What is new in CoCoA? (slides) Conference Paper July 2015 CITATIONS - - PDF document

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What is new in CoCoA? (slides)

Conference Paper · July 2015

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1 author: Some of the authors of this publication are also working on these related projects: CoCoA and CoCoALib: software presentations, tutorials, demos View project SC-square: Symbolic Computation and Satisfiability Checking View project John Abbott Università degli Studi di Genova

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What is new in CoCoA?

http://cocoa.dima.unige.it/

• J. Abbott & A.M. Bigatti

Università di Genova, Italy

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 1 / 11

Overview

The CoCoA project dates back to 1987 under the lead of L. Robbiano. The aim was to create a mathematician-friendly software laboratory for studying Computational Commutative Algebra (multivariate polynomials). Main functions/operations Gröbner bases of ideals/modules with many term orderings Hilbert series, resolutions, Betti numbers special handling for ideals of points, monomial ideals, . . . ideal operations (e.g. elim, colon, primary decomp.) polynomial factorization basic exact linear algebra (LinSolve, LinKer, eigenvectors, det) approx points, border bases, approx polynomial relations

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 2 / 11

Software

Starting in 2000: brand new design and implementation in C++. Software: 3 interfaces an open source C++ software library, CoCoALib a new interpreter for the interactive system, CoCoA-5 a prototype OpenMath-based server, CoCoAServer CoCoA software relies on GMP library.

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 3 / 11

Software

CoCoA is designed to be easy to use . . . free and open source C++ code (GPL3 licence) source code is clean and portable (C++03) well-documented, incl. many examples function interfaces are natural for mathematicians design respects underlying mathematical structures execution speed is good with helpful error reporting . . . and easy to use correctly!

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 4 / 11

CoCoA-5 Language

The Interactive CoCoA System Natural mathematical syntax for lists

[ N in 1..50 | IsPrime(N) and IsPrime(N+2) ];

No declaration of variables (dynamically typed)

use QQ[x,y,z]; A := (x-y)^3 * (z-2); // type(A) is RINGELEM factor(A); A := mat([[x,y,z], [1,2,3], [0,x-1,1/2]]); // type(A) is MAT det(A); A := ideal(x^3 + x*y^2 - 2*z, ........ ); // type(A) is IDEAL GBasis(A);

Summary: simple imperative language; easy syntax;

• n-line manual.
• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 5 / 11

CoCoA-5 Language

New features in CoCoA-5 CoCoA-5 is mostly compatible with CoCoA-4, yet provides greater expressibility and a more solid mathematical basis. create different types of ring:

use R ::= QQ[a]; k := NewQuotientRing(R, ideal(a^2-2)); // k is QQ[a]/(a^2-2) P ::= k[x,y,z]; // P is QQ(sqrt(2))[x,y,z]

create ring homomorphisms

phi := CanonicalHom(R,k); // phi: QQ[a] --> QQ[a]/(a^2-2) theta := CanonicalHom(k,P)(phi); // composition R --> k --> P f := a^2 + 2*a - 1; // poly in R = QQ[a] 1/phi(f); // gives ((2/7)*a -1/7) in k 1/theta(f); // gives ((2/7)*a -1/7) in P

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 6 / 11

Extensions and Contributions

The openness and clean design of CoCoALib and CoCoA-5 are intended to encourage external contributions. Extending CoCoA-5 There are several ways of extending CoCoA-5: Add a new function written in CoCoA-5 language; Collect functions into a new CoCoA-5 package; Write the new functions in C++ inside CoCoALib, and then make them “visible” to CoCoA-5.

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 7 / 11

Extensions and Contributions

Direct contributions to CoCoALib Mathematical support and feedback (L. Robbiano) Gröbner bases structure and ideal/module operations (M. Caboara, 2005) Approximate points (M. L. Torrente and C. Fassino, 2008) Mayer-Vietoris trees (E. Sáenz de Cabezón, 2009) Janet and Pommaret Bases (M. Albert and W. Seiler, 2015) External libraries integrated with CoCoALib

• B. Roune: Frobby (monomial ideals)

experimental interface with GSL (GNU Scientific Library)

• C. Söger: Normaliz (affine monoids or rational cones)
• A. N. Jensen: GFan (Gröbner fans and tropical varieties) — soon
• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 8 / 11

Converting CoCoA-5 code to C++

Heron’s formula for the area of a triangle:

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 9 / 11

Converting CoCoA-5 code to C++

CoCoA-5 code:

use QQ[x[1],x[2],y, a,b,c,area]; defns := ideal(a^2 - (x[2]^2+y^2), // define length a b^2 - (x[1]^2+y^2), // define length b c

• (x[2]-x[1]),

// define length c area - (1/2)*c*y); // define area J := elim([x[1],x[2],y], defns); F := gens(J)[1]; println factor(F+16*area^2).factors;

CoCoALib/C++ code:

ring P = NewPolyRing(RingQQ(),SymbolList("x[1],x[2],y,a,b,c,area")); ideal defns(ReadExpr(P,"a^2 - (x[2]^2+y^2)"), ReadExpr(P,"b^2 - (x[1]^2+y^2)"), ReadExpr(P,"c

• (x[2]-x[1])"),

ReadExpr(P,"area - (1/2)*c*y")); ideal J = elim(SymbolList("x[1],x[2],y"), defns); RingElem F = gens(J)[0]; RingElem area = RingElem(P, symbol("area")); cout << factor(F+16*area*area).myFactors() << endl;

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 10 / 11

Web site

Web site:

http://cocoa.dima.unige.it/

Downloads: CoCoALib v. 0.99536 (July 2015) CoCoA v. 5.1.2 (?July 2015?) Also... redmine (bug/feature tracker), & other useful bits and bobs. Have fun with CoCoA! Thanks!

• J. Abbott & A.M. Bigatti

What is new in CoCoA? Bath, ISSAC 2015 11 / 11

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