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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[, LB:INT, OFlag: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 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. About 2 optional parameters: in some situations, truncated Groebner basis is too large for our applications. Thus, instead of computing the whole truncated Groebner basis, the function uses two optional parameters to control the enumerating procedure. Note that at the moment all of the following 2 optional parameters must be used at the same time.

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