Completion of Linear Differential Systems to Involution
Vladimir P. Gerdt
TL;DR
This work extends involutive completion methods from polynomial ideals to differential ideals generated by linear polynomials, providing a rigorous framework for linear involutive bases via rankings and constructive, noetherian divisions. It introduces the MinimalLinearInvolutiveBasis algorithm, proving correctness and termination and showing that the output is a minimal involutive differential basis under orderly rankings; starting from a Gröbner basis, the method augments it with irreducible nonmultiplicative prolongations. The paper also develops explicit Hilbert-function and Hilbert-polynomial formulas $HF_{[F]}(s)$, $HP_{[F]}(s)$ for the differential ideal, and demonstrates two key applications: (i) posing initial-value problems with guaranteed uniqueness for linear systems and (ii) enabling Lie symmetry analysis of nonlinear PDEs via an involutive determining system, which yields the structure and size of symmetry groups. Together, these results provide a unified, algorithmic approach to analysis and integration of linear differential systems and their symmetry properties, with practical implications for PDE theory and computer algebra implementations.
Abstract
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutive algorithms. We present an algorithm for computation of a minimal involutive differential basis. Its correctness and termination hold for any constructive and noetherian involutive division. As two important applications we consider posing of an initial value problem for a linear differential system providing uniqueness of its solution and the Lie symmetry analysis of nonlinear differential equations. In particular, this allows to determine the structure of arbitrariness in general solution of linear systems and thereby to find the size of symmetry group.