Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems

Abstract

Summary. This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.

Computing power series solutions of a nonlinear PDE system

“…We refer the reader to [12] for an expanded tutorial presentation of this material that is fully consistent with the present notations and definitions. In particular we shall denote ÂΣÃ the radical differential ideal generated by Σ.…”

“…Then, there are algorithms [2,3,10,4,27,12] that computes for any pair (Σ, H) of finite sets of differential polynomials in R ÂY Ã a finite set of regular differential systems (A1, H1), . .…”

Algebraic and Differential Invariants

“…Complete lecture notes [25,26] are available for a detailed understanding of the concepts involved and the algorithms underlying the diffalg library in Maple. A representative sample of the original articles is [6,7,24,38,41,55,56] All these address systems of classical differential equations, i.e.…”

“…The Maple library diffalg was extended to handle this new type of differential polynomial ring [5,27]. The concepts and algorithms underlying it are described in great details in [25,26].…”

Differential Invariants of Conformal and Projective Surfaces

“…Thus, there are two basic fourth order differential invariants: Let P 1 , P 2 , P 3 be the differential polynomials obtained from ∆ 6 , ∆ 9 , ∆ 15 after substitution of κ and τ : To obtain τ and σ we proceed with a differential elimination [4,12,13,14] on {P 1 , P 2 , P 3 }. We use a ranking where ψ < φ < ψ 01 < φ 01 < ψ 10 < φ 10 < ψ 02 < ψ 11 < φ 11 < φ 20 < · · · · · · < τ < σ < τ 01 < σ 01 < τ 10 < σ 10 < τ 02 < σ 02 < τ 11 < σ 11 < τ 20 < σ 20 < · · · .…”