Rational simplification modulo a polynomial ideal

ACM Communications in Computer Algebra, 2017

We employ two techniques to dramatically improve Maple's performance on the Fermat benchmarks for simplifying rational expressions. First, we factor expanded polynomials to ensure that gcds are identified and cancelled automatically. Second, we replace all expanded polynomials by new variables and normalize the result. To undo the substitutions, we use a C routine for sparse multivariate division by a set of polynomials. The resulting times for the first Fermat benchmark are a factor of 17x faster than Fermat and 39x faster than Magma.

Singular is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, alge-braic geometry, and singularity theory. Singular's main computational objects are polynomials, ideals and modules over a large variety of rings. Singular features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. The new, upcoming version 2-2 includes also algorithms for a wide class of non-commutative algebras (Plural) and the possiblity for dynamic extension of the program at run-time (dynamic modules). Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root– finding, visualisation, and many more related functionalities.

The algorithms for linear algebra in the Magma and Axiom computer algebra systems work over an arbitrary ring. For example, the implementation of Gaussian elimination for reducing a matrix to (reduced) row Echelon form works over any field that the user constructs. In contrast, Maple's facilities for linear algebra in its LinearAlgebra package only work for specific rings. If the input matrix contains general expressions, the algorithms may work incorrectly. Motivated by a need to do linear algebra over quotient rings and finite fields in Maple, we have designed a simple to use facility that permits the Maple user to define a field, Euclidean domain, integral domain or ring so that our "generic" algorithms for linear algebra are immediately available. These algorithms comprise a package called GenericLinearAlgebra which is being integrated into Maple 11. We have also implemented a package for computing in quotient rings. This package, called QuotientRings, exports all the necessary operations so that one can immediately do linear algebra over a quotient ring, for example, the trigonometric polynomial ring Q[s, c]/ s 2 + c 2 − 1. The package also includes a new algorithm for simplifying fractions over a quotient ring to canonical form that we discuss.

Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994

Mark. Encarnacion@risc. Uni-linz. ac. at Modular methods for computing the gcd of two univariate polynomials over an algebraic number field require a prim-i knowledge about the denominators of the rational numbers in the representation of the gtd. We derive a multiplicative bound for these denominators without assuming that the number generating the field is an algebraic integer. Consequently, the gcd algorithm of Langemyr and McCallum [J. Symbolic Computation, 8:429-448, 1989] can now be applied directly to polynomials that are not necessarily represented in terms of an algebraic integer. Worst-case analyses and experiments with an implementation show that by avoiding a conversion of representation the reduction in the computing time can be significant. We also suggest the use of an algorithm for recovering a rational number from its modular residue so that the denominator bound need not be computed explicitly. Experiments and analyses indicate that this is a good practical alternative.

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.

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.

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.

Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07, 2007

We present a first sparse modular algorithm for computing a greatest common divisor of two polynomials f1, f2 ∈ L[x] where L is an algebraic function field in k ≥ 0 parameters with r ≥ 0 field extensions. Our algorithm extends the dense algorithm of Monagan and van Hoeij from 2004 to support multiple field extensions and to be efficient when the gcd is sparse. Our algorithm is an output sensitive Las Vegas algorithm. We have implemented our algorithm in Maple. We provide timings demonstrating the efficiency of our algorithm compared to that of Monagan and van Hoeij and with a primitive fraction-free Euclidean algorithm for both dense and sparse gcd problems.

Central European Journal of Mathematics, 2011

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.

2016

Proceedings of the 1991 international symposium on Symbolic and algebraic computation - ISSAC '91, 1991

We present two algorithms for interpolating sparse rational functions. The first is the interpolation algorithm in a sense of sparse partial fraction representation of rational functions. The second is the algorithm for computing the entier and the remainder of a rational function. The first algorithm works without apriori known bound on the degree of a rational function, the second one is in the parallel class NC provided that the degree is known. The presented algorithms complement the sparse interpolation results of [GKS 90].

ACM Sigsam Bulletin, 2009

One of the main successes of the computer algebra community in the last 30 years has been the discovery of algorithms, called modular methods, that allow to keep the swell of the intermediate expressions under control. Without these methods, many applications of computer algebra would not be possible and the impact of computer algebra in scientific computing would be severely

2007

This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enable...

Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009

We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1,. .. , xv] where L is an algebraic function field with k ≥ 0 parameters t1,. .. , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization of f when using Hensel lifting. We have implemented our algorithm in Maple 13. We provide timings demonstrating the efficiency of our algorithm.

2010

We discuss the algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, compute a polynomial U(x) ∈ k[x] (a universal denominator) such that the denominator of each of rational solutions (if exist) of the given equation divides U(x). We consider two types of such algorithms. One of them is based on constructing a