In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems, which is a key relation for many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems. In 2009, Muhi\v{c} and Plestenjak extended the above relation to a class of singular two-parameter eigenvalue problems with coprime characteristic polynomials and such that all finite eigenvalues are algebraically simple. They introduced a way to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems. Using new tools, in particular the stratification theory, we extend this connection to singular two-parameter eigenvalue problems with possibly multiple eigenvalues and such that characteristic polynomials can have a nontrivial common factor.

翻译：1960年代, Atkinson 引入了多参数电子价值问题的抽象代数设置。 他显示,非单数多参数电子价值问题相当于通用电子价值问题的相关系统,这是许多非单数多参数电子价值问题理论结果和数字方法的关键关系。 2009年, Muhi\v{c} 和 Plestenjak 将上述关系扩展至一类单数双参数电子价值问题,其中含有多个特性的多数值,因此所有有限的电子价值都是简单的。他们引入了一种方法,通过计算两个单一通用电子价值问题相关系统共同的正常电子价值来解决单数电子价值问题。我们使用新的工具,特别是分层理论,将这种联系扩展至单一的两参数电子价值问题,其中可能含有多个电子价值,而且特性的多数值可能具有非三维共同因素。