  We consider essential self-adjointness on the space
$C_0^{\infty}((0,\infty))$ of even order, strongly singular, homogeneous
differential operators associated with differential expressions of the type \[
\tau_{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0,
\; n \in \mathbb{N}, \; c \in \mathbb{R}, \] in $L^2((0,\infty);dx)$. While the
special case $n=1$ is classical and it is well-known that
$\tau_2(c)\big|_{C_0^\infty((0,\infty))}$ is essentially self-adjoint if and
only if $c \geq 3/4$, the case $n \in \mathbb{N}$, $n \geq 2$, is far from
obvious. In particular, it is not at all clear from the outset that \[ \text{
there exists } c_n \in \mathbb{R}, \, n \in \mathbb{N}, \text{ such that }
\tau_{2n}(c)\big|_{C_0^\infty((0,\infty))} \, \text{ is essentially
self-adjoint if and only if } c \geq c_n. \tag{*}\label{0.1} \] As one of the
principal results of this paper we indeed establish the existence of $c_n$,
satisfying $c_n \geq (4n-1)!!\big/2^{2n}$, such that property \eqref{0.1}
holds.
  In sharp contrast to the analogous lower semiboundedness question, \[ \text{
for which values of } c \, \text{\it is }
\tau_{2n}(c)\big|_{C_0^{\infty}((0,\infty))} \, \text{ bounded from below?}, \]
which permits the sharp (and explicit) answer $c \geq [(2n
-1)!!]^{2}\big/2^{2n}$, $n \in \mathbb{N}$, the answer for \eqref{0.1} is
surprisingly complex and involves various aspects of the geometry and
analytical theory of polynomials. For completeness we record explicitly, \[ c_1
= 3/4, \quad c_2= 45, \quad c_3 = 2240 \big(214+7 \sqrt{1009}\,\big)\big/27, \]
and remark that $c_n$ is the root of a polynomial of degree $n-1$. We
demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as
radicals over $\mathbb{Q}$ (and conjecture this is in fact true for general $n
\geq 6$).
