Basic global relative invariants for nonlinear differential equations

Abstract

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. Generalization 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.