On the computation of parametric Gröbner bases for modules and syzygies

Abstract

This paper presents algorithms to compute parametric Grobner bases for modules and parametric syzygies. The theory of Grobner basis is by far the most important tool for computations in commutative algebra and algebraic geometry. The theory of parametric Grobner basis is also important to solve problems of ideals generated by parametric polynomials and submodule generated by parametric vectors. Several algorithms are known for computing parametric Grobner bases in polynomial rings. However, nobody has studied the extension of parametric Grobner bases to modules yet. In this paper we extend the theory of parametric Grobner bases to modules, and we describe an algorithm for computing syzygies of parametric polynomials (vectors). These algorithms have been implemented in the computer algebra system Risa/Asir.