We consider essential self-adjointness of strongly singular, homogeneous, polyharmonic operators of the form \( T_m = \left( (-\Delta )^m +c|x|^{-2\,m}\right) \big |_{C_0^{\infty }({\mathbb {R}}^n \backslash \{0\})}, \quad m \in {\mathbb {N}}, \; n \in {\mathbb {N}}, n\ge 2, \; c \in {\mathbb {R}}, \) in \(L^2({\mathbb {R}}^n; d^n x)\) , with special emphasis on the biharmonic case \(m=2\) and the case \(m=3\) . For instance, in the biharmonic case \(m=2\) we prove the sharp result \(\begin{aligned}&T_2 \text { is essentially self-adjoint if and only if} \\&\quad c\ge {\left\{ \begin{array}{ll} 3(n+2)(6-n) & \text { for } 2\le n\le 5; \\ {\displaystyle -\frac{(n+4)n(n-4)(n-8)}{16}}& \text {for } n\ge 6. \end{array}\right. } \end{aligned}\) In particular, in the special (nonsingular) case \(c=0\) , \((-\Delta )^2\big |_{C_0^{\infty }({\mathbb {R}}^n \backslash \{0\})}\) is essentially self-adjoint in \(L^2({\mathbb {R}}^n; d^n x)\) if and only if \(n \ge 8\) . Similarly, we derive the analogous sharp essential self-adjointness result for \(T_3\) (i.e., for \(m=3\) ) for all space dimensions \(n \in {\mathbb {N}}\) , \(n \ge 2\) . Our methods generalize to homogenous polyharmonic differential operators; however, there are some nontrivial subtleties that arise. For example, the natural expectation that for \(m, n \in {\mathbb {N}}\) , \(n \ge 2\) , there exist \(c_{m,n} \in {\mathbb {R}}\) such that \(\left( (-\Delta )^m +c|x|^{-2m}\right) \big |_{C_0^{\infty }({\mathbb {R}}^n \backslash \{0\})}\) is essentially self-adjoint in \(L^2({\mathbb {R}}^n; d^n x)\) if and only if \(c \ge c_{m,n}\) , turns out to be false. Indeed, for \(n=20\) , we prove that the differential operator \( \left( (-\Delta )^5 +c|x|^{-10}\right) \big |_{C_0^{\infty }({\mathbb {R}}^{20} \backslash \{0\})}, \quad c \in {\mathbb {R}}, \) is essentially self-adjoint in \(L^2\big ( {\mathbb {R}}^{20}; d^{20} x\big )\) if and only if \(c\in [0,\beta ]\cup [\gamma ,\infty )\) , where \(\beta \approx 1.0436\times 10^{10}\) , and \(\gamma \approx 1.8324 \times 10^{10}\) are the two real roots of a particular quartic equation with integer coefficients (see Theorem 4.4, eq. (4.29)).
