$A$ is a Koszul algebra. Its Hilbert series $h(z)=1+az+bz^2$. Prove that $h(z)$ has real roots.
I know that $A=A_0\oplus A_1\oplus A_2$ and $\dim A_1=a$, $\dim A_2=b$. And it's needed to prove that $a^2\geqslant 4b$. But don't know how to prove it. Thanks in advance!
This is proved, for example, in Theorem $3.1$ on page $207$ in the paper Koszul property for points in projective spaces. There it is proved that for the Hilbert series $1+nz+mz^2$ we have Koszulness if and only if $m\le \frac{n^2}{4}$. In the proof it is used that the coefficients of the series $\frac{1}{h(-z)}$ are positive in this case.