Difference between revisions of "Category:ApCoCoA-1:Package gbmr"

Package gbmr is designed to provide basic operations over monoid rings and compute Groebner bases of finite generated ideals.

For the field of rationals Q and a monoid M presented by a string rewriting system (Alphabet, Rs), where Alphabet is finite alphabet and Rs is set of relations, let Q[M] denote the ring of all finite formal sums (called Polynomials) a_{1}*w_{1}+ a_{2}*w_{2} +...+a_{n}*w_{n} with coefficients a_{i} in Q\{0} and terms w_{i} in M. This ring is called the monoid ring of M over Q.

Notice that in this package, each Polynomial is represented as a LIST of LISTs, which are pairs of term and corresponding coefficient. Polynomial:=[[Term,Coefficient],...,[Term, Coefficient]]. For example, polynomial F:=a+1 is represented as F:=[[1,"a"], [1,""]].

Let p, f be two non-zero polynomials in Q[M]. We say f prefix reduces p to q at a monomial a*t of p in one step, denoted by p-->_{f}q if

 (1) LT(f)w = t for some w in M, i.e., LT(f) is a prefix of t, and
 (2) q = p-a*LT(f)^{-1}*f*w.

A set G is said to be a Groebner basis with respect to the reduction -->, if <-->_{G} = Equiv_{Ideal(G)} and -->_{G} is confluent.

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.