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

• 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

• 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

• 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

• 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 $m \geq2$

• The class of equations specified by $y''(z)$$y'(z)$
• Formulations of greater generality
• Invariants for simple equations unlike (29.1)
• Bibliography
• Index