• Corpus ID: 119631303

Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces

@article{Guilfoyle2018UmbilicPO,
  title={Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces},
  author={Brendan Guilfoyle and Adriana Ortiz-Rodr'iguez},
  journal={arXiv: Differential Geometry},
  year={2018}
}
The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincar\'e-Hopf type formula where the sum over all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of $f$. Moreover, we study the projective extension of these fields and prove, under generic conditions, that every umbilic point at… 
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References

SHOWING 1-10 OF 22 REFERENCES
On the geometric structure of certain real algebraic surfaces
In this paper we study the affine geometric structure of the graph of a polynomial $$f \in \mathbb {R}[x,y]$$f∈R[x,y]. We provide certain criteria to determine when the parabolic curve is compact and
On binary differential equations and umbilics
Synopsis In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation where a and b are smooth functions which we suppose vanish at 0 ∈
Locally Stable Singularities for Positive Quadratic Differential Forms
Abstract Every positive C ∞ -quadratic differential form defined on an oriented surface has two transverse C ∞ -one-dimensional foliations with common singularities associated to it. In this article
On Invariant Theory
TLDR
In this paper it is shown how the invariants of n−ary forms can be produced from the discriminant of multilinear forms (determinants of multidimensional matricies), which should be considered as the generalization of the operation of taking classical hessians and resultants.
Classical Invariant Theory and the Equivalence Problem for Particle Lagrangians
The problem of equivalence of binary forms under linear changes of variables is shown to be a special case of the problem of equivalence of particle Lagrangians under the pseudogroup of
Structurally stable configurations of lines of principal curvature
Sufficient conditions are established for the stability of the configuration defined by the umbilical points and the families of lines of principal curvature of a compact orientable 3 surface
Positive quadratic differential forms : linearization, finite determinacy and versal unfolding
Des problemes locaux autour des singularites tels que linearisation, determination finie, et deploiements versels sont consideres pour un type de forme differentielle quadratique C∞ sur des surfaces
Lines of curvature on algebraic surfaces
Dans ce travail sont etudiees les configurations stables des lignes de courbure sur les surfaces algebriques. Les conditions suffisantes pour la stabilite structurelle des feuilletages principaux sur
Master Dissertation
TLDR
This work presents the extension that adds English support to the framework, which is achieved with the modification of ErauzOnt, the tool that enables the acquisition of learning resources, definitions, examples, exercises, etc. used in the learning process.
...
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