I started to read about Projective spaces.
Homogenous polynomials work perfectly. But the natural question that I asked here is: Are there any non homogenous polynomials for which $(x_0, \cdots ,x_n)$ is a root IFF $(\lambda x_0 , \cdots \lambda x_n)$ is a root for all non zero $\lambda$. In other words can $\mathcal{V}(f)$ be a $\textbf{projective variety}$?
An immediate obesrvation in this direction is that for such a polynomial $f$ all its roots must also be roots of its homogenous components. This is true regardless of algebraic closure. This also generalises to saying that every projective variety the variety of an ideal generated by homogenous polynomials. Obviously this variety will contain non homogenous elements as well but even the variety cut out will be a collection of lines for any ideal that is generated by homogenous polynomials.
But coming back to our original question, does there exist a non homogenous $f$ for which $\mathcal{V}(f)$ is union of lines? We have a necessary condition that is if $f = f_1 + f_2 +\cdots f_n$ then every root of $f$ must be a root of each $f_i$. So, we don't want a $(x_0, \cdots x_n)$ such that $f_i(x_0, \cdots x_n) \neq 0$ but the values cancel out and $f(x_0, \cdots x_n) = 0$.
Looking at it this gives me the idea that this is definitely possible in the $\mathbb{R}$ case, in any place where there is a field order. I have constructed a simple example:
$$f = (x+y)(x^{98}+1)x$$
I started with $f = (x+y)x^{100}+(x+y)x^2$. The two pieces cannot cancel. Hence, $f$ vanishes iff $f_1$ and $f_2$ vanish and hence $f$ vanishes on lines since $f_1$ and $f_2$ vanish on lines.
So, there are exist non homogenous polynomials $f$ with a zero set, which is a projective variety.
But the real question is: Can this happen in algebraically closed field? The intuition of having zeroes, more common zeroes not having any field order says that the answer should be no!
My question is is there a non-homogenous polynomial $f$ such that $\mathcal{V}(f)$ is union of lines, in other words is it a projective variety?
I don't know how to resolve it...I tried to add some variables and use Nullstellensatz but didn't work.
Please help
