# Basic global relative invariants for homogeneous linear differential
equations

### About this Title

**Roger Chalkley**

Publication: Memoirs of the American Mathematical Society

Publication Year
2002: Volume 156, Number 744

ISBNs: 978-0-8218-2781-9 (print); 978-1-4704-0337-9 (online)

DOI: http://dx.doi.org/10.1090/memo/0744

MathSciNet review: 1880800

MSC (2000): Primary 34A30; Secondary 12H20, 34A25, 34M99, 35G05

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Given any fixed integer $m \ge 3$, we present simple formulas
for $m - 2$ algebraically independent polynomials over
$\mathbb{Q}$ having the remarkable property, with respect to
transformations of homogeneous linear differential equations of order
$m$, that each polynomial is both a semi-invariant of the first kind
(with respect to changes of the dependent variable) and a semi-invariant of
the second kind (with respect to changes of the independent variable). These
relative invariants are suitable for global studies in several different
contexts and do not require Laguerre-Forsyth reductions for their evaluation.
In contrast, all of the general formulas for basic relative invariants that
have been proposed by other researchers during the last 113 years are merely
local ones that are either much too complicated or require a Laguerre-Forsyth
reduction for each evaluation. Unlike numerous studies of relative
invariants from 1888 onward, our global approach completely avoids
infinitesimal transformations and the compromised rigor associated with them.
This memoir has been made completely self-contained in that the proofs for all
of its main results are independent of earlier papers on relative invariants.
In particular, rigorous proofs are included for several basic assertions from
the 1880's that have previously been based on incomplete arguments.

Readership

Graduate students and research mathematicians interested in
ordinary differential equations.

### Table of Contents

**Chapters**

- 1. Introduction
- 2. Some problems of historical importance
- 3. Illustrations for some results in Chapters 1 and 2
- 4. $L_n$ and $I_{n,i}$ as semi-invariants of the first kind
- 5. $V_n$ and $J_{n,i}$ as semi-invariants of the second kind
- 6. The coefficients of transformed equations
- 7. Formulas that involve $L_n(z)$ or $I_{n,n}(z)$
- 8. Formulas that involve $V_n(z)$ or $J_{n,n}(z)$
- 9. Verification of $I_{n,n} \equiv J_{n,n}$ and various observations
- 10. The local constructions of earlier research
- 11. Relations for $G_i$, $H_i$, and $L_i$ that yield equivalent formulas
for basic relative invariants
- 12. Real-valued functions of a real variable
- 13. A constructive method for imposing conditions on Laguerre-Forsyth
canonical forms
- 14. Additional formulas for $K_{i,j}$, $U_{i,j}$, $A_{i,j}$, $D_{i,j}$, …
- 15. Three canonical forms are now available
- 16. Interesting problems that require further study