# Difference between revisions of "ApCoCoA-1:Weyl.WRGB"

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− | This function converts a Weyl Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If <tt>GB</tt> is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list <tt>GB</tt> of Weyl polynomials using <ref>Weyl.WNR</ref> into a new list <tt>L</tt> such that <tt>Ideal(L) = Ideal(GB)</tt>, every polynomial is reduced with respect to the remaining polynomials in the list <tt>L</tt> and leading coefficient of each polynomial in <tt>L</tt> is 1. | + | This function converts a Weyl Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If <tt>GB</tt> is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list <tt>GB</tt> of Weyl polynomials using <ref>ApCoCoA-1:Weyl.WNR|Weyl.WNR</ref> into a new list <tt>L</tt> such that <tt>Ideal(L) = Ideal(GB)</tt>, every polynomial is reduced with respect to the remaining polynomials in the list <tt>L</tt> and leading coefficient of each polynomial in <tt>L</tt> is 1. |

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− | <see>Weyl.WNormalForm</see> | + | <see>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</see> |

− | <see>Weyl.WGB</see> | + | <see>ApCoCoA-1:Weyl.WGB|Weyl.WGB</see> |

− | <see>Weyl.WRGBS</see> | + | <see>ApCoCoA-1:Weyl.WRGBS|Weyl.WRGBS</see> |

− | <see>Weyl.WRedGB</see> | + | <see>ApCoCoA-1:Weyl.WRedGB|Weyl.WRedGB</see> |

− | <see>Introduction to Groebner Basis in CoCoA</see> | + | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |

</seealso> | </seealso> | ||

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## Revision as of 08:47, 7 October 2020

## Weyl.WRGB

Reduced Groebner basis of an ideal `I` in Weyl algebra `A_n`.

### Syntax

Weyl.WRGB(GB:LIST):LIST

### Description

This function converts a Weyl Groebner basis `GB` computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If `GB` is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list `GB` of Weyl polynomials using Weyl.WNR into a new list `L` such that `Ideal(L) = Ideal(GB)`, every polynomial is reduced with respect to the remaining polynomials in the list `L` and leading coefficient of each polynomial in `L` is 1.

@param

*GB*Groebner Basis of an ideal in the Weyl algebra.@result The reduced Groebner Basis of the given ideal.

#### Example

A1::=QQ[x,d]; --Define appropriate ring Use A1; L:=[x,d,1]; Weyl.WRGB(L); [1] -------------------------------

#### Example

A2::=ZZ/7[x[1..2],y[1..2]]; -- define appropriate ring Use A2; I:=Ideal(2x[1]^14y[1]^7,x[1]^2y[1]^3+x[1]^2-1,y[2]^7-1,x[2]^3y[2]^2-x[2]y[2]-3x[2]-1); GbI:=Weyl.WGB(I,0);Len(GbI); ------------------------------- -- CoCoAServer: computing Cpu Time = 0.485 ------------------------------- 42 -- size of complete GB of the ideal I ------------------------------- Time GbI:=Weyl.WRGB(GbI);Len(GbI); Cpu time = 9.61, User time = 10 ------------------------------- 11 ------------------------------- -- Done. ------------------------------- Time GbI:=Weyl.WRGBS(GbI);Len(GbI); -- Weyl.WRGBS() can now be used for calling same implementation in ApCoCoALib -- note that this speeds up the computations ------------------------------- -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Cpu time = 0.04, User time = 0 ------------------------------- 11 -- this is now size of reduced GB of the ideal I ------------------------------- -- Done. -------------------------------

### See also

Introduction to Groebner Basis in CoCoA