The following situation occurs in a proof that I would like to understand: we have polynomials $F_1,\ldots, F_N$ in $k[X_1,\ldots,X_M]$, where $k$ is of characteristic zero and algebraically closed. The polynomials $F_i$ are homogeneous of positive degree. And we have $M>N$.
Now I would like to conclude that these polynomials have a common zero $\neq (0,\ldots,0)$.
(the following seems to be wrong) By Hilbert's Nullstellensatz it would be enough to show that the ideal $(F_1,\ldots,F_N)$ is proper. This should be easy (probably by looking at dimensions and using $M>N$?), but somehow I don't see a nice argument for that.