Gröbner Bases and Syzygy Modules

Abstract

The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger\u27s algorithm, and syzygy modules. Specially, we present several equivalent definitions for Gröbner bases and prove how to compute a Gröbner basis for an ideal I of A = k[x1, x2, • • • , xn] generated by {fl, f2, • • • , f8} through Buchberger\u27s algorithm. As an application of Gröbner bases, we present a standard method (see [1]) to compute the syzygy module of a set {f1, f2, • • • , f8} of polynomials, illustrated with original examples. Finally, we implement these examples on the computer using the Mathematica package of [4]