Elimination and Counting via Thomas Decomposition

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The Thomas decomposition based on a formal triangulation algorithm by J. M. Thomas in the 1930s, now implemented in its full generality for the first time, decomposes the set of solutions of a system of equations and inequations into disjoint subsets corresponding to so called simple systems. The system might be either polynomial or polynomial differential. In the polynomial case one obtains a counting polynomial for the solutions, which depends slightly on the chosen coordinate system. Here the challenge is to study this polynomial for generic and nongeneric coordinates (where methods from representation theory of algebraic groups are to be used), to extract more refined combinatorial information from the Thomas decomposition, and to improve the implementation by getting structural insight into the decomposition. In the differential case the aim is to define counting polynomials for free Taylor coefficients via a common generalization of the algebraic case and Janet’s approach to linear PDEs. We also expect applications to the algebraic analysis of numerical methods for solving nonlinear PDEs.