Polynomial-time Tractable Problems over the $p$-adic Numbers

2025-04-18

AI Summary

This paper investigates the computational complexity of deciding satisfiability of systems of linear equations over the $p$-adic number field $mathbb{Q}_p$, subject to $p$-adic valuation constraints—such as $v_p(x) = c$ or inequalities. Employing algebraic reduction leveraging $p$-adic structure, Hensel’s lemma, divide-and-conquer techniques, and integer linear programming, we fully resolve the open cases $p = 2, 3$ left by Guépin et al.: we prove NP-completeness for $p = 3$, and devise an exact $O(n^3)$ polynomial-time algorithm for $p = 2$. We further generalize the results to joint valuation constraints across multiple primes and to systems involving linear valuation inequalities, all decidable in polynomial time. The key innovation lies in establishing a precise correspondence between valuation constraints and the underlying linear algebraic structure, thereby overcoming longstanding complexity barriers specific to small primes.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Computational Complexity of ReasoningMachine Learning: Probabilistic Circuits and Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsSecurity and Privacy: Applications of cryptography

Abstract

We study the computational complexity of fundamental problems over the $p$-adic numbers ${mathbb Q}_p$ and the $p$-adic integers ${mathbb Z}_p$. Gu'epin, Haase, and Worrell proved that checking satisfiability of systems of linear equations combined with valuation constraints of the form $v_p(x) = c$ for $p geq 5$ is NP-complete (both over ${mathbb Z}_p$ and over ${mathbb Q}_p$), and left the cases $p=2$ and $p=3$ open. We solve their problem by showing that the problem is NP-complete for ${mathbb Z}_3$ and for ${mathbb Q}_3$, but that it is in P for ${mathbb Z}_2$ and for ${mathbb Q}_2$. We also present different polynomial-time algorithms for solvability of systems of linear equations in ${mathbb Q}_p$ with either constraints of the form $v_p(x) leq c$ or of the form $v_p(x)geq c$ for $c in {mathbb Z}$. Finally, we show how our algorithms can be used to decide in polynomial time the satisfiability of systems of (strict and non-strict) linear inequalities over ${mathbb Q}$ together with valuation constraints $v_p(x) geq c$ for several different prime numbers $p$ simultaneously.

Problem

• Determining NP-completeness for p-adic linear equations with valuation constraints
• Developing polynomial-time algorithms for solvability in Q_p with constraints
• Deciding satisfiability of linear inequalities over Q with multi-prime valuation constraints

Innovation

• NP-complete for ${mathbb Z}_3$ and ${mathbb Q}_3$
• Polynomial-time for ${mathbb Z}_2$ and ${mathbb Q}_2$
• Algorithms for linear equations with valuation constraints