Prove this:
For every $z\in\mathbb{C}$ with nonzero imaginary part, there exists a polynomial $p(x)=a_0+a_1x+a_2x^2+\cdots + a_nx^n$ such that $a_k>0$ for every nonnegative $k\leqslant n$, with $p(z)=0$.
For instance, $i$ is a zero of $1+x+x^2+x^3.$
I could reduce the problem to $z=a+i.$ That is, the statement is proved if the following can be shown:
For every $a\in\mathbb{R},$ there exists a polynomial $p(x)=a_0+a_1x+a_2x^2+\cdots + a_nx^n$ such that $a_k>0$ for every nonnegative $k\leqslant n$, with $p(a+i)=0$.
But I could not show this. Perhaps, the general statement is easier to prove! Any help will be appreciated.
Best Answer
We start from the simple observation: if $a< 0$, then $$ p(z) = z^2 -2az+a^2+1 $$ satisfies the condition. If $a\geq 0$, we can find $n\in \mathbb{N}$ such that $\Re[(a+i)^n]< 0.$ Let $b = (a+i)^n$. Then, $$ (z^{2n} -2\Re[b]z^n +|b|^2)(z^n+z^{n-1} +\cdots +1) $$ satisfies the condition.