Abstract:

The result of Whitney on even functions gives the canonical form of invariant C∞ function germs under Z₂ group {±1} in one variable: If f∈E₁ and f(-x)=f(x),then there exists h∈E₁ such that f(x)=h(x²).In this paper, by means of Malgrange preparation theorem and the related computation, the authors obtain the canonical from of invariant C∞ function germs under group {±I_n} in Rⁿ at origin.

Key words: Malgrange preparation theorem, Z₂ invariant, canonical form.