Index of binary differential equations

Author(s)

Lizandro Sanchez Challapa

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2006-03-31

Examining board members

Maria Aparecida Soares Ruas; Ronaldo Alves Garcia; Marcelo José Saia; Marcio Gomes Soares; Marco Antonio Teixeira

• Advisor: Maria Aparecida Soares Ruas; Farid Tari

Abstract

In this work we study binary differential equations in a neighborhood of an isolated singular point. Following the geometric approach of Bruce and Tari in their work on multiplicity of a binary differential equation, we introduce a new definition of index for this class of equations, which coincides with the classical definition by Hopf for positive binary differential equations. The main result is a formula expressing the index in terms of information obtained from the coefficients of the original equation. The invariance of the index by smooth equivalences is also proved. Some results relating the index with the indices of 1-forms and vector fields in singular varieties are given for a special class of implicit differential equations (AU)