Gröbner Systems Conversion

Testing Binomiality of Chemical Reaction Networks Using Comprehensive Gröbner Systems

“…Roughly speaking, the varieties V i and W i are typically obtained by considering the monomials that are vanished by specialisations, and simultaneously, using a Gröbner basis computation algorithm, a Gröbner basis under the conditions imposed by the specialisations is computed. Below we present a modified version of Kapur, et al's algorithm by Dehghani and Hashemi from [29]. "Other cases" in line 16 of the algorithm refers to those cases that the Gröbner basis is 1.…”

“…Dehghani and Hashemi group all those cases together with the aim of speeding the computations up. In line 13, MDBasis computes a minimal Dickson basis for a given set of polynomials in S. For more details, refer to [29].…”

“…We also computed a comprehensige Groöbner system of 2-phosphorylation in Maple, using Dehghani and Hashemi's PWWG package 8 , which uses a modification of Kapur et al's algorithm so that the branches with Gröbner basis {1} are ignored [29]. According to the authors' experiments in [29], this modification results in speed-up of the computations. However, even for 2-phosphorylation the computations did not finish in six hours in Maple.…”

“…Among the most efficient algorithms for computing comprehensive Gröbner bases are the algorithms given by Kapur et al [34,33]. Dehghani and Hashemi studied Gröbner walk and FGLM for comprehensive Gröbner bases [10,29] and implemented several algorithms for computing comprehensive Gröbner bases and related topics in Maple [29]. 4 To the best of our knowledge, to this date, comprehensive Gröbner bases have not been used in chemical reaction networks theory.…”