Is it true that $$p(x)\,e^{-x^2} \leq A e^{-Bx^2},$$ for any $x\in \mathbb{R}$ and some positive constants $A$ and $B$ where $p$ is any polynomial of a given degree? I guess $A$ and $B$ depend on the degree of $p$ of course.
Comment: I even think one can assume $A$ is the biggest coefficient of $p$ and here $B$ gets smaller as the degree of $p$ gets bigger.
Thanks a lot!
For any $\varepsilon>0$, $f(x)=p(x)\,e^{-\varepsilon x^2}$ is a continuous function whose limits at $\pm\infty$ are zero.
It follows that such a function has an absolute maximum over the real line and $ f(x)\leq M$ can be re-written in the wanted form (with $A=M$ and $B=1-\varepsilon$).