ApCoCoA-1:NCo.TruncatedGB

NCo.TruncatedGB

Compute a truncated Groebner basis of a finitely generated homogeneous two-sided ideal in a free monoid ring.

Syntax

NCo.TruncatedGB(G:LIST, DB:INT):LIST

Description

Given a word ordering and a homogeneous two-sided ideal I, a set of non-zero polynomials G is called a Groebner basis of I if the leading word set LW{G} generates the leading word ideal LW(I). Note that it may not exist finite Groebner basis of the ideal I. Moreover, let D be a positive integer. Then the set {g in G | Deg(g)<=D} is a Groebner basis of the ideal <f in I | Deg(f)<=D> and is called a D-truncated Groebner basis of I.

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.

Please set ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

• @param G: a LIST of non-zero homogeneous polynomials generating a two-sided ideal in K<X>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].

• @param DB: a positive INT, which gives a degree bound of S-polynomials (or obstructions) during the Buchberger enumerating procedure. The procedure will discard S-polynomials (or obstructions) whose degrees are larger than DB.

• @return: a LIST of polynomials, which is a truncated Groebner basis at degree DB with respect to the current word ordering if the enumerating procedure doesn't terminate due to reaching the loop bound LB, and is a partial Groebner basis otherwise.

Example

NCo.SetX(<quotes>xyz</quotes>);
F1:=[[1,<quotes>yxy</quotes>],[-1,<quotes>zyz</quotes>]];
F2:=[[1,<quotes>xyx</quotes>],[-1,<quotes>zxy</quotes>]];
F3:=[[1,<quotes>zxz</quotes>],[-1,<quotes>yzx</quotes>]];
F4:=[[1,<quotes>xxx</quotes>],[1,<quotes>yyy</quotes>],[1,<quotes>zzz</quotes>],[1,<quotes>xyz</quotes>]];
G:=[F1,F2,F3,F4];
NCo.TruncatedGB(G,6);
[[[1, <quotes>yzx</quotes>], [-1, <quotes>zxz</quotes>]], [[1, <quotes>yxy</quotes>], [-1, <quotes>zyz</quotes>]], [[1, <quotes>xyx</quotes>], [-1, <quotes>zxy</quotes>]], [[1, <quotes>xxx</quotes>], [1, <quotes>xyz</quotes>], [1, <quotes>yyy</quotes>], [1, <quotes>zzz</quotes>]], 
[[1, <quotes>zxzy</quotes>], [-1, <quotes>zzxz</quotes>]], [[1, <quotes>xzyz</quotes>], [-1, <quotes>zxyy</quotes>]], [[1, <quotes>xxyz</quotes>], [1, <quotes>xyyy</quotes>], [-1, <quotes>xzxz</quotes>], [1, <quotes>xzzz</quotes>], [-1, <quotes>yyyx</quotes>], 
[-1, <quotes>zzzx</quotes>]], [[1, <quotes>zzxyy</quotes>], [-1, <quotes>zzxzz</quotes>]], [[1, <quotes>yzzxz</quotes>], [-1, <quotes>zxzzy</quotes>]], [[1, <quotes>yzzxy</quotes>], [-1, <quotes>zzxzx</quotes>]], [[1, <quotes>yzyyy</quotes>], 
[1, <quotes>yzzzz</quotes>], [1, <quotes>zxzxx</quotes>], [1, <quotes>zzxzz</quotes>]], [[1, <quotes>yxzxz</quotes>], [-1, <quotes>zyzzx</quotes>]], [[1, <quotes>xzzxz</quotes>], [-1, <quotes>zxyyx</quotes>]], [[1, <quotes>xyyyy</quotes>], [1, <quotes>xyzzz</quotes>], 
[1, <quotes>zxyyz</quotes>], [1, <quotes>zzzxy</quotes>]], [[1, <quotes>xxzxz</quotes>], [1, <quotes>xyyyx</quotes>], [-1, <quotes>xzxzx</quotes>], [1, <quotes>xzzzx</quotes>], [-1, <quotes>yyyxx</quotes>], [-1, <quotes>zzzxx</quotes>]], [[1, <quotes>xxzxy</quotes>], 
[1, <quotes>xyzyx</quotes>], [1, <quotes>yyyyx</quotes>], [1, <quotes>zzzyx</quotes>]], [[1, <quotes>xxyyy</quotes>], [1, <quotes>xxzzz</quotes>], [-1, <quotes>xyzyz</quotes>], [-1, <quotes>xzxzx</quotes>], [-1, <quotes>yyyxx</quotes>], [-1, <quotes>yyyyz</quotes>], 
[-1, <quotes>zzzxx</quotes>], [-1, <quotes>zzzyz</quotes>]], [[1, <quotes>zxzzyz</quotes>], [-1, <quotes>zzxzxy</quotes>]], [[1, <quotes>yzzzxz</quotes>], [-1, <quotes>zxzzyy</quotes>]], [[1, <quotes>yzzzxy</quotes>], [-1, <quotes>zzxzxx</quotes>]], 
[[1, <quotes>xzzzxz</quotes>], [-1, <quotes>zxyzyz</quotes>]], [[1, <quotes>xyyzyz</quotes>], [1, <quotes>xzxyyx</quotes>], [-1, <quotes>xzxzxy</quotes>], [1, <quotes>xzzzxy</quotes>], [-1, <quotes>yyyxxy</quotes>], [-1, <quotes>zzzxxy</quotes>]], 
[[1, <quotes>xxzzzy</quotes>], [1, <quotes>xyyyzz</quotes>], [-1, <quotes>xyzyzy</quotes>], [-1, <quotes>xzxyyz</quotes>], [-1, <quotes>xzxzxy</quotes>], [-1, <quotes>xzxzzz</quotes>], [-1, <quotes>xzzzxy</quotes>], [1, <quotes>xzzzzz</quotes>], 
[-1, <quotes>yyyxxy</quotes>], [-1, <quotes>yyyxzz</quotes>], [-1, <quotes>yyyyzy</quotes>], [-1, <quotes>zzzxxy</quotes>], [-1, <quotes>zzzxzz</quotes>], [-1, <quotes>zzzyzy</quotes>]], [[1, <quotes>xxzzxy</quotes>], [1, <quotes>xyzyxx</quotes>], 
[1, <quotes>yyyyxx</quotes>], [1, <quotes>zzzyxx</quotes>]]]
-------------------------------

See also

NCo.GB

NCo.IsGB

NCo.LW

NCo.ReducedGB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer