标题： 关于参数线性方程组求解 显示英文标题

标题： On Parametric Linear System Solving

摘要： 参数线性系统是方程组的线性系统，其中某些符号参数出现，即这些符号在分析问题的过程中不被视为消去或求解的候选者。 在本工作中，我们假设符号参数以多项式形式出现在系统的系数中，并且要解的变量仅为该线性系统的变量。 系统的相容性和解的表达可能取决于参数的值。 众所周知，可以指定一个覆盖集的区域，每个区域都是参数上的半代数条件以及在该条件下有效的解描述。 我们提供了一种求解方法，当有最多三个参数时，该方法所需的时间在矩阵维度和多项式的次数上是多项式的。 在以前的方法中，所需的区域数量在系统维度和参数的多项式次数上是指数级的。 我们的方法利用了Hermite和Smith标准形式，当系统系数域映射到适当构造的域上的单变量多项式域时，可以计算这些形式。 我们的方法有效地识别了内在奇点和分支点，在这些点上矩阵的代数和几何结构发生变化。 显示英文摘要

摘要： Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. In this work we assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. The consistency of the system and expression of the solutions may vary depending on the values of the parameters. It is well-known that it is possible to specify a covering set of regimes, each of which is a semi-algebraic condition on the parameters together with a solution description valid under that condition. We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. In previous methods the number of regimes needed is exponential in the system dimension and polynomial degree of the parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our approach effectively identifies intrinsic singularities and ramification points where the algebraic and geometric structure of the matrix changes.