Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal F}(n)=\chi(\mathcal F \otimes L^n).$ Is there any way to prove that $P$ is polynomial not using Riemann-Roch and Chern classes. I think that there should be some argument similar to mdeland's argument(https://mathoverflow.net/questions/889/why-is-the-euler-characteristic-of-powers-of-a-line-bundle-a-polynomial-in-the-p), which works for locally free $\mathcal F$.