# On property of monic polynomial with integer coefficients

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 $\partial f$)?

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 $f(p/q)=1/q^4$.