We describe three ways to generalise 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...
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1989 (v1)PublicationUploaded on: March 27, 2023
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2014 (v1)Publication
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 ...
Uploaded on: April 14, 2023 -
2006 (v1)Publication
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.
Uploaded on: April 14, 2023 -
2013 (v1)Publication
We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in $\ZZ[x]$; we include a new bound which was latent in a paper by Mignotte, and a few improvements to some existing bounds. We compare these bounds, and for each bound give explicit examples where that bound is the best; thus...
Uploaded on: March 31, 2023 -
2002 (v1)Publication
We answer a question left open in an article of Coppersmith and Davenport (Acta Arithmetica LVIII.1) 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...
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1991 (v1)Publication
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...
Uploaded on: April 14, 2023 -
2012 (v1)Publication
This article presents a finite precision (floating point) arithmetic with heuristic guarantees of correctness
Uploaded on: March 31, 2023 -
2006 (v1)Publication
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...
Uploaded on: April 14, 2023 -
2001 (v1)Publication
For a real square matrix $M$, Hadamard's inequality gives an upper bound $H$ for the determinant of $M$. This upper bound is sharp if and only if the rows of $M$ are orthogonal. In this paper 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...
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2010 (v1)Publication
Robbiano, Lorenzo; Abbott, John (Eds.); Series: Texts and Monographs in Symbolic Computation; Springer-Verlag Wien; ISBN 978-3-211-99313-2
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2009 (v1)Publication
Computer Algebra System (CAS) Open Source (GPL 3)
Uploaded on: April 14, 2023 -
2014 (v1)Publication
CoCoA is a ``well-established'' Computer Algebra System dating back to 1989. It was created as a laboratory for studying Computational Commutative Algebra, and specifically Groebner bases... and still today maintains this tradition. In the last few years CoCoA has undergone a profound change: the code has been totally re-written in C++, and is...
Uploaded on: March 27, 2023 -
2008 (v1)Publication
C++ Library Open Source (GPL 3) For Computations in Commutative Algebra
Uploaded on: April 14, 2023 -
2006 (v1)Publication
No description
Uploaded on: April 14, 2023 -
2017 (v1)Publication
C++ library for Computations in Commutative Algebra
Uploaded on: April 14, 2023 -
2012 (v1)Publication
First released in 1988 under the scientific direction of Lorenzo Robbiano, CoCoA is a special-purpose system for doing Computations in Commutative Algebra: i.e. it is an interactive system specialized in the algorithmic treatment of polynomials, and is freely available for most common platforms. About 10 years ago, a new initiative began:...
Uploaded on: April 14, 2023 -
2014 (v1)Publication
CoCoALib is a portable, high-quality, open source C++ software library released under the GPL licence; it forms the mathematical core of the CoCoA-5 computer algebra system. Both software products are available from http://cocoa.dima.unige.it/. The design and development of this software requires a solid knowledge of both basic and advanced...
Uploaded on: April 14, 2023 -
2013 (v1)Publication
CoCoALib is a portable, high-quality, open source C++ software library released under the GPL licence; it forms the mathematical core of the CoCoA-5 computer algebra system. Both software products are available from http://cocoa.dima.unige.it/. The design and development of this software requires a solid knowledge of both basic and advanced...
Uploaded on: April 14, 2023 -
2012 (v1)Publication
Open source C++ library for Computations in Commutative Algebra
Uploaded on: March 31, 2023 -
2010 (v1)Publication
First released in 1988, CoCoA is a freely available special-purpose system for doing Computations in Commutative Algebra. It belongs to an elite group of highly specialized systems (like "Macaulay2" and "Singular") having as their main forte the capability to calculate Groebner bases. This means that CoCoA is optimized for working...
Uploaded on: March 31, 2023 -
2011 (v1)Publication
Open source C++ library for Computations in Commutative Algebra
Uploaded on: March 31, 2023 -
2009 (v1)Publication
First released in 1988, CoCoA is a freely available special-purpose system for doing Computations in Commutative Algebra. It belongs to an elite group of highly specialized systems (like "Macaulay2" and "Singular") having as their main forte the capability to calculate Groebner bases. This means that CoCoA is optimized for working...
Uploaded on: March 31, 2023