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Full Description
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.
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 semi-invariants of the first kind $\boldsymbol{V}_n$ and $\boldsymbol{J}_{n,\boldsymbol{i}}$ as semi-invariants 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 Real-valued functions of a real variable A constructive method for imposing conditions on Laguerre-Forsyth 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 self-containment Appendix B. Related studies for a class of nonlinear equations Appendix C. Polynomials that are linear in a key variable Appendix D. Rational semi-invariants and relative invariants Appendix E. Generating additional relative invariants Bibliography Index.
