Bounds on polynomial roots I am looking to find the minimum absolute value of the roots of the following polynomial:
$$ux^M - x + 1$$
where $u$ and $M$ are constants. Does a closed form upper or lower bound expression exist?
M is a positive integer.

Consider the substitution $x\mapsto1/z$ to get
$$z^M-z^{M-1}+u$$
and hence the roots $z$ are bounded above by
$$1+\max(1,|u|)$$
and your roots are bounded below by
$$\frac1{1+\max(1,|u|)}$$
