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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) $\chi(L^k)$ of tensor powers of L $L$ is a polynomial in k. $k$. 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 $0 \to \mathcal O(-1) \to \mathcal O \to K \to 00$. More generally, if L $L$ (or L*) $L^*$) is an ideal sheaf, the same argument should work. Why does the result still hold for arbitrary L?$L$?

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!
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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 O(-1) \to 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!
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Why is the Euler characteristic of powers of a line bundle a polynomial in the power?

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 O(-1) \to 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?