ApCoCoA-1:NC.TruncatedGB
NC.TruncatedGB
Degree truncated Groebner basis of a finitely generated two-sided ideal in a free associative K-algebra.
Syntax
NC.TruncatedGB(G:LIST, D:INT):LIST NC.TruncatedGB(G:LIST, D:INT, LoopBound:INT, IFlag:INT):LIST
Description
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 indeterminates) X and ordering Ordering through the functions NC.SetFp(Prime), NC.SetX(X) and NC.SetOrdering(Ordering), respectively, before calling the function. Default coefficient field is Q. Default ordering is 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 LISTs, i.e. as [[C1,W1],...,[Cs,Ws]] where C1,...,Cs are coefficients and W1,...,Ws are terms. Each term is represented as a STRING. For example, xy^2x is represented as "xyyx", unit is represented as an empty string "". Polynomial F=xy-yx is represented as F:=[[1,"xy"], [-1, "yx"]]. 0 polynomial is represented as an empty LIST [].
@param D: a positive integer which is the truncated degree of Groebner basis.
About the optional parameters: in some situations, D-truncated Groebner basis is huge. Thus instead of computing the whole D-truncated Groebner basis, the function has 2 optional parameters to control the computation. Note that at the moment all of the following 2 additional optional parameters must be used at the same time.
@param LoopBound: a positive integer which gives a bound for the main loop of Groebner basis computation. When it runs through the main loop LoopBound times, the function stops and returns a partial Groebner basis.
@param IFlag: a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If IFlag=0, the server prints nothing on the screen. If IFlag=1, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If IFlag=2, the server additionally prints current partial Groebner basis before each loop as well as the information when IFlag=1. Note that the initial idea of Flag is to use it as a tool for debugging and tracing the computing process.
@return: a LIST of polynomials, which is a D-truncated Groebner basis if (1)finite Groebner basis of degree no more than DegreeBound exists or (2)the computation reaches the degree D, and which is a partial Groebner basis otherwise.
Example
NC.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]; NC.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
