Memoirs of the American Mathematical Society 2002; 204 pp; softcover Volume: 156 ISBN10: 0821827812 ISBN13: 9780821827819 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/156/744
 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 semiinvariant of the first kind (with respect to changes of the dependent variable) and a semiinvariant 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 LaguerreForsyth 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 LaguerreForsyth 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 selfcontained 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  Introduction
 Some problems of historical importance
 Illustrations for some results in Chapters 1 and 2
 \(\boldsymbol{L}_n\) and \(\boldsymbol{I}_{n,\boldsymbol{i}}\) as semiinvariants of the first kind
 \(\boldsymbol{V}_n\) and \(\boldsymbol{J}_{n,\boldsymbol{i}}\) as semiinvariants of the second kind
 The coefficients of transformed equations
 Formulas that involve \(L_{n}(z)\) or \(I_{n,n}(z)\)
 Formulas that involve \(V_{n}(z)\) or \(J_{n,n}(z)\)
 Verification of \(\boldsymbol{I}_{n,n} \equiv \boldsymbol{J}_{n,n}\) and various observations
 The local constructions of earlier research
 Relations for \(\boldsymbol{G}_{i}\), \(\boldsymbol{H}_{i}\), and \(\boldsymbol{L}_{i}\) that yield equivalent formulas for basic relative invariants
 Realvalued functions of a real variable
 A constructive method for imposing conditions on LaguerreForsyth canonical forms
 Additional formulas for \(\boldsymbol{K}_{i,j}\), \(\boldsymbol{U}_{i,j}\), \(\boldsymbol{A}_{i,j}\), \(\boldsymbol{D}_{i,j}, \dots\)
 Three canonical forms are now available
 Interesting problems that require further study
 Appendix A. Results needed for selfcontainment
 Appendix B. Related studies for a class of nonlinear equations
 Appendix C. Polynomials that are linear in a key variable
 Appendix D. Rational semiinvariants and relative invariants
 Appendix E. Generating additional relative invariants
 Bibliography
 Index
