[Paper Review] Near NP-Completeness for Detecting p-adic Rational Roots in One Variable

Martín Avendaño, Ashraf Ibrahim+2 more|arXiv (Cornell University)|Jan 24, 2010

Cryptography and Residue Arithmetic21 references3 citations

TL;DR

This paper establishes that detecting p-adic rational roots for sparse univariate polynomials is in NP for most inputs, and provides a polynomial-time algorithm for trinomials with generic p-adic Newton polygons, improving the prior EXPTIME bound. It also proves NP-hardness under randomized reductions, offering the first unconditional complexity lower bound for this problem.

ABSTRACT

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an unconditional complexity lower bound of NP-hardness with respect to randomized reductions for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers.

Motivation & Objective

• To close the complexity gap for detecting p-adic rational roots in univariate polynomials, which previously had only an EXPTIME upper bound.
• To establish tighter complexity bounds by showing that the problem lies in NP for most inputs, particularly for trinomials with generic Newton polygons.
• To prove an unconditional NP-hardness result under randomized reductions, removing reliance on unproven number-theoretic hypotheses.
• To extend and parallel algorithmic results from real algebraic geometry to the p-adic setting, particularly for sparse polynomials.
• To provide a certificate-based decision procedure using Hensel's Lemma and modular root lifting, enabling efficient verification of p-adic roots.

Proposed method

• Uses Hensel's Lemma to lift roots modulo $p^\ell$ for $\ell = 4 \cdot \mathrm{size}(f)$, ensuring that a root in $\mathbb{Z}_p$ exists iff a root exists modulo $p^\ell$.
• Employs a certificate of size polynomial in $\mathrm{size}(f)$, consisting of a root modulo $p^\ell$, to place the problem in NP.
• Applies the $p$-adic Newton polygon to characterize the structure of roots and identify cases where the discriminant $\mathcal{D}_\mathcal{A}(f)$ vanishes.
• Uses recursive squaring and efficient $\mathbb{F}_p$-arithmetic to test whether $p$ divides the discriminant $\mathcal{D}_\mathcal{A}(f)$, which determines whether the certificate applies.
• Constructs a randomized polynomial-time reduction from 3CNFSAT to $\texttt{FEAS}_{\mathbb{Q}_{\text{primes}}}$ using algebraic number theory and properties of cyclotomic polynomials.
• Replaces randomized reductions with brute-force search under Wagstaff's Conjecture to achieve deterministic NP-hardness, avoiding reliance on randomness.

Experimental results

Research questions

• RQ1 Can the complexity of detecting p-adic rational roots in univariate polynomials be improved from EXPTIME to NP?
• RQ2 Under what conditions on the Newton polygon does the problem admit a polynomial-time solution?
• RQ3 Is the problem NP-hard unconditionally, or does it depend on unproven number-theoretic assumptions?
• RQ4 How do the complexity bounds for p-adic root detection compare to those over the reals, especially in the sparse polynomial setting?
• RQ5 Can a succinct certificate for p-adic roots be constructed using modular lifting and discriminant conditions?

Key findings

• The problem $\texttt{FEAS}_{\mathbb{Q}_{\text{primes}}}(\mathbb{Z}[x_1] \times \mathbb{P})$ lies in NP for most inputs, specifically when $p \nmid \mathcal{D}_\mathcal{A}(f)$, which holds for a density-1 fraction of inputs.
• For trinomials with generic $p$-adic Newton polygons, the problem can be solved in polynomial time, improving the prior EXPTIME bound.
• The problem is NP-hard under randomized reductions, with the first unconditional lower bound for this problem, removing the need for unproven hypotheses on primes in arithmetic progression.
• A certificate for a $p$-adic rational root consists of a root modulo $p^\ell$ with $\ell = 4 \cdot \mathrm{size}(f)$, which can be verified in polynomial time.
• The reduction from 3CNFSAT to the $p$-adic feasibility problem uses a sequence of algebraic reductions involving cyclotomic polynomials and the AGP Theorem, establishing randomized NP-hardness.
• Under Wagstaff's Conjecture, the NP-hardness result can be made deterministic by replacing the randomized reduction with brute-force search, maintaining polynomial complexity.