I've been thinking about this problem for a while, and could use a hint!
Let $f_1, \dots, f_m$ be positive degree homogeneous polynomials in $k[x_1, \dots, x_n]$. Prove that if $n > m$, then $f_i$ share a non-trivial common root in the algebraic closure: there is a point $c := (c_1, \dots, c_n) \in \overline{k}^{\oplus n}$ such that $f_i(c) = 0$ for each $i$.