I am studying about the (Stability on a curve). Suppose $C$ is a smooth curve of genus g. The Riemann-Roch theorem asserts that if $E$ is a coherent sheaf on $C$ then the Euler characteristic of $E $ is $\chi(E)= deg(E)+ r(E)(1-g)$
Polarizing $C$ with $H = p$ a point, we find
Then The Hilbert polynomial $P(E,m)= \chi(E(m))$ $=deg(E(m))+r(E)(1-g)$.
My question is how can we rewrite these equation to become $r(E)m +(deg(E)+r(E)(1-g))$? Where $E(m)$ Is the twisted sheaf
