The Center and Focus Problem: Algebraic Solutions and Hypotheses (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

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Introduction
1 Lie Algebra of Operators of Centro-Affine Group Representation in the Coefficient Space of Polynomial Differential Systems
	1.1 Two-dimensional Polynomial Differential Systems
		1.1.1 Affine System
		1.1.2 System with Quadratic Nonlinearities
		1.1.3 Quadratic System
		1.1.4 System with Cubic Nonlinearities
		1.1.5 Cubic System
	1.2 One-Parameter Linear Groups of Transformations of the Phase Plane of System (1.1)-(1.2)
	1.3 Centro-Affine and Unimodular Transformation Groups of the Phase Plane of System (1.1)-(1.2)
	1.4 Lie Operators of One-Parameter Linear Groups and Their Representations in the Coefficient Space of System (1.1)-(1.2)
	1.5 Operators of Representation of the Linear Groups (1.12), (1.14), (1.16) and (1.17) in the Space of Variables and Coefficients of System (1.1)-(1.2)
	1.6 Lie Algebra of Operators of Centro-Affine Group Represen-tation in the Space of Variables and Coefficients of System (1.1)-(1.2)
	1.7 Comments to Chapter One
2 Differential Equations for Centro-Affine Invariants and Comitants of Differential Systems and Their Applications
	2.1 Concept of Centro-Affine Comitant and an Invariant of Differential System
	2.2 Centro-Affine Transformations of System (1.1) (1.2)
	2.3 Differential Equations for Centro-Affine Invariants and Comitants
	2.4 Rational Absolute Centro-Affine Invariants and Comitants and Their Applications
	2.5 Examples of Algebraic Bases of Centro-Affine Comitants and Invariants for Some Differential Systems
	2.6 Comments to Chapter Two
3 Generating Functions and Hilbert Series for Sibirsky Graded Algebras of Comitants and Invariants of Differential Systems
	3.1 Formulas for Weights of Centro Affine Comitants and Invariants of Given Type
	3.2 Initial form of Generating Function for Centro-Affine Comitants of Differential Systems
	3.3 Examples of Reduced Forms of Generating Functions for Centro-Affine Comitants of Differential Systems
	3.4 Hilbert Series for Graded Algebras of Unimodular Comitants and Invariants of Differential Systems
	3.5 Comments to Chapter Three
4 Hilbert Series for Sibirsky Algebras S[sub(Г)] (SI[sub(Г)] ) and Krull Dimension for Them
	4.1 Krull Dimension for Sibirsky Graded Algebras
	4.2 Hilbert Series for Sibirsky Graded Algebras S[sub(1)],m[sub(1)],m[sub(2)]....m[sub(l)],SI[sub(1)],m[sub(1)],m[sub(2)]....m[sub(l)]
	4.3 Hilbert Series for Sibirsky Algebras S[sub(1,2)] , SI[sub(1,2)] and Their Krull Dimensions
	4.4 Hilbert Series for Sibirsky Algebras S[sub(1,3)] , SI[sub(1,3)] and Their Krull Dimensions
	4.5 Hilbert Series for Sibirsky Algebras S[sub(1,4)] , SI[sub(1,4)] and Their Krull Dimensions
	4.6 Hilbert Series for Sibirsky Algebras S[sub(1,5)] , SI[sub(1,5)] and TheirKrull Dimensions
	4.7 Obtaining Ordinary Hilbert Series for Sibirsky Algebras S[sub(1,2,3)]SI[sub(1,2,3)] Using, Residues, and Calculating Krull Dimensions for Them
	4.8 Comments to Chapter Four
5 About the Center and Focus Problem
	5.1 On a New Formulation of the Center and Focus Problem for Differential Systems s(1,m[sub(1)]m[sub(2)],m[sub(l)] )
	5.2 Sibirsky Invariant Variety for Center and Focus
	5.3 Focus Quantities L[sub(k)] and Constants G[sub(k)] on Sibirsky Invariant Variety of the System s(1,m[sub(1)]),...,m[sub(l)] and Null Focus Pseudo-Quantity
	5.4 Polynomials in Coefficients of Differential Systems that Have Weight Isobarity (h,g)
	5.5 Comments to Chapter Five
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,m[sub(1)]),...,m[sub(2)]
	6.1 Applications of Generating Functions and Hilbert Series to the Center and Focus Problem for the Differential System s(1,2)
	6.2 Type of Generalized Focus Pseudo-Quantities for the Differential System s(1; 3)
	6.3 On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities, that Take Part in Solving the Center and Focus Problem for the Differential System s(1; 3)
	6.4 The Differential System s(1, 4) and Algebraically Independent Generalized Focus Pseudo-Quantities
	6.5 On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities for the Differential System s(1,5)
	6.6 Comitants that Have Generalized Focus Pseudo-Quantities of the Systems(1, 2, 3) as Coefficients, and Their Sibirsky Graded Algebra
	6.7 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Differential System s(1,m[sub(1)],m[sub(l)])
	1 ` 6.8 Comments to Chapter Six
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
	7.1 Lie Operators of Representation of the Rotation Group SO(2,R) in the Space of Coefficients of Lyapunov System (5.6)
	7.2 Comitants of System (7:1)  (7:2) for the Rotation Group and Concept of Functional Basis
	7.3 General Formulas that Interconnect Coefficients of Comitants of the Lyapunov System s(1,m[sub(1)] ,....,m[sub(l)] ) Among Themselves with Respect to the Rotation Group
	7.4 On the Invariance of Focus Quantities in the Center and Focus Problem with Respect to the Rotation Group
	7.5 On the Upper Bound of the Number of Algebraically Indepen-dent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Lyapunov System sL(1,m[sub(1)] ,....,m[sub(l)] )
	7.6 Comments to Chapter Seven
Appendix 1
Appendix 2
Appendix 3
Appendix 4
Appendix 5
Appendix 6
Appendix 7
Appendix 8
Appendix 9
Appendix 10
Bibliography
Index