CHAPTER 1
Introduction
We have discovered new constructions of basic relative invariants that are far
superior to the best of those previously published. Before presenting them in terms
of precise definitions, we shall first indicate why a radically different approach has
been needed.
1.1. General summary
After relative invariants of weights 3 and 4 were obtained by E. Laguerre and
G.-H. Halphen in [39, 40, 29, 30] during the years 1879-1884 and related studies
had been made in [25, 8, 1, 2, 26, 27, 28], the evidence suggested that: for each
integer ra 3, there are ra 2 algebraically independent relative invariants of re-
spective weights 3, ..., ra for homogeneous linear differential equations of order ra.
The main problem has been to prove the correctness of this statement by explicitly
constructing such relative invariants. A thoroughly satisfactory solution is provided
by the polynomials Xmj3, ..., Xm5m over the field Q of rational numbers that we
shall present in Section 1.4. In fact, for any integers ra, n satisfying 3 n ra,
we shall introduce two polynomials /
n ? n
and Jn,n over Q that depend on both n
and ra. Our proof that In^n and Jn,n are relative invariants of weight n for homo-
geneous linear differential equations of order m will consist in showing that: In^n
is a semi-invariant of the first kind, Jn,n is a semi-invariant of the second kind,
and J
n? n
= J
n
,
n
. Then, as a convenience, we set Xm?n = In,ni for 3 n ra.
The general idea for this argument was first advanced in [16] of 1993. But the
development of our remarkably simple formulas for In^n and J
n
,
n
required a com-
plete break with earlier investigations. The procedure is as explicit and satisfactory
as the construction of analogous algebraic invariants stemming from the work of
G. Boole, A. Cayley, and J. J. Sylvester. Research on relative invariants can now
focus on other aspects of the subject. To encourage new activity, several unsolved
problems of considerable interest are presented in Chapter 16 on pages 130-132 and
in Section E.4 on pages 195-196. The relative invariants of Section 1.4 should have
numerous applications in other areas of research. The need for instructions about
their use has influenced the style of this memoir. Historical perspective is also
required to bridge the gap between previous advancements and current activity.
Because few persons currently recognize the inadequacy of prior efforts to discover
a general construction for basic relative invariants, we shall summarize next the
various deficiencies of the earlier procedures.
To evaluate a basic relative invariant of weight n at a given homogeneous linear
differential equation of order ra when 3 n ra, it became a standard recommen-
dation in [52, 7, 62, 55, 57, 44, 45, 61, 54] and elsewhere that: one should
first transform the differential equation to a Laguerre-Forsyth canonical form (by
solving a corresponding second-order differential equation) and then apply explicit
l
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