The Basic Polynomial Algebra Subprograms

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.

ACM Communications in Computer Algebra, 2011

We demonstrate new routines for sparse multivariate polynomial multiplication and division over the integers that we have integrated into Maple 14 through the expand and divide commands. These routines are currently the fastest available, and the multiplication routine is parallelized with superlinear speedup. The performance of Maple is significantly improved. We describe our polynomial data structure and compare it with Maple's. Then we present benchmarks comparing Maple 14 with Maple 13, Magma, Mathematica, Singular, Pari, and Trip.

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

SIAM Journal on Computing, 1983

It is shown that any multivariate polynomial of degree d that can be computed sequentially in C steps can be computed in parallel in O((log d)(log C + log d)) steps using only (Cd) 1) processors.

[Now twelve years old, but still worth a look.] Exact symbolic computation with polynomials and matrices over polynomial rings has wide applicability to many fields. By "exact symbolic" we mean computation with polynomials whose coefficients are integers (of any size), rational numbers, or finite fields, as opposed to coefficients that are "floats" of a certain precision. Such computation is part of most computer algebra systems ("CA systems"). Over the last dozen years several large CA systems have become widely available, such as Axiom, Derive, Macsyma, Magma, Maple, Mathematica, and Reduce. They tend to have great breadth, be produced by profit-making companies, and be relatively expensive. However, most if not all of these systems have difficulty computing with the polynomials and matrices that arise in actual research. Real problems tend to produce large polynomials and large matrices that the general CA systems cannot handle. In the last few years several smaller CA systems focused on polynomials have been produced at universities by individual researchers or small teams. They run on Macs, PCs, and workstations. They are freeware or shareware. Several claim to be much more efficient than the large systems at exact polynomial computations. The list of these systems includes CoCoA, Fermat, MuPAD, Pari-GP, and Singular.

This report describes our work on implementation of effective numerical routines for polynomials and polynomial matrices in the MATHEMATICA software. Such operations are recalled during the controller design process if the so called polynomial or algebraic design methods are employed. This research is also motivated by the fact that MATHEMATICA developers pay attention to control engineers needs and produce the Control Systems Professional package for use with MATH-EMATICA and, as we believe, a set of routines for algebraic approach could conveniently complement the existing bunch of programs primarily intended for state-space representations.

We continue the work in and present our maple implementation of an algebraic toolbox capable of doing computations with one and two real algebraic numbers and real solving bivariate polynomial systems. In addition we describe new functions of the subpackage of the C++ library synaps for root isolation of univariate and multivariate polynomials. For this implementation we combine symbolic and numeric tools and illustrate their behavior on some classical family of polynomials.

Solving polynomial systems with CoCoALib (a C++ library from algebra to applications) ============================================================= We present the algebraic and exact methods for solving polynomial systems and analyzing their structure, and also the opposite problem i.e. finding polynomials vanishing on a given set of points; and then we discuss the recent results about the interaction between these algebraic techniques with approximation issues. We show how to perform these computations using the Computer Algebra System CoCoA, and also with its core C++ library, CoCoALib.

Computers & Mathematics with Applications, 1997

A new algorithm for splitting polynomials is presented. This algorithm requires O(d log −1) 1+δ floating point operations, with O(log −1) 1+δ bits of precision. As far as complexity is concerned, this is the fastest algorithm known by the authors for that problem. An important application of the method is factorizing polynomials or polynomial root-finding.

2010

We present a Las Vegas algorithm for interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of Ben-Or and Tiwari in 1988 for interpolating polynomials over rings with characteristic zero to characteristic p by doing additional probes.