Bifurcation set and index at infinity of polynomials

Author(s)

Gabriel Esteban Perico Monsalve

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2023-12-19

Examining board members

Raimundo Nonato Araújo dos Santos; Alexandre César Gurgel Fernandes; Aurélio Menegon Neto; Maria Aparecida Soares Ruas

• Advisor: Raimundo Nonato Araújo dos Santos; Mihai Marius Tibar

Abstract

In the context of polynomial functions f : R2 → R of degree d > 0, we tackle three fundamental challenges: effective detection of atypical values, computation of the index at infinity, and estimation of the upper bound of the index in terms of the degree d. We demonstrate that the presence of specific phenomena at infinity in the fibers, such as the vanishing and splitting of fiber components, leads to the emergence of atypical values, also known as bifurcation values. To identify these phenomena, we leverage the connected components of the Milnor set of the polynomial f outside a compact set in R2, allowing us to describe the topological behavior of fibers in proximity to infinity. Furthermore, we provide a detailed characterization of atypical values and apply our approach to compute them for two polynomials exhibiting intriguing phenomena at infinity. In our study of the index at infinity ind∞f for polynomial functions f : R2 → R with isolated singularities, we define this index as the winding number of the gradient vector field grad f restricted to a circle C encompassing all singular points of f . We present a formula that unveils how the behavior of fibers at infinity influences this index. Lastly, we investigate the phenomena contributing to the gap between ind∞f and the upper bound of the index, previously established by Durfee. (AU)

• FAPESP's process: 19/24377-2 - Index of the gradient at infinity and bifurcation locus
• Grantee: Gabriel Esteban Perico Monsalve
• Support Opportunities: Scholarships in Brazil - Doctorate