Hypersurfaces of bounded Cohen-Macaulay type

Let R = k[[x₀,...,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x₀,...,x_d]]. We investigate the question of which rings of this form have bounded Cohen-Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen-Macaulay modules. As with finite Cohen-Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen-Macaulay type if and only if R≅k[[x₀,...,x_d]]/ (g+x₂² + ⋯ + x_d²), where g ∈ k[[x₀, x₁]] and k[[x₀, x₁]]/(g) has bounded Cohen-Macaulay type. We determine which rings of the form k[[x₀, x ₁]]/(g) have bounded Cohen-Macaulay type.