[Paper Review] Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

Jordi Guàrdia, Jesús Sanz Montes+1 more|ArXiv.org|Jul 25, 2008

Algebraic Geometry and Number Theory4 references6 citations

TL;DR

This paper presents a highly efficient algorithm for computing discriminants and prime ideal factorizations in number fields using higher-order Newton polygons. By leveraging p-adic techniques and optimizing computation order, the method computes the p-adic valuation of the discriminant and factorization of p in number fields of degree up to 1000 in seconds, even for polynomials with large indices, demonstrating superior performance over classical integral basis methods.

ABSTRACT

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer.

Motivation & Objective

• To develop a fast, direct algorithm for computing discriminants and prime ideal factorizations in number fields without computing a global integral basis.
• To address the computational bottleneck in algebraic number theory where traditional methods for integral basis computation scale poorly with field degree.
• To improve efficiency in p-adic factorization by using higher-order Newton polygons that avoid field extensions and work solely with residue fields.
• To demonstrate that the algorithm scales to high-degree number fields (e.g., degree 6912) and large indices (e.g., 77 million), outperforming existing approaches.

Proposed method

• The algorithm uses higher-order Newton polygons to analyze the p-adic factorization of a defining polynomial f(x) of a number field K=Q(θ).
• It computes the p-adic valuation of the discriminant and the decomposition type of p in OK by analyzing types and representatives in successive orders of Newton polygons.
• A refinement process is introduced that lowers the order of computation, significantly reducing complexity and improving performance (Theorem 3.1).
• The method computes the p-index of the polynomial f(x), which is essential for determining the discriminant when combined with disc(f).
• It computes generators of prime ideals above p by extending the algorithm with extended GCD computations.
• The algorithm is highly parallelizable, enabling efficient scaling to very high-degree number fields.

Experimental results

Research questions

• RQ1 Can higher-order Newton polygons be used to compute discriminants and prime ideal factorizations in number fields more efficiently than classical integral basis methods?
• RQ2 How does the refinement of the Newton polygon computation order affect the algorithm’s complexity and runtime?
• RQ3 What is the performance limit of the algorithm in terms of field degree and p-index for practical computation on standard hardware?
• RQ4 Can the algorithm handle polynomials with very large indices that are 'hidden' from simpler factorization methods?
• RQ5 To what extent does the algorithm’s efficiency scale with field degree and number of prime ideal factors?

Key findings

• The algorithm computes the p-adic valuation of the discriminant and factorization of p in a number field of degree 1000 in under a second, even for p-index up to 200,000.
• For a degree-6912 polynomial with 2-index 77,673,504, the factorization of 2 is computed in 787 seconds on a personal computer.
• The algorithm runs in under 0.2 seconds for 32 test polynomials from [FPR02], even after perturbation with high-degree p^1000 terms.
• The time to compute generators of prime ideals increases significantly due to extended GCD computations, with one example exhausting Magma’s virtual memory due to coefficient explosion.
• The algorithm is highly parallelizable, enabling it to scale to number fields of extremely high degree.
• The method provides a significant improvement over classical integral basis computation, especially when only discriminant or factorization is needed.