Returns a positive integer N such that $\&verbar;r\&verbar;<N$ for all complex roots r of p.  In general, this bound is better than Cauchy's bound of

$⌈1+\frac{\mathrm{maxnorm}\left(p\right)}{\left|\mathrm{lcoeff}\left(p\right)\right|}⌉$.

$p≔x\mapsto x^4-10\cdot x^2+1\&colon;$

$\mathrm{rootbound}\left(p\left(x\right)\&comma;x\right)$

$4$

$\mathrm{ceil}\left(1+\frac{\mathrm{maxnorm}\left(p\left(x\right)\right)}{\mathrm{abs}\left(\mathrm{lcoeff}\left(p\left(x\right)\right)\right)}\right)$

$11$

$\mathrm{fsolve}\left(p\left(x\right)=0\&comma;x\right)$

$\mathrm{-3.146264370},\mathrm{-0.3178372452},0.3178372452,3.146264370$

$q≔x\mapsto \left(x-3.2\right)\cdot \left(x-2.5\right)\cdot \left(x-0.3\right)\cdot \left(x+1.3\right)\cdot \left(x+2.5\right)\cdot \left(x+3.6\right)\&colon;$

$\mathrm{rootbound}\left(q\left(x\right)\&comma;x\right)$

$6$

$\mathrm{ceil}\left(1+\frac{\mathrm{maxnorm}\left(\mathrm{expand}\left(q\left(x\right)\right)\right)}{\mathrm{abs}\left(\mathrm{lcoeff}\left(q\left(x\right)\right)\right)}\right)$

$78$

Monagan, M.B. "A Heuristic Irreducibility Test for Univariate Polynomials." J. of Symbolic Comp. Vol. 13 No. 1. Academic Press, (1992): 47-57.