Solutions of a polynomial equation modulo a prime power

TL;DR

The paper tackles solving polynomial congruences $P(x) \equiv 0$ modulo prime powers by developing a $p$-adic tree framework. It introduces the trunk as a compact subtree of the full solution tree, with thickness labels on trunk vertices and a tree-top function $\varphi$ that encodes lifting potential. A main theorem shows that the full solution set modulo $p^e$ can be reconstructed from the trunk via fans, yielding a closed-form counting formula $N_e = \sum_{(r,k)\in\mathrm{Trunk}(P),\; \varphi(r,k)-t_k<e\le\varphi(r,k)} p^{e-k}$ and enabling polynomial-time description of all solutions. The work connects to Hensel lifting, residual degree, and Igusa zeta-function perspectives, and provides explicit algorithms for trunk construction and solution counting, with detailed treatments for degree two polynomials. Overall, the trunk/solution-tree framework offers an efficient, structural approach to understand and compute all solutions to polynomial congruences modulo prime powers, with broader implications for $p$-adic analysis and arithmetic geometry.

Abstract

How do you find the integer solutions of a polynomial equation modulo an integer?

Figures (11)

• Figure 1: The trunk and the tree of solutions of $P(X) = (X^2+3)(X^2+3X+9)$, $p=3$.
• Figure 2: Here $p=3$. Left: the $p$-adic congruence tree $\Omega_p$, each vertex is labeled by in integer $x \in[0,p^e-1]$. Right: the decomposition of $x=11$ in base $p$, each edge is labeled by a integer $a_i \in[0,p-1]$.
• Figure 3: Thickness and the tree-top function $\varphi$.
• Figure 4: A fan.
• Figure 5: The tree from the trunk.

Theorems & Definitions (29)

• Definition 2.1
• Example 2.2
• Theorem 3.1
• Example 3.2
• Theorem 3.3: Hensel's Lemma
• Example 3.4
• Remark
• Lemma 4.1
• proof
• Lemma 4.2