When does $\sum_{k=0}^M a_k x^{2k}$, $a_k \in \mathbb{R}$, $M \in \mathbb{N}$ have no roots in $[0,1]$? There is nothing special about the $1$, the question can be generalized to $[0,c]$ but that would add potential unnecessary complications.
Obviously since if $a_0 = 0$ we always have a root at $x=0$ that is a necessary condition but it is also obviously not sufficient.
A really "overpowered" sufficient condition can easily be found, such as $|a_0| - |\sum_{k\neq0} a_k| > 0$ but this is obviously not necessary.
I'm looking ideally for necessary and sufficient conditions but also realize this may be a difficult question, so any better ideas for necessary (but not sufficient) or sufficient (but not necessary) conditions on the $a_k$ would also be of interest to me.