Let K be a field and let f ∈ K[[x1, x2,. . . , xr]] and g ∈ K[[y1, y2, . . . , ys]] be non-zero and non-invertible elements. If X (resp. Y) is a matrix factorization of f (resp. g), then we can construct the matrix factorization X ⊗ Y of f + g over K[[x1, x2, . . . , xr, y1, y2, . . . , ys]], which we call the tensor product of X and Y. After showing several general properties of tensor products, we will prove theorems which give bounds for the number of indecomposable components in the direct decomposition of X ⊗ Y.
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