Counting fixed points and rooted closed walks of the singular map $x \mapsto x^{x^n}$ modulo powers of a prime

TL;DR

This paper investigates the fixed points and closed walks of the generalized self-power map modulo prime powers using advanced $p$-adic techniques, providing new methods for lifting singular solutions in number theory.

Contribution

It introduces a novel $p$-adic lifting method for singular solutions of equations related to the self-power map modulo prime powers.

Findings

Counted fixed points and closed walks for the map $x o x^{x^n}$ modulo prime powers.

Developed a new technique for lifting singular solutions using the Jacobian's left kernel.

Enhanced understanding of the structure of solutions to self-power maps in modular arithmetic.

Abstract

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ 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.