Gröbner bases for everyone with CoCoA-5 and CoCoALib

Abstract

We present a survey on the developments related to Gröbner bases, and show explicit examples in CoCoA.The CoCoA project dates back to 1987: its aim was to create a "mathematician"-friendly computational laboratory for studying Commutative Algebra, most especially Gröbner bases. Always maintaining this "friendly" tradition, the project has grown and evolved, and the software has been completely rewritten.CoCoA offers Gröbner bases for all levels of interest: from the basic, explicit call in the interactive system , to problem-specific optimized implementations, to the computer-computer communication with the open source C++ software library, , or the prototype OpenMath-based server.The openness and clean design of CoCoALib and CoCoA-5 are intended to offer different levels of usage, and to encourage external contributions.

Extremal behavior in sectional matrices

“…Then rgin(I) = (x 3 , x 2 y 2 , xy 4 , y 6 ). See [3] for details about the computation of gin in CoCoA.…”

New characterizations of freeness for hyperplane arrangements

“…Then d min = 6 and d max = 9, and β 0,6 = 3, β 0,7 = 2 and β 0,8 = β 0,9 = 1. Using the construction of Theorem 9.8, we obtain (e 1 , e 2 , e 3 ) = (1,2,4). Consider now the arrangement A in K 3 defined by Q = x(x−y)(x−2y)(x−z)(x−2z)(x−3z)(x−4z), then B = rgin(J(A)).…”

Hyperplane arrangements in CoCoA

“…We describe the new package arrangements that computes several combinatorial invariants (like the lattice of intersections and its flats, the Poincaré, the characteristic and the Tutte polynomials) and algebraic ones (like the Orlik-Terao and the Solomon-Terao ideals) of hyperplane arrangement for the software CoCoA ( [1], [2] and [3]). Moreover, several functions for the class of free hyperplane arrangements are implemented.…”

“…** / use S ::= QQ[x,y,z]; / ** / A:=[x,y,z,x-y,x-y-z,x-y+2 * z]; / ** / A_1:=MultiArrRestrictionZiegler(A,z);A_1;[[y[1], 1], [y[2], 1], [y[1]-y[2], 3]] / ** / MultiArrDerMod(A_1); matrix( / * RingWithID(18, "QQ[y[1],y[2]]") * / [[y[1] * y[2], y[1]ˆ3], [y[1] * y[2], 3 * y[1]ˆ2 * y[2]-3 * y[1] * y[2]ˆ2+y[2]ˆ3]]) / ** / MultiArrExponents(A_1); [2, 3]/ ** / ArrExponents(A);[1,2,3] …”