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We present a survey on the developments related to Groebner 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 Groebner bases. Always maintaining this friendly tradition, the project has grown and evolved, and the software has been completely rewritten. CoCoA offers Groebner bases for all levels of interest: from the basic, explicit call in the interactive system CoCoA-5, to problem-specific optimized implementations, to the computer--computer communication with the open source C++ software library, CoCoALib, 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.
A set of polynomials G in a polynomial ring S over a field is said to be a universal Gru007foebner basis, if G is a Gru007foebner basis with respect to every term order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that the set o
In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.
The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Grobner bases pass to the quotient and when they can be lifted. The main difficulty to overco
We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals
Let $R=oplus_{igeq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $dim R_2leq 3$ and that $K$ is algebraically closed of characteristic $ eq 2$. We show that $R$ is defined by a Grobner basis of quadrics with, ess