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.