2
1. INTRODUCTION
formulas that A. R. Forsyth presented in [24] of 1888. This procedure will generally
yield only local results (because only local solutions can generally be found for the
second-order differential equation). Actually, Forsyth had merely advanced this
procedure in [24] as a first step toward the goal of eventually obtaining satisfactory
formulas for basic relative invariants; but meager progress caused the technique to
be reproduced in [52, 7, 62, 55, 57, 44, 45, 61, 54] as if it were a satisfactory final
result. Because many second-order differential equations do not possess convenient
solutions, there are many situations in which Forsyth's plan is unable to be effec-
tively applied. For this reason, G. Metzler in [43], F. Brioschi in [9], and Z. Husty
in [34, 35] outlined methods to carry out the computations without having to solve
any differential equations. However, their techniques are so complicated that they
received little attention by later researchers. Another major deficiency of earlier
research was caused by the building blocks used in constructions. Namely, from
Forsyth's work in [24] of 1888 until recent years, all basic relative invariants were
formulated in terms of the semi-invariants of the first kind that J. Cockle developed
during the 1860's. Those semi-invariants are unsuitable for that purpose because,
as we shall demonstrate in Example 3.1 on page 21, they do not form a decisive
set for homogeneous linear differential equations. Thus, the research on basic rel-
ative invariants from 1888 until recent years is seriously impaired. For additional
perspective about this, see Chapter 10.
Whenever the coefficients of (1.1) can be represented in a system of computer
algebra, machine computations can be employed to efficiently evaluate our formulas
at those coefficients. For this, the techniques presented in Sections 3.4-3.6 are quite
useful. There is certainly no need to consider Laguerre-Forsyth reductions or solve
differential equations. However, such reductions are essential for some investigations
of other researchers. To aid these endeavors, we include Proposition 9.7 on page 75
and the techniques of Chapter 13.
Throughout Chapters 1-11, we employ a traditional context and assume that
each function is meromorphic on some region of the complex plane. Nonetheless,
our arguments can be applied to real-valued functions of a real variable when var-
ious differentiability conditions are satisfied; e.g. see Context 12.1 on page 94 and
Theorem 12.3 on page 95. These details in Chapter 12 show that our formulas for
basic relative invariants are directly applicable to the interesting global investiga-
tions of F. Neuman in [46]. Numerous mathematicians should find this significant
because it is well known that a general method of constructing global relative invari-
ants has not previously been available. In particular, Laguerre-Forsyth reductions
merely give local results. Our formulas can also play an important role for other
current areas of research described in Chapter 16. We begin by developing precise
definitions.
1.2. The ring 7£m of polynomials to be employed throughout
Our subject is concerned with polynomials into which one can substitute the
coefficients of various linear homogeneous differential equations of order m and their
derivatives. To introduce a ring of such polynomials, let m denote a fixed positive
integer. We assume that the symbols w\J , for 1 i m and j 0, represent
algebraically independent variables over the field Q of rational numbers and we let
IZm denote the ring of polynomials in these variables over Q. We recall from [6] or
[41] that, in analogy to a derivative, a derivation ' for 7£m is a mapping P \-^ P'
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