Malgrange preparation theorem

Malgrange preparation theorem In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by B. Malgrange (1962–1963, 1964, 1967).

Contents 1 Statement of Malgrange preparation theorem 2 Proof of Malgrange preparation theorem 3 Algebraic version of the Malgrange preparation theorem 4 References Statement of Malgrange preparation theorem Suppose that f(t,x) is a smooth complex function of t∈R and x∈Rn near the origin, and let k be the smallest integer such that {displaystyle f(0,0)=0,{partial f over partial t}(0,0)=0,dots ,{partial ^{k-1}f over partial t^{k-1}}(0,0)=0,{partial ^{k}f over partial t^{k}}(0,0)neq 0.} 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, {displaystyle f(t,x)=c(t,x)left(t^{k}+a_{k-1}(x)t^{k-1}+cdots +a_{0}(x)right)} 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 {displaystyle g=qf+r} where q and r are smooth, and as a function of t, r is a polynomial of degree less than k. This means that {displaystyle r(x)=sum _{0leq j

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