Basic global relative invariants for homogeneous linear differential equations (Q2783392)

Let \(F\) be a differential field of characteristic zero with differentiation \(\theta\), \(y\) be a differential indeterminate and NEWLINE\[NEWLINEL_{\theta}(y)\equiv\theta^{m}y + a_{1}\theta^{m-1}y +\cdots+ a_{m}y \tag{1}NEWLINE\]NEWLINE be a homogeneous linear differential operator over \(F\). For any \(\lambda \in F^{*}\) put \(u=\lambda^{-1}y\) then \(u\) is also indeterminate over \(F\) and the substitution \(y=\lambda u\) in (1) gives NEWLINE\[NEWLINE\lambda^{-1}L_{\theta}(\lambda u)=L^{*}_{\theta}(u)\equiv\theta^{m}u + a_{1}^{*}\theta^{m-1}u +\cdots+ a_{m}^{*}u.NEWLINE\]NEWLINE So, we get a transformation \(\tau\) on \(F^{m} mapping a \mapsto a^{*}\). \(\tau(a)=a^{*}\) and \(\tau\) is a regular differentially algebraic mapping over \(F\). Analogously, for any \(\alpha \in E^{*}\) (some extension of \(F\)) put \(\delta=\alpha^{-1}\theta\) then \(\delta\) is also a differentiation of \(E\) and the substitution (2) \(\theta=\alpha\delta\) in (1) gives NEWLINE\[NEWLINE\alpha^{-1}L_{\alpha\delta}(y)=L^{**}_{\delta}(y)\equiv\delta^{m}y + a_{1}^{**}\delta^{m-1}y +\cdots+ a_{m}^{**}yNEWLINE\]NEWLINE and the transformation \(\tau_{\alpha}\) on \(E^{m}\) maps \( a \mapsto a^{**}\), \(\tau_{\alpha}(a)=a^{**}\) and \(\tau_{\alpha}\) is a regular differentially algebraic mapping over \(E\). Let \(Q\) be a field of rational numbers and \(Q(w)=Q(w_1,\dots,w_m)\) be a ring of differential polynomials in the differential indeterminates \(w_{1}, \dots , w_{m}\). A differential polynomial \(P \in Q(w)\) is called relative (or semi-)invariant if \(P(\tau(w))=P(w)\) and \(P(\tau_{\alpha}(w))=\alpha^{q}P(w)\) (for some \(q \in \mathbb{N}\)). NEWLINENEWLINENEWLINEThe author presents a full solution of the problem to find all relative invariants in \(Q(W)\). He uses a classical technique and discovers new constructions of basic relative invariants that are far superior to the best of those previously published. This memoir is entirely self-contained in regard to the proofs of its main results. It contains a short historical review of the problem and relative references. MATHEMATICA is used for computing the invariants. A connection with so-called decisive sets and algebraic dependencies between zeros of (1) and canonical forms of (1) for the transformations \(\tau\) and \(\tau_{\alpha}\) are discussed. NEWLINENEWLINENEWLINEWe note that substitution (2) is too general. For example, any operator (1) of second order can be reduced to operator \(\theta^{2}y\) by means of (2). It is clear such transformation is useless for operators of second order. It is more naturally to demand that the Galois group of \(L^{**}_{\delta}(y)\) be isomorphic to the Galois group of \(L_{\theta}(y)\) or to suppose in (2) that \(\alpha \in F^{*}\). This last approach was used by \textit{U. D. Bekbaev} [Yusupdjan Khakimdjanov (ed.) et al., Algebra and operator theory. Proceedings of the colloquium, Tashkent, Uzbekistan, September 29--October 5, 1997. Dordrecht: Kluwer Academic Publishers, 145-156 (1998; Zbl 0931.12010)].