Abstract. Several basic relative invariants for homogeneous linear differential
equations were discovered during the years shortly after 1878. Also, a basic
relative invariant was found by Paul Appell in 1889 for a type of nonlinear
differential equation. There was little progress during the years 1892--1988 as
researchers who worked with homogeneous linear differential equations were
unknowingly handicapped by the standard practice of introducing binomial
coefficients in the writing of their equations. They thereby failed to develop
adequate formulas for the coefficients of equations resulting from a change of
the independent variable. Consequently, for relative invariants as the most
important kind of invariant, progress was stymied.
The notation was simplified in 1989, adequate transformation formulas
were developed, and explicit expressions were deduced in 2002 for all of the
basic relative invariants of homogeneous linear differential equations. In 2007,
explicit formulas were obtained for all of the basic relative invariants of a
type of ordinary differential equation involving two parameters m and n that
represent positive integers. When n = 1 and m >= 3, the formulas specialize to
provide all of the basic relative invariants for homogeneous linear differential
equations of order m; and, when m = n = 2, they yield all three of the basic
relative invariants for the equations of Paul Appell.
A general method developed in 2014 combines two relative invariants of
weights p and q for the same type of equation to explicitly obtain a relative
invariant of weight p+q +r, for any r >= 0. With that, the principal problems
about relative invariants have now been solved.
This monograph provides clear perspective about the reformulation begun
after 1988 and recently completed. Chapters 15 and 18 show how the major
difficulties confronting earlier researchers have been overcome.