Mumford's book Abelian Varieties asserts that for a line bundle L on a projective variety (if necessary, you can assume it is as nice as possible), the Euler characteristic $\chi(L^k)$ of tensor powers of $L$ is a polynomial in $k$. If L is very ample, this is just the Hilbert polynomial, and this can be proven by an induction argument twisting the short exact sequence $0 \to \mathcal O(-1) \to \mathcal O \to K \to 0$. More generally, if $L$ (or $L^*$) is an ideal sheaf, the same argument should work. Why does the result still hold for arbitrary $L$?
Edit: I'd be particularly interested in an elementary proof that does not involve proving an entire Riemann-Roch theorem--Mumford is using this result to prove Riemann-Roch for abelian varieties!