Elimination and Counting via Thomas Decomposition

Thomas分解是基于J. M. Thomas在20世纪30年代提出的正式三角测量算法，现在第一次实现了它的全面普遍性，它将方程和不等式系统的一组解分解成与所谓的简单系统相对应的不相交的子集。系统可以是多项式的，也可以是多项式微分的。在多项式的情况下，我们得到了解的计数多项式，它稍微依赖于所选择的坐标系。这里的挑战是研究这个多项式的泛型和非泛型坐标（其中将使用代数群表示理论的方法），从Thomas分解中提取更精细的组合信息，并通过对分解的结构洞察来改进实现。在微分情况下，目的是通过代数情况和Janet的线性偏微分方程方法的共同推广来定义自由泰勒系数的计数多项式。我们也期望应用于求解非线性偏微分方程的数值方法的代数分析。

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.