Thomas decomposition and its applications

Abstract

Nowadays, triangular decomposition algorithms and software implementing them have become powerful tools for symbolic treatment of systems of multivariate polynomial and (non-linear) partial differential equations. In our ISSAC 2010 poster, we present the algebraic and differential Thomas decomposition and applications of our new Maple implementation [BLH10]. Algebraic Thomas DecompositionAmong the various triangular decompositions the Thomas one stands by itself. It was suggested by the American mathematician J.M.Thomas [Tho37, Tho62] and decomposes a finite system of polynomial equations and inequations into finitely many triangular subsystems that he called simple. The work was continued by Wang [Wan98, Wan01] and Gerdt [Ger08]. Unlike other decomposition algorithms it yields a disjoint decomposition of the zeroes. Every simple system is a regular chain, it defines a characterizable radical ideal and reduction decides membership of polynomials in this ideal. CountingThe disjointness of the Thomas decomposition and the properties of simple systems allow constructing a polynomial -the counting polynomial -which counts the (possibly infinite) set of solutions of a polynomial system via iterated fibrations of projections [Ple09]. Unlike the Hilbert polynomial, the counting polynomial disregards multiplicities of solutions. Differential Thomas DecompositionThe differential Thomas decomposition is concerned with manipulations of polynomial differential equations and inequations. It makes systems simple by treating them as algebraic systems for the differential variables. Moreover, the systems are completed to involution in the sense of Janet [Jan29] to find all integrability conditions. As above, the disjointness allows counting of solutions. Furthermore, a suitable ranking enables the solving of differential elimination problems. Algorithmic AspectsUnlike other triangular decompositions, the Thomas decomposition requires disjointness of solutions and several restrictions regarding the inequations. This forces us to treat inequations as an