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 $\partial f$)?
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$\begingroup$ Bounds depending of degree and height of f can be found in wwwirma.ustrasbg.fr/~bugeaud/travaux/Polpos1.pdf $\endgroup$– dujeMay 29, 2016 at 8:05
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No. If $p,q$ satisfy Pell's equation $p^22q^2=1$, then minimum of $f(x)=(x^22)^2+(qxp)^2$ is at most $f(p/q)=1/q^4$.