Gröbner bases in Laurent polynomial rings and their applications

In this paper, we review the concept of a Laurent polynomial ring and present some new definitions and notations that help in computations within this ring. Next, we introduce the concept of Gröbner bases in the Laurent setting and discuss the challenges encountered during the computation of these bases. Finally, we explore the practical applications of this study, demonstrating how it can enhance the solving of a polynomial system, as well as the saturation of a binomial ideal with respect to the product of all variables.