Skip to Main content Skip to Navigation
Theses

THE SHAPES OF LEVEL CURVES OF REAL POLYNOMIALS NEAR STRICT LOCAL MINIMA

Abstract : We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex, as was shown by Coste. The aim of the present thesis is twofold. Firstly, to construct examples of non-Morse strict local minima whose sufficiently small level curves are far from being convex. And secondly, to study a combinatorial object measuring this non-convexity, namely the Poincaré-Reeb tree of the restriction of the first coordinate to the region bounded by a given level curve. These planar trees are rooted and their vertices roughly speaking correspond to points on the curve with vertical tangent lines. The main objective of our study is to characterise all possible topological types of Poincaré-Reeb trees. To this end, we construct a family of examples realising a large class of such trees. As a preliminary step, we restrict our attention to the univariate case, using a tool inspired by Ghys’ work. One of our main results gives a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold’s snake”) is separable.
Complete list of metadata

Cited literature [98 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01909028
Contributor : Miruna-Stefana Sorea Connect in order to contact the contributor
Submitted on : Tuesday, October 30, 2018 - 5:20:20 PM
Last modification on : Sunday, November 29, 2020 - 3:24:05 AM

File

SoreaMirunaStefana_PhDThesis.p...
Files produced by the author(s)

Identifiers

  • HAL Id : tel-01909028, version 1

Collections

Citation

Miruna-Stefana Sorea. THE SHAPES OF LEVEL CURVES OF REAL POLYNOMIALS NEAR STRICT LOCAL MINIMA. Algebraic Geometry [math.AG]. Université de Lille / Laboratoire Paul Painlevé, 2018. English. ⟨tel-01909028⟩

Share

Metrics

Les métriques sont temporairement indisponibles