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NC.TruncatedGB

Compute truncated Groebner bases of finitely generated homogeneous two-sided ideals in a non-commutative polynomial ring.
Syntax
          
NC.TruncatedGB(G:LIST, DB:INT[, Optimize:INT, OFlag:INT]):LIST

          

Description
Given a word ordering and a homogeneous two-sided ideal I, a set of non-zero polynomials Gb is called a Groebner basis of I if the leading word set LW{Gb} 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 Gb | 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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions. About 2 optional parameters:

Example
Use QQ[x,y,z];
F1:=[[y,x,y],[-z,y,z]];
F2:=[[x,y,x],[-z,x,y]];
F3:=[[z,x,z],[-y,z,x]];
F4:=[[x,x,x],[y,y,y],[z,z,z],[x,y,z]];
G:=[F1,F2,F3,F4];
NC.TruncatedGB(G,6);

[[[y, z, x], [-z, x, z]], [[y, x, y], [-z, y, z]], [[x, y, x], [-z, x, y]], [[x^3], [x, y, z], [y^3], [z^3]], 
[[z, x, z, y], [-z^2, x, z]], [[x, z, y, z], [-z, x, y^2]], [[x^2, y, z], [x, y^3], [-x, z, x, z], [x, z^3], 
[-y^3, x], [-z^3, x]], [[z^2, x, y^2], [-z^2, x, z^2]], [[y, z^2, x, z], [-z, x, z^2, y]], [[y, z^2, x, y], 
[-z^2, x, z, x]], [[y, z, y^3], [y, z^4], [z, x, z, x^2], [z^2, x, z^2]], [[y, x, z, x, z], [-z, y, z^2, x]], 
[[x, z^2, x, z], [-z, x, y^2, x]], [[x, y^4], [x, y, z^3], [z, x, y^2, z], [z^3, x, y]], [[x^2, z, x, z], 
[x, y^3, x], [-x, z, x, z, x], [x, z^3, x], [-y^3, x^2], [-z^3, x^2]], [[x^2, y^3], [x^2, z^3], 
[-x, y, z, y, z], [-x, z, x, z, x], [-y^3, x^2], [-y^4, z], [-z^3, x^2], [-z^3, y, z]], [[x^2, z, x, y], 
[x, y, z, y, x], [y^4, x], [z^3, y, x]], [[z, x, z^2, y, z], [-z^2, x, z, x, y]], [[y, z^3, x, z], [-z, x, z^2, y^2]], 
[[y, z^3, x, y], [-z^2, x, z, x^2]], [[x, z^3, x, z], [-z, x, y, z, y, z]], [[x, y^2, z, y, z], [x, z, x, y^2, x], 
[-x, z, x, z, x, y], [x, z^3, x, y], [-y^3, x^2, y], [-z^3, x^2, y]], [[x^2, z^2, x, y], [x, y, z, y, x^2], 
[y^4, x^2], [z^3, y, x^2]], [[x^2, z^3, y], [x, y^3, z^2], [-x, y, z, y, z, y], [-x, z, x, y^2, z], 
[-x, z, x, z, x, y], [-x, z, x, z^3], [-x, z^3, x, y], [x, z^5], [-y^3, x^2, y], [-y^3, x, z^2], [-y^4, z, y], 
[-z^3, x^2, y], [-z^3, x, z^2], [-z^3, y, z, y]]]
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See Also