Abstract
The "self-power" map x ↦ x^x modulo m and its generalized form x ↦ x^xⁿ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel's lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.