All Questions

Consider a polynomial $P_n(x)\in\mathbb{R}[x]$, of degree $n\geq1$, of the form $$P_n(x)=c_0+c_1x+c_2x^2+\cdots+c_{n-1}x^{n-1}+x^n.$$ To illustrate the question, take $P_1(x)=c_0+x$ so that $P_1(0)=...

This question is inspired by the MO query here, although it has no direct implications. Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and ...

Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$. Question. Is it true that $D^nf(z)=0$ has only real roots ...