I am right now battling with algebraic geometry and can't wrap my head around one problem.
Problem: Assume that the field $K$ is the field of complex numbers. Let $V \in \mathbb{A}^n_K$ be an affine variety, let $f \in K[V]$. Suppose that for all $P \in V$ we have that $f(P)\neq 0$. If $x_1, ..., x_n$ denote the coordinate functions on $V$, prove that there exists a unique $g \in K[V]$ such that there is an equality
$$(f \cdot g) = \sum^n_{m = 1} (-1)^m \cdot \frac{m!}{(2m)!}\cdot x_m$$
in $K[V]$. You should use the Nullstellensatz in your solution.
Unfortunately, I have no idea even where to start with this and would appreciate any help. Thank you.
