Upper Bounds for the Betti Numbers of a given Hilbert Function

From a Macaulay's paper it follows that a lex-segment ideal has the greatest number of generators (the 0-th Betti number \b_0) among all the homogeneous ideals with the same Hilbert function. In this paper we prove that this fact extends to every Betti number, in the sense that all the Betti numbers of a minimal free resolution of a lex segment ideal are bigger than or equal to the ones of any homogeneous ideal with the same Hilbert function.Comment: 18 pages, plain te

Discovery of statistical equivalence classes using computer algebra

Discrete statistical models supported on labelled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.Comment: 26 pages, 9 figure

CoCoA and CoCoALib: Fast prototyping and flexible C++ library for computations in commutative Algebra

The CoCoA project began in 1987, and conducts research into Computational Commutative Algebra (from which its name comes) with particular emphasis on Gr\uf6bner 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. The software is open-source (GPL v.3), and under continual, active development. We give a summary of the features of the software likely to be relevant to the SC-Square community

New in CoCoA-5.2.2 and CoCoALib-0.99560 for SC-square

CoCoA-5 is an interactive Computer Algebra System for Computations in Commutative Algebra, particularly Gr\uf6bner bases. It offers a dedicated, mathematician-friendly programming language, with many built-in functions. Its mathematical core is CoCoALib, an opensource C++ library, designed to facilitate integration with other software. We give an overview of the latest developments of the library and of the system, in particular relating to the project SC-Square

Ideals modulo p

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 many primes are good for all term orderings. We relate our notion of sigma-good primes to some other similar notions already in the literature. One characteristic of our approach is that enables us to detect some bad primes, a distinct advantage when using modular methods

New in Cocoa-5.2.4 and Cocoalib-0.99600 for SC-square

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 SC2project. We give an overview of the latest developments of the library, especially those relating to the project SC2. The CoCoA software suite includes also the programmable, interactive system CoCoA-5. Most of the operations in CoCoALib are also accessible via CoCoA-5. The programmability of CoCoA-5 together with its interactivity help in fast prototyping and testing conjectures

Computing and Using Minimal Polynomials

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 be a “resolved problem”. But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality