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We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product method. For k-parameter problem with matrices of sizes $$n_1,\ldots ,n_k = O(n)$$ , requires deformation O(1) linear equations, while existing methods MEPs require O(n) nonlinear equations. show that theoretically finds all eigenpairs an MEP probability one. It is especially well-suit...