For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have
$$
\textrm{inf}(f(x)) > 0 \implies 
\textrm{inf}(f(x)) \geq \frac{3}{4} .
$$
Could we generalize this (for common $\partial f$)?



On property of monic polynomial with integer coefficients
No. If $p,q$ satisfy Pell's equation $p^2-2q^2=1$, then minimum of $f(x)=(x^2-2)^2+(qx-p)^2$ is at most $f(p/q)=1/q^4$.


