Let $N$ be a big integer number 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 $\lim_{h \to 0} O(h)/h =0$. Assume that all coefficients $a_j$ have a big norm : $|| a_j||\approx N^{j-1}$ except $a_{1}$ which is asymptotically a nonzero constant.

Let $x_s$ be a complex root of above polynomial with the smallest norm. I want to show something similar to $|| x_s || \approx O(\frac{1}{N}) $ or any upper bound like $|| x_s || \leq \frac{1}{N} $ or even smaller than that. It's obvious that if we don't have the term $O(\frac{1}{N})$ then smallest norm root of the polynomial $$ x^{N} + a_{N-1} x^{N-1} + ...+a_{1} x=0$$ is $x_s=0$. So intuitively makes sense to say that for the first polynomial $|| x_s ||$ is also small but I can't show how small it is .

I do not expect someone gives me the exact solution of this problem because I understand we need more details, but please let me know how you tackle with these kind of questions.

Let $N$ be a big integer number 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 $\lim_{h \to 0} o(h)/h =0$. Assume that all coefficients $a_j$ have a big norm : $|| a_j||\approx N^{j-1}$ except $a_{1}$ which is asymptotically a nonzero constant.

Let $x_s$ be a complex root of above polynomial with the smallest norm. I want to show something similar to $|| x_s || \approx o(\frac{1}{N}) $ or any upper bound like $|| x_s || \leq \frac{1}{N} $ or even smaller than that. It's obvious that if we don't have the term $o(\frac{1}{N})$ then smallest norm root of the polynomial $$ x^{N} + a_{N-1} x^{N-1} + ...+a_{1} x=0$$ is $x_s=0$. So intuitively makes sense to say that for the first polynomial $|| x_s ||$ is also small but I can't show how small it is .

I do not expect someone gives me the exact solution of this problem because I understand we need more details, but please let me know how you tackle with these kind of questions.