[Paper Review] An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field

Yao Sun, Dingkang Wang|arXiv (Cornell University)|Jul 14, 2009

Polynomial and algebraic computation12 references4 citations

TL;DR

This paper presents a new efficient algorithm for factoring polynomials over algebraic extension fields defined by a maximal ideal in a polynomial ring. By using a generic linear map to reduce the problem to univariate factorization over the ground field and leveraging existing Gröbner bases without recomputation, the method achieves significant speedups—especially in complex cases—demonstrating superior performance over existing methods like Wang's algorithm.

ABSTRACT

A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra Groebner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a polynomial over the ground field by a generic linear map. Then this polynomial is factorized over the ground field. From these factors, the factorization of the polynomial over the extension field is obtained. The new algorithm has been implemented and computer experiments indicate that the new algorithm is very efficient, particularly in complicated examples.

Motivation & Objective

• To develop a more efficient algorithm for factoring multivariate polynomials over algebraic extension fields, particularly in cases where existing methods are computationally expensive.
• To eliminate the need for additional Gröbner basis computations when the extension field is defined by a maximal ideal with a known Gröbner basis.
• To transform the factorization problem over the extension field into univariate factorization over the ground field using linear algebra techniques.
• To ensure the irreducible factors are verifiable without extra computation, overcoming limitations in prior probabilistic approaches.
• To improve performance in both characteristic zero and finite fields, especially for high-dimensional or complex systems.

Proposed method

• The algorithm uses a generic linear map to lift a polynomial over the extension field to a polynomial over the ground field, reducing the factorization problem to univariate factorization over the base field.
• It relies on the linear map $ m_r $ on the quotient ring $ \mathcal{A}_k(Q) = k[x_1,\dots,x_n]/I $, whose characteristic polynomial encodes algebraic relations in the extension field.
• The minimal polynomial of $ m_r $ is irreducible when $ I $ is maximal, ensuring the field structure is preserved in the transformation.
• The method avoids recomputing Gröbner bases by reusing the input Gröbner basis of the maximal ideal $ I $, which defines the extension field.
• Factorization over the ground field is performed using standard univariate factorization, and the results are lifted back to the extension field via the linear map.
• The algorithm is implemented in Magma and compared against Wang’s algorithm in Maple 7, showing substantial performance gains in complex examples.

Experimental results

Research questions

• RQ1 Can polynomial factorization over algebraic extension fields be reduced to univariate factorization over the ground field without recomputing Gröbner bases?
• RQ2 How can the efficiency of factorization over algebraic extensions be improved by leveraging linear algebra and existing Gröbner bases?
• RQ3 What is the impact of using a generic linear map on the complexity and correctness of the factorization process?
• RQ4 Can the irreducible factors obtained via this method be verified without additional computation, ensuring completeness and correctness?
• RQ5 How does the performance of this algorithm compare to existing methods like Wang’s algorithm in terms of runtime and scalability?

Key findings

• The new algorithm outperforms Wang’s algorithm significantly, with efactor completing the factorization of $ f^{(9)} $ in 63.578 seconds compared to Wang’s algorithm timing out after 2 hours.
• For $ f^{(3)} $, the new algorithm runs in 0.734 seconds versus 12.171 seconds for Wang’s method, showing a 16x speedup.
• The algorithm is particularly efficient for high-dimensional systems: $ f^{(10)} $ with 7 variables and dimension 64 is solved in 96.344 seconds, while Wang’s method exceeds 2 hours.
• The use of a generic linear map avoids coefficient expansion issues common in Wang’s method, which relies on variable substitution and pseudo-remainder maps.
• The complexity of computing characteristic polynomials is polynomial, whereas Wang’s method involves exponential-time characteristic set computation, explaining the performance gap.
• The algorithm is complete and verifiable: irreducible factors are guaranteed without extra verification steps, unlike Monico’s incomplete primary decomposition approach.