The Center and Focus Problem

Algebraic Solutions and Hypotheses

M.N. Popa, V.V. Pricop

• Publisher: Chapman and Hall/CRC
• Year: 2021
• eBook ISBN: 9781000426670
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

The Center and Focus Problem: Algebraic Solutions and Hypotheses, M. N. Popa and V.V. Pricop, ISBN: 978-1-032-01725-9 (Hardback)

This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades.

A rather essential result is obtained in solving Poincaré's problem. Namely, there are given the upper estimations of the number of Poincaré-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Authors
• Introduction
• 1 Lie Algebra of Operators of Centro-Affine Group Representation in the Coefficient Space of Polynomial Differential Systems
• 2 Differential Equations for Centro-Affine Invariants and Comitants of Differential Systems and Their Applications
• 3 Generating Functions and Hilbert Series for Sibirsky Graded Algebras of Comitants and Invariants of Differential Systems
• 4 Hilbert Series for Sibirsky Algebras Sm(SIm) and Krull Dimension for Them
• 5 About the Center and Focus Problem
• 6 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the System s(1,m1,..., mℓ)
• 7 On the Upper Bound of the Number of Functionally Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for Lyapunov System
• Appendix 1
• Appendix 2
• Appendix 3
• Appendix 4
• Appendix 5
• Appendix 6
• Appendix 7
• Appendix 8
• Appendix 9
• Appendix 10
• Bibliography
• Index