The Malgrange preparation theorem has two forms; one says that where the functions c and a are smooth and c is nonzero at the origin, that is a function of its polynomial degree of K; the other Mather division is a sort of addition with remainder theorem.
Then one form of the preparation theorem states that near the origin f can be written as the product of a smooth function c that is nonzero at the origin and a smooth function that as a function of t is a polynomial of degree k. In other words, where the functions c and a are smooth and c is nonzero at the origin. A second form of the theorem, occasionally called the Mather division theorem, is a sort of "division with remainder" theorem: it says that if f and k satisfy the conditions above and g is a smooth function near the origin, then we can write where q and r are smooth, and as a function of t, r is a polynomial of degree less than k. This means that 
