I'm reading Shafarevich's Basic Algebraic Geometry in an attempt to understand more of the algebraic perspective of projective varieties. I'm confused by this problem:
(Ex 10 Section 4.4 Shafarevich Basic Algebraic Geometry 1)
Prove that the Veronese image $v_m(\mathbb{P}^n)⊂\mathbb{P}^N$ is not contained in any linear subspace of $\mathbb{P}^N$.
My attempt: A linear subspace of $\mathbb{P}^N$ is the image under the standard projection $\pi: k^{N+1} \to \mathbb{P}^N$ of a linear subspace of $k^{N+1}$. Here, it suffices to prove this for hyperplanes. Unwinding definitions, the goal is to show that homogenous degree $m$ monomials are all linearly independent. In other words, there is no nontrivial homogenous degree $m$ polynomial which vanishes everywhere on $\mathbb{P}^n$.
Over $k=\mathbb{C}$, this is easy, since one can extract the coefficients of a polynomial by taking suitable derivatives and evaluating at $0$. However, this obviously can't work in general, and it's not really in the spirit of my current learning objectives.
Over an infinite field, it seems like one can argue by counting roots. It's enough to just do this over an affine chart, and fix $\beta\in k^n$. Then $p(\beta,x_n)$ has too many roots to be a nonzero polynomial, and so this determines some linear constraints on the coefficients of $p$. It seems obvious that if there are a lot of choices of $\beta$ then this would force the answer, but I'm also not sure how to effectively choose $\beta$. In particular, if the entries of $\beta$ are all $0$ or $1$, then there are finitely many constraints for a result which supposedly holds for all $m$.
I think somehow I'm missing a fundamental point about what linear independence for polynomials over positive characteristic actually means. If $k=\mathbb{F}_2$, then the image of $v_3(\mathbb{P}^1)$ is all points of the form $[x^3,x^2y,xy^2,y^3]$ for $x,y$ not both $0$, but $x^2y+xy^2$ is a homogenous polynomial that evaluates to $0$ at every point in $\mathbb{F}_2^2$! Doesn't that mean that the image of this Veronese map is contained in a hyperplane?
PS: I have seen this virtually identical question already but my specific question is how to show the linear independence (and whether or not there is some extra restriction on the field $k$ that I'm missing).