Geodesic
The application of the rank-factorization method to the analysis of spectral characteristics of a polynomial multiparameter matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIV, Tome 268 (2000), pp. 115-144

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The method of rank-factorization (the $\Delta W$-$q$ method), previously suggested by the author as a method for solving multiparameter algebraic problems for matrix $F$ polynomially dependent on parameters, is applied to analyze the finite spectrum of the matrix $F$. Special attention is paid to the part of the spectrum $\sigma[F]$ of the $q$-parameter matrix $F$ whose points are independent of at least one of the spectral parameters. 
@article{ZNSL_2000_268_a7,
     author = {V. N. Kublanovskaya},
     title = {The application of the rank-factorization method to the analysis of spectral characteristics of a polynomial multiparameter matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {115--144},
     publisher = {mathdoc},
     volume = {268},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_268_a7/}
}
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V. N. Kublanovskaya. The application of the rank-factorization method to the analysis of spectral characteristics of a polynomial multiparameter matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIV, Tome 268 (2000), pp. 115-144. http://geodesic.mathdoc.fr/item/ZNSL_2000_268_a7/