Memoirs of the American Mathematical Society 2007; 365 pp; softcover Volume: 190 ISBN10: 0821839918 ISBN13: 9780821839911 List Price: US$108 Individual Members: US$64.80 Institutional Members: US$86.40 Order Code: MEMO/190/888
 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 123year 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 \(n\)thdegree 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 LaguerreForsyth canonical forms
Part 2. The Basic Relative Invariants for \(Q_{m} = 0\) when \(m\geq 2\)  Formulas that involve \(L_{i,j}(z)\)
 Basic semiinvariants of the first kind for \(m \geq 2\)
 Formulas that involve \(V_{i,j}(z)\)
 Basic semiinvariants of the second kind for \(m \geq 2\)
 The existence of basic relative invariants
 The uniqueness of basic relative invariants
 Realvalued 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 Fanotype 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 semiinvariants 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 \(n \geq 2\)
 LaguerreForsyth 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
