A formula giving the known relative invariants for homogeneous linear differential equations
Each of the known relative invariants has been found by selecting a suitable combination of particular semi-invariants of the first kind [change of the dependent variable \(y = \rho (z)v]\) in order to obtain an expression that is both a semi-invariant of the first kind and a semi-invariant of the second kind [change of the independent variable \(z = f(\zeta)]\). Because this procedure has been severely limited by computational difficulties, we use results from a previous paper [J. Differ. Equations 80, No. 1, 107-153 (1989; Zbl 0693.34005)] to significantly improve the situation. Our new procedure enables us to verify that, for any integers \(m,s\) satisfying \(3 \leq s \leq 10\) and \(s \leq m\), a certain polynomial given in this paper is a relative invariant of weights \(s\) for homogeneous linear differential equations of order \(m\). This result yields all of the basic relative invariants previously known, and it yields new relative invariants for \(s = 8 \leq m\), \(s = 9 \leq m\), and \(s = 10 \leq m\).
scientific article; zbMATH DE number 147083