Resolving Zero Divisors Using Hensel Lifting

2000

We consider the problem of computing the monic gcd of two polyno- mials over a number eld

Symposium on Discrete Algorithms, 1998

We compare four fast methods for univariate polynomial GCD computation over an algebraic number field. The first two are the modular method of Langemyr and McCallum (1987), and the heuristic method of Smedley et al. (1988). Because of recent improvements to the modular method by Encarnacion (1994), we expected it now to always be the better of the two in practice, provided it is implemented "properly". This turned out to be the case in our Maple implementations. We also implemented a quadratic lifting Hensel based method and a more direct method which we call the prime-power method. Like the heuristic and quadratic Hensel methods, the prime-power method is simple to implement efficiently, but it is usually better. Due to the large effort required to implement the modular method efficiently, we recommend the prime-power method as a practical alternative.

2018

Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a radical triangular set overQ. Our modular algorithm naturally generalizes previous work from algebraic number theory. We have implemented our algorithm using Maple’s recden package. We compare our implementation with the procedure RegularGcd in the RegularChains package.

Proceedings of the 2005 international symposium on Symbolic and algebraic computation - ISSAC '05, 2005

Let G = (4y 2 + 2z)x 2 + (10y 2 + 6z) be the greatest common divisor (gcd) of two polynomials A, B ∈ Z[x, y, z]. Because G is not monic in the main variable x, the sparse modular gcd algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of G in x consistently. We call this the normalization problem. We present two new sparse modular gcd algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippel's algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic gcd x 2 + (5y 2 + 3z)/(2y 2 + z) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm.

Annals of mathematics and physics, 2022

Based on the Bezout approach we propose a simple algorithm to determine the gcd of two polynomials which doesn't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only n steps for polynomials of degree n. Formal manipulations give the discriminant or the resultant for any degree without needing division nor determinant calculation.

IEEE Transactions on Signal Processing, 1992

The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.

2016

Four new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is a simple improvement of PRS (polynomial remainder sequence) algorithms. The second is to calculate a Groebner basis with a certain term ordering. The third is to calculate subresultant by treating the coefficients as truncated power series. The fourth is to calculate PRS by treating the coefficients as truncated power series. The first and second algorithms are not important practically, but the third and fourth ones are quite efficient and seem to be useful practi-cally. 1.

Journal of Symbolic Computation, 1992

Three new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is to calculate a GrSbner basis with a certain term ordering. The second is to calculate the subresultant by treating the coefficients w.r.t, the main variable as truncated power series. The third is to calculate a PRS (polynomial remainder sequence) by treating the coefficients as truncated power series. The first algorithm is not important praetioaUy, but the second and third ones are efficient and seem to be useful practically. The third algorithm has been implemented naively and compared with the trial-division PRS algorithm and the EZGCD algorithm. Although it is too early to derive a definite conclusion, the PRS method with power series coefficients is very efficient for calculating low degree GCD of high degree non-sparse polynomials.

Journal of Algorithms, 2005

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U , V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC '93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111-116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111-122] when U and V are close to the same size, that replaces U by (U − bV )/p, where p is one of the prime moduli and b is the unique integer in the interval (−p/2, p/2) such that b ≡ UV −1 (mod p). When the algorithm is executed on a bit common CRCW PRAM with O(n log n log log log n) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/ log n) time on the same number of processors.  2004 Elsevier Inc. All rights reserved.

LMS Journal of Computation and Mathematics, 2014

We develop algorithms to turn quotients of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.