Is the zero polynomial a homogenous polynomial?

Question

I think this is just tying up a definition for me, but the definition I have for homogeneous polynomial is that all its non-zero terms have the same degree so what happens in the case of the zero polynomial?

Answer 1

If you want the space of degree-$n$ homogeneous polynomials to be a vector space, for every $n$, which is the usual convention, then yes you must consider the zero polynomial to be homogeneous. Moreover, you must consider it to be homogeneous of every degree. This is a bit strange but acceptable as it indeed satisfies $f(ax)=a^n f(x)$ for every $n$.