John Abbott

Computing Ideals of Points

We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger?Moller algorithm, best suited for the computation over Q, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct approach and a new stopping criterion. The described algorithms are implemented in CoCoA, and we report some experimental timings.

Quadratic interval refinement for real roots

We present a new algorithm for refining a real interval containing a single real root: the new method combines the robustness of the classical Bisection algorithm with the speed of the Newton-Raphson method; that is, our method exhibits quadratic convergence when refining isolating intervals of simple roots of polynomials (and other well-behaved functions). We assume the use of arbitrary precision rational arithmetic. Unlike Newton-Raphson our method does not need to evaluate the derivative.

Approximate Commutative Algebra

Approximate Commutative Algebra is an emerging field of research which endeavours to bridge the gap between traditional exact Computational Commutative Algebra and approximate numerical computation. The last 50 years have seen enormous progress in the realm of exact Computational Commutative Algebra, and given the importance of polynomials in scientific modelling, it is very natural to want to extend these ideas to handle approximate, empirical data deriving from physical measurements of phenomena in the real world. In this volume nine contributions from established researchers describe various approaches to tackling a variety of problems arising in Approximate Commutative Algebra.

Computing zero-dimensional schemes

This paper is a natural continuation of Abbott et al. [Abbott, J., Bigatti, A., Kreuzer, M., Robbiano, L., 2000. Computing ideals of points. J. Symbolic Comput. 30, 341-356] further generalizing the Buchberger-Moller algorithm to zero-dimensional schemes in both affine and projective spaces. We also introduce a new, general way of viewing the problems which can be solved by the algorithm: an approach which looks to be readily applicable in several areas. Implementation issues are also addressed, especially for computations over Q where a trace-lifting paradigm is employed. We give a complexity analysis of the new algorithm for fat points in affine space over Q. Tables of timings show the new algorithm to be efficient in practice.

The design of CoCoALib

We describe some of the more important aspects of the design of CoCoALib, a new C++ library for effecting Computations in Commutative Algebra. Special effort has been invested in making the code clean and portable while not neglecting run-time performance; one of the primary goals is to offer freely available reference implementations of the most important algorithms in the field.

Fault-Tolerant Modular Reconstruction of Rational Numbers

Abstract In this paper we present two efficient methods for reconstructing a rational number from several residue-modulus pairs, some of which may be incorrect. One method is a natural generalization of that presented by Wang et al. in (Wang et al., 1982) (for reconstructing a rational number from correct modular images), and also of an algorithm presented in Abbott (1991) for reconstructing an integer value from several residue-modulus pairs, some of which may be incorrect. The other method is heuristic, but much easier to apply; it may be viewed as a generalization of Monagans MQRR ( Monagan, 2004 ). We compare our heuristic method with that of Bohm et al. (2015) . Our method is clearly preferable when the rational to be reconstructed is unbalanced.

SC2 : Satisfiability Checking Meets Symbolic Computation

Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted \(\mathsf {SC}^\mathsf{2} \) project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified \(\mathsf {SC}^\mathsf{2} \) community.Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted SC 2 project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified SC 2 community.

SC2: Satisfiability Checking Meets Symbolic Computation (Project Paper)

Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted \(\mathsf {SC}^\mathsf{2} \) project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified \(\mathsf {SC}^\mathsf{2} \) community.Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted SC 2 project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified SC 2 community.

Twin-float arithmetic

We present a heuristically certified form of floating-point arithmetic and its implementation in CoCoALib. This arithmetic is intended to act as a fast alternative to exact rational arithmetic, and is developed from the idea of paired floats expounded by Traverso and Zanoni (2002). As prerequisites we need a source of (pseudo-)random numbers, and an underlying floating-point arithmetic system where the user can set the precision. Twin-float arithmetic can be used only where the input data are exact, or can be obtained at high enough precision. Our arithmetic includes a total cancellation heuristic for sums and differences, and so can be used in classical algebraic algorithms such as Buchbergers algorithm. We also present a (new) algorithm for recovering an exact rational value from a twin-float, so in some cases an exact answer can be obtained from an approximate computation. The ideas presented here are implemented as a ring in CoCoALib, called RingTwinFloat, allowing them to be used easily in a wide variety of algebraic computations (including Grobner bases).

CoCoA: computations in commutative algebra

CoCoA is a special-purpose system for doing Computations in Commutative Algebra. It runs on all common platforms.