A new approach to the family of singularities Re(x+iy)ᵐ

Original · EN

Assume that m≥ 2 and let l be a nonnegative integer with l≥ m-4. We give an alternative proof of the fact that any smooth function defined locally around (0,0)∈ R² with the Taylor power series at (0,0) beginning with Re(x+iy)ᵐ+0+...+0 (l zeros) is diffeomorphically equivalent to Re(x+iy)ᵐ at (0,0). For m≥ 5 and C≠ 0 we show that the function Re(x+iy)ᵐ+C(x²+y²)ᵐ⁻² is not diffeomorphically equivalent to Re(x+iy)ᵐ at (0,0).