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Basic Global Relative Invariants for Nonlinear Differential Equations
Roger Chalkley, University of Cincinnati, OH
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Memoirs of the American Mathematical Society
2007; 365 pp; softcover
Volume: 190
ISBN-10: 0-8218-3991-8
ISBN-13: 978-0-8218-3991-1
List Price: US$102
Individual Members: US$61.20
Institutional Members: US$81.60
Order Code: MEMO/190/888
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The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order \(m\) was initiated in 1879 with Edmund Laguerre's success for the special case \(m = 3\). It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any \(m \geq3\), each of the \(m - 2\) basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations.

With respect to any fixed integer \(\,m \geq 1\), the author begins by explicitly specifying the basic relative invariants for the class \(\,\mathcal{C}_{m,2}\) that contains equations like \(Q_{m} = 0\) in which \(Q_{m}\) is a quadratic form in \(y(z), \, \dots, \, y^{(m)}(z)\) having meromorphic coefficients written symmetrically and the coefficient of \(\bigl( y^{(m)}(z) \bigr)^{2}\) is \(1\). Then, in terms of any fixed positive integers \(m\) and \(n\), the author explicitly specifies the basic relative invariants for the class \(\,\mathcal{C}_{m,n}\) that contains equations like \(H_{m,n} = 0\) in which \(H_{m,n}\) is an \(n\)th-degree form in \(y(z), \, \dots, \, y^{(m)}(z)\) having meromorphic coefficients written symmetrically and the coefficient of \(\bigl( y^{(m)}(z) \bigr)^{n}\) is \(1\). These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equations.

Table of Contents

Part 1. Foundations for a General Theory
  • Introduction
  • The coefficients \(c_{i,j}^{*}(z)\) of (1.3)
  • The coefficients \(c_{i,j}^{**}(\zeta)\) of (1.5)
  • Isolated results needed for completeness
  • Composite transformations and reductions
  • Related Laguerre-Forsyth canonical forms
Part 2. The Basic Relative Invariants for \(Q_{m} = 0\) when \(m\geq 2\)
  • Formulas that involve \(L_{i,j}(z)\)
  • Basic semi-invariants of the first kind for \(m \geq 2\)
  • Formulas that involve \(V_{i,j}(z)\)
  • Basic semi-invariants of the second kind for \(m \geq 2\)
  • The existence of basic relative invariants
  • The uniqueness of basic relative invariants
  • Real-valued functions of a real variable
Part 3. Supplementary Results
  • Relative invariants via basic ones for \(m \geq 2\)
  • Results about \(Q_{m}\) as a quadratic form
  • Machine computations
  • The simplest of the Fano-type problems for (1.1)
  • Paul Appell's condition of solvability for \(Q_{m} = 0\)
  • Appell's condition for \(Q_{2} = 0\) and related topics
  • Rational semi-invariants and relative invariants
Part 4. Generalizations for \(H_{m, n} = 0\)
  • Introduction to the equations \(H_{m, n} = 0\)
  • Basic relative invariants for \(H_{1,n} = 0\) when \(n \geq 2\)
  • Laguerre-Forsyth forms for \(H_{m, n} = 0\) when \(m \geq 2\)
  • Formulas for basic relative invariants when \(m \geq 2\)
  • Extensions of Chapter 7 to \(H_{m,n} = 0\), when \(m \geq 2\)
  • Extensions of Chapter 9 to \(H_{m,n} = 0\), when \(m \geq 2\)
  • Basic relative invariants for \(H_{m, n} = 0\) when \(m \geq2\)
Additional Classes of Equations
  • The class of equations specified by \(y''(z)\)\(y'(z)\)
  • Formulations of greater generality
  • Invariants for simple equations unlike (29.1)
  • Bibliography
  • Index
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