We consider essential self-adjointness on the space C0∞((0,∞)) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type (Formula presented.) in L2((0,∞);dx). While the special case n=1 is classical and it is well known that τ2(c)|C0∞((0,∞)) is essentially self-adjoint if and only if c≥3/4, the case n∈N, n≥2, is far from obvious. In particular, it is not at all clear from the outset that (Formula presented.) As one of the principal results of this paper we indeed establish the existence of cn, satisfying cn≥(4n-1)!!/22n, such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question, (Formula presented.) which permits the sharp (and explicit) answer c≥[(2n-1)!!]2/22n, n∈N, the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, (Formula presented.) and remark that cn is the root of a polynomial of degree n-1. We demonstrate that for n=6,7, cn are algebraic numbers not expressible as radicals over Q (and conjecture this is in fact true for general n≥6).