# Upper bound at the smallest root (smallest norm) of a complex polynomial

Leave $$N$$ be a large integer and consider the equation:

$$x ^ {N} + a_ {N-1} x ^ {N-1} + … + a_ {1} x + O ( frac {1} {N}) = 0,$$
where $$O (h)$$ is by definition a term such that $$lim_ {h a 0} O (h) / h = 0$$. Suppose that all the coefficients $$a_j$$ have a great standard: $$|| a_j || approx N ^ {j-1}$$ except $$a_ {1}$$ which is asymptotically a constant other than zero.

Leave $$x_s$$ Be a complex root of the previous polynomial with the smallest standard. I want to show something similar to $$|| x_s || approx O ( frac {1} {N})$$ or any upper limit as $$|| x_s || leq frac {1} {N}$$ or even smaller than that.
It is obvious that if we do not have the term $$O ( frac {1} {N})$$ then the root of the smallest polynomial norm
$$x ^ {N} + a_ {N-1} x ^ {N-1} + … + a_ {1} x = 0$$ is $$x_s = 0$$. So intuitively it makes sense to say that for the first polynomial $$|| x_s ||$$ It is also small, but I can not show how small it is.

I do not expect someone to give me the exact solution to this problem because I understand that we need more details, but let me know how you approach these types of questions.