Hello there I am investigating catastrophe theory although I have discovered something in a book I am reading that I dont understand

we consider a smooth function of two variables with , i.e. there is a critical point at the origin, so the Taylor series is

+ higher order terms

where

We consider the case when the hessian , and that not all the derivatives are zero

So the book says that this means is a perfect square and allows us to write

+ higher order terms

BUT!

if we do , all the conditions are satisfied but

could anyone explain? thanks very much