04.2013
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| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | Consider a polynomial fR[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. [PUBLICATION ABSTRACT] 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. |
| Author | Sánchez-Bringas, Federico Hernández Martínez, Lucía Ivonne Ortiz Rodríguez, Adriana |
| Author_xml | – sequence: 1 givenname: Lucía Ivonne surname: Hernández Martínez fullname: Hernández Martínez, Lucía Ivonne organization: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F 04510, México – sequence: 2 givenname: Adriana surname: Ortiz Rodríguez fullname: Ortiz Rodríguez, Adriana organization: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F 04510, México – sequence: 3 givenname: Federico surname: Sánchez-Bringas fullname: Sánchez-Bringas, Federico organization: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F 04510, México |
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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... Consider a polynomial fR[x; y] whose Hessian curve is compact and the unbounded connected component of its complement is hyperbolic. We study the fields of... |
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| Title | On the Hessian geometry of a real polynomial hyperbolic near infinity |
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