## polynomial of two variables proof

Hi! Nice to be here! I have a theorem from the article: Local and global extrema for functions of several variables:

Theorem: Theorem Let $p(x,y)$ be a cubic in two variables. Suppose $p$ has a local minimum at $0$. Then $p$ has other critical points.

Proof: Proof: there are no repeated roots in $p_{3}$ the cubic term, theorem 3 gives the result. Thus we let $p_{3}$ have double or triple root, and after a linear transformation we have either:

$(i) \quad f(x,y)=x^{2}y+ax^{2}+2bxy+cy^{2}$
$(ii) \quad f(x,y)=x^{3}+ax^{2}+2bxy+cy^{2}$

I understand the remainder of the proof...

I have a question for the bold part. How received these two polynomials?? I would ask for launching.