John Abbott

• PhD (mathematics, computer algebra)
• Ricercatore a tempo determinato at University of Genoa

We consider the question of certifying that a polynomial in \({\mathbb Z}[x]\) or \({\mathbb Q}[x]\) is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv. that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory...

We consider the question of certifying that a polynomial in ${\mathbb Z}[x]$ or ${\mathbb Q}[x]$ is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv.~that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory beca...

The main focus of this paper is on the problem of relating an ideal I in the polynomial ring ℚ[x1,…,xn] to a corresponding ideal in 𝔽p[x1,…,xn] where p is a prime number; in other words, the reduction modulop of I. We first define a new notion of σ-good prime for I which does depends on the term ordering σ, but not on the given generators of I. We...

Download the source code from http://cocoa.dima.unige.it/cocoalib

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to...

Manual for the C++ library CoCoALib v 0.99600

Manual for CoCoA System, v. 5.2.4 -- Download CoCoA from http://cocoa.dima.unige.it

CoCoALib is a C++ software library offering operations on polynomials, ideals of polynomials, and related objects. The principal developers of CoCoALib are members of the SC 2 project. We give an overview of the latest developments of the library, especially those relating to the project SC 2. The CoCoA software suite includes also the programmable...

Given a zero-dimensional ideal $I$ in a polynomial ring, many computations start by finding univariate polynomials in $I$. Searching for a univariate polynomial in $I$ is a particular case of considering the minimal polynomial of an element in $P/I$. It is well known that minimal polynomials may be computed via elimination, therefore this is cons...

The main focus of this paper is on the problem of relating an ideal I in the polynomial ring Q[x_1,..., x_n] to a corresponding ideal in F_p[x_1, ..., x_n] where p is a prime number; in other words, the reduction modulo p of I. We define a new notion of sigma-good prime for I which depends on the term ordering sigma, and show that all but finitely...

Documentation for CoCoALib: C++ library for Computations in Commutative Algebra Release CoCoALib-0.99560 January 2018

Documentation for CoCoA: Computations in Commutative Algebra Release CoCoA-5.2.2 January 2018

CoCoA-5 is an interactive Computer Algebra System for Computations in Commutative Algebra, particularly Gröbner bases. It offers a dedicated, mathematician-friendly programming language, with many built-in functions. Its mathematical core is a user-friendly C++ library, called CoCoALib; being a software library facilitates integration with other so...

We give an overview of the latest developments in the library and system: specifically for the recent release of CoCoA-5.2.0/CoCoALib-0.99550 (May 2017), and the upcoming release (autumn 2017).

Presentation of some new features which are of interest for the SC-Square project

CoCoA-5 is an interactive Computer Algebra System for Computations in Commutative Algebra, particularly Gröbner bases. It offers a dedicated, mathematician-friendly programming language, with many built-in functions. Its mathematical core is CoCoALib, an open-source C++ library, designed to facilitate integration with other software. We give an ove...

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to...

Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the two communities are currently only weakly connected. We introduce a new project SC 2 to build a...

We present a survey on the developments on Groebner bases showing explicit examples in CoCoA. The CoCoA project dates back to 1987: its aim was to create a mathematician-friendly laboratory for studying Commutative Algebra, most especially Groebner bases. Since then, always maintaining this "friendly" tradition, it has evolved and has been complete...

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, “ElimTH”, has as main step the computation of an elimination ideal...

This is a pdf version for online viewing only. For running it in CoCoA download the ".cocoa5" version.

Sofware demonstration part of the presentation at workshop SC-SQUARE - SYNASC 2016 [the ”cocoa5” extension indicates a plain-text file written in CoCoA-5 language ]

The CoCoA project began in 1987, and conducts research into Computational Commutative Algebra (from which its name comes) with particular emphasis on Gröbner bases of ideals in multivariate polynomial rings, and related areas. A major output of the project is the CoCoA software, including the CoCoA-5 interactive system and the CoCoALib C++ library....

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...

Presentation of paper ”Implicitization of Hypersurfaces”

Tutorial CoCoA/CoCoALib at ICMS2016: introduction

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...

Presentation of CoCoA-5 and CoCoALib side by side (2016)

2016 SC-Square Project Poster

CoCoA is a well-established Computer Algebra System for Computations in Commutative Algebra, and specifically for Gröbner bases. In the last few years the CoCoA software has undergone a profound change: the code has been totally re-written in C++, and includes CoCoA-5 (an interactive system) and CoCoALib (an open source C++ library). The new CoCoA-...

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, "ElimTH", has as main step the computation of an elimination ideal...

An overview of several main ”modular methods” in computer algebra.

The CoCoA project dates back to 1987 under the lead of L.Robbiano: the aim was to create a mathematician-friendly laboratory for studying Commutative Algebra, especially Groebner bases. Since then, always maintaining this tradition, it has evolved and has been rewritten, and now offers: - an open source C++ software library, CoCoALib - a new inter...

[Full distribution] C++ open source library for Computations in Commutative Algebra. Also contain source code for the Computer Algebra System CoCoA-5, version 5.1.1. http://cocoa.dima.unige.it/cocoalib/

CoCoA 5.1.1 == A few new functions have been added == -- ApproxSolve, RationalSolve of a polynomial system -- BettiDiagram, BettiMatrix -- CheckArgTypes -- GradingMat (grading of a polynomial RING) -- ID of a RING -- IsInjective, IsSurjective, ker, PreImage of a RINGHOM -- MayerVietorisTreeN1 -- NmzDiagInvariants, NmzEhrhartRing, NmzFiniteDiagInvar...

libnormaliz is a C++ library for computations with rational cones and affine monoids and CoCoALib/CoCoA-5 offers a general environment for computations in Commutative Algebra. For mutual benefit we have developed a simple and fast interface between the two software libraries. We present how this integration was designed, and then describe in detail...

CoCoA is a well-established Computer Algebra System for Computations in Commutative Algebra, and specifically for Gröbner bases. In the last few years CoCoA has undergone a profound change: the code has been totally re-written in C++, and includes an integral open source C++ library, called CoCoALib. The new CoCoA-5 language still resembles the CoC...

C++ open source library for Computations in Commutative Algebra. Also contain source code for the Computer Algebra System CoCoA-5, version 5.0.9. http://cocoa.dima.unige.it/cocoalib/

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 th...

The CoCoA Project (freely available at http://cocoa.dima.unige.it) offers two products: -- CoCoA System: CAS (Computer Algebra System) for doing Computations in Commutative Algebra -- CoCoALib: open source (GPL) C++ library for doing Computations in Commutative Algebra CoCoA and CoCoALib are freely downloadable at http://cocoa.dima.unige.it: CoCo...

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 presente...

We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in Z[x]; we include a new bound which was latent in a paper by Mignotte, and a few minor improvements to some existing bounds. We compare these bounds and show that none is universally better than the others. In the second part of the paper we...

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 a...

We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when refining isolating intervals of simple roots of polynomials (and other well-behaved functions). We assume the use o...

First released in 1988,CoCoAis a special-purpose system for doing Computations in Commutative Algebra: i.e. it is a system specialized in the algorithmic treatment of polynomials. It is freely available and offers a textual interface, an Emacs mode, and a graphical user interface common to most platforms ([6]).

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 scien...

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

Let X be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal l(X) independent of the data uncertainty. We present a method to compute, starting from X, a polynomial basis B of l(X) which exhibits structural stability, that is, if (X) over tilde is any set of points differin...

Given a set $X$ of "empirical" points, whose coordinates are perturbed by errors, we analyze whether it contains redundant information, that is whether some of its elements could be represented by a single equivalent point. If this is the case, the empirical information associated to $X$ could be described by fewer points, chosen in a suitable way....

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 i...

We describe the first complete implementation of Davenport's algorithm [Davenport86] for the solution of the Risch differential equation. Our code forms part of a new integration package written in REDUCE which operates over algebraic number fields.

In this paper we consider a number of challenges from the point of view of the CoCoA project one of whose tasks is to develop software specialized for computations in commutative algebra. Some of the challenges extend considerably beyond the boundary of commutative algebra, and are addressed to the computer algebra community as a whole. @InProceedi...

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-Möller algorithm to zero-dimensional schemes in both affine and projective spaces. We also introduce a new, general way of viewing the probl...

The CoCoA program together with Singular and Macaulay 2 form an elite group of highly specialized systems having as their main forte the capability to calculate Gröbner bases. Although a number of general purpose symbolic computation systems (e.g. REDUCE and Maple) do offer the possibility to compute Gröbner bases, their non-specialist nature impli...

We answer a question left open in an article of Coppersmith and Davenport which proved the existence of polynomials whose powers are sparse, and in particular polynomials whose squares are sparse (i.e., the square has fewer terms than the original polynomial). They exhibit some polynomials of degree 12 having sparse squares, and ask whether there a...

For a real square matrix $M$, Hadamard's inequality gives an upper bound $H$ for the determinant of $M$; the bound is sharp if and only if the rows of $M$ are orthogonal. We study how much we can expect that $H$ overshoots the determinant of $M$, when the rows of $M$ are chosen randomly on the surface of the sphere. This gives an indication of the...

Syntax] is strictly in the eyes of the beholder(s)." (op.cit.) In the CaminoReal implementation, mathematical expressions in abstract syntax are converted to the target systems' isomorphic syntactic equivalents before transmission; CaminoReal codes for mathematical symbols are translated to their closest match in the target systems' name spaces in...

We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger–Möller 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 stop...

In this paper we describe ideas used to accelerate the Searching Phase of the Berlekamp{Zassenhaus algorithm, the algorithm most widely used for computing factorizations in Z[x]. Our ideas do not alter the theoretical worst-case complexity, but they do have a signi- cant eect in practice: especially in those cases where the cost of the Searching Ph...

In this paper we consider deterministic computation of the exact determinant of a dense matrix M of integers. We present a new algorithm with worst case complexity O Gamma n 4 (log n + log jjM jj) + n 3 log 2 jjM jj Delta , where n is the dimension of the matrix and jjM jj is a bound on the entries in M , but with average expected complexity O Gamm...

OpenMath aims at providing a universal means of communicating mathematical information between applications. In this paper we set out the objectives and design goals of OpenMath, and sketch the framework of a model that meets these requirements. On the basis of this model, we propose a structured approach for further development and implementation...

This report will detail the steps taken to date by the OpenMath Consortium to achieve its goals. In particular, section 2 will provide an overview of OpenMath. Emphasis is placed on OpenMath's design goals, initial target applications, and Consortium structure. In section 3, we provide an exposition of the levels of the OpenMath model which maps ma...

This document summarises the work and conclusions of the OpenMath Communications Committee. The committee was formed to discuss and resolve problems relating to the OpenMath "transport layer": this is the layer concerned solely with the transmission of the byte string encodings of OpenMath objects. The basis of the discussion was to be the OpenMath...

OpenMath aims at providing a universal means of communicating mathematical infor- mation between applications. In this paper we set out the objectives and design goals of OpenMath , and sketch the framework of a model that meets these requirements. Based upon this model, we propose a structured approach for further development and imple- mentation...

OpenMath aims at providing a universal means of communicating mathematical infor­mation between applications. In this paper we set out the objectives and design goals of OpenMath, and sketch the framework of a model that meets these requirements. Based upon this model, we propose a structured approach for further development and implementation of O...

We present some algorithms for performing Chinese Remaindering allowing for the fact that one or more residues may be erroneous — we suppose also that an a priori upper bound on the number of erroneous residues is known. A specific application would be for residue number codes (as distinct from quadratic residue codes). We generalise the method of...

We describe three ways to generalize Lenstra's algebraic integer recovery method. One direction adapts the algorithm so that rational numbers are automatically produced given only upper bounds on the sizes of the numerators and denominators. Another direction produces a variant which recovers algebraic numbers as elements of multiple generator alge...

We give a counterexample to a bound quoted in Wang's paper on polynomial factorization over algebraic number fields. We also give an alternative to that bound which seems not to have been published before.

We give a counterexample to a bound quoted in P. S. Wang’s paper on polynomial factorization over algebraic number fields [ibid. 30, 324-336 (1976; Zbl 0333.12003)]. We also give an alternative to that bound which seems not to have been published before.

We describe various design decisions and problems encountered during the implementation of the Lenstra factoriser [Lenstra82] in REDUCE. A practical viewpoint is taken with descriptions of both successful and unsuccessful attempts at tackling some of the problems. Particular areas considered include bounding coefficients of factors, the Cantor-Zass...

The problem of factorising polynomials: that is to say, given a polynomial with integer coefficients, to find the irreducible polynomials that divide it, is one with a long history. While the last word has not been said on the subject, we can say that the past 15 years have seen major break-throughs, and many computer algebra systems now include ef...

This paper describes a package implemented in REDUCE 3.2 for the manipulation of algebraic numbers. The package regards algebraic numbers as elements of abstract extensions of the rational numbers, not as particular real or complex numbers. We describe in this paper the various design choices that were made, and the current state of the package, as...

We are concerned with factoring polynomials over algebraic extensions of the rationals, and have implemented a variant of Trager's [1976] algorithm, which reduces this problem to that of factoring polynomials over the integers, for which we use Wang's [1978] algorithm (as implemented in REDUCE [Hearn 82] by Moore and Norman [1981]). However, Trager...