Difference between revisions of "ApCoCoA-1:Weyl.WRGBS"
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<title>Weyl.WRGBS</title> | <title>Weyl.WRGBS</title> | ||
<short_description>Convert a Groebner basis of an ideal in Weyl algebra <tt>A_n</tt> in to its reduced Groebner Basis using corresponding implementation in ApCoCoALib.</short_description> | <short_description>Convert a Groebner basis of an ideal in Weyl algebra <tt>A_n</tt> in to its reduced Groebner Basis using corresponding implementation in ApCoCoALib.</short_description> | ||
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<seealso> | <seealso> | ||
| − | <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.WRGB</see> | + | <see>ApCoCoA-1:Weyl.WRGB|Weyl.WRGB</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> | ||
<types> | <types> | ||
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<key>Weyl.WRGBS</key> | <key>Weyl.WRGBS</key> | ||
<key>wrgbs</key> | <key>wrgbs</key> | ||
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Latest revision as of 10:39, 7 October 2020
| This article is about a function from ApCoCoA-1. |
Weyl.WRGBS
Convert a Groebner basis of an ideal in Weyl algebra A_n in to its reduced Groebner Basis using corresponding implementation in ApCoCoALib.
Syntax
Weyl.WRGBS(GB:LIST):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.
This function converts Groebner basis GB of and ideal I computed by ApCoCoAServer into the reduced Groebner Basis by using the corresponding implementation in ApCoCoaLib. 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 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. Therefore, if GB is a Weyl Groebner basis then output of this function will be reduced Groebner basis.
@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.WRGBS(L); ------------------------------- -- CoCoAServer: computing Cpu Time = 0 ------------------------------- [1] ------------------------------- -- Done. -------------------------------
Example
A2::=ZZ/(13)[x[1..2],y[1..2]]; -- define appropriate ring and activate it with Use command. Use A2; I:=Ideal(x[1]^13-1,x[1]^3+x[1]y[1]^2+y[1]-2); -- Done. ------------------------------- GbI:=Weyl.WGB(I,0); --computes complete GB of the ideal I ------------------------------- -- CoCoAServer: computing Cpu Time = 0.063 ------------------------------- -- Done. ------------------------------- Len(GbI); 11 ------------------------------- -- Done. ------------------------------- GbI:=Weyl.WRGBS(GbI);GbI;Len(GbI); ------------------------------- -- CoCoAServer: computing Cpu Time = 0.031 ------------------------------- [ y[1]^10 - 5x[1]^2y[1]^6 - 3x[1]y[1]^7 - y[1]^8 + 6x[1]^2y[1]^5 - x[1]y[1]^6 + y[1]^7 + 3x[1]^2y[1]^4 - 6x[1]y[1]^5 + 2y[1]^6 + x[1]^2y[1]^3 - 6x[1]y[1]^4 + 4y[1]^5 + x[1]^2y[1]^2 + 2x[1]y[1]^3 + 2x[1]^2y[1] + 2x[1]y[1]^2 - 2y[1]^3 + 3x[1]^2 - 3x[1]y[1] + 4y[1]^2 - 6x[1] - 6y[1] + 5, x[1]^2y[1]^7 - 4x[1]y[1]^8 + y[1]^9 - 6x[1]^2y[1]^6 - x[1]y[1]^7 + 2y[1]^8 + 4x[1]^2y[1]^5 - 3y[1]^7 - x[1]^2y[1]^4 - 2x[1]y[1]^5 - 5x[1]^2y[1]^3 - 4x[1]y[1]^4 + 4x[1]^2y[1]^2 - 3x[1]y[1]^3 - 6y[1]^4 - 5x[1]^2y[1] + 6x[1]y[1]^2 + y[1]^3 + 3x[1]^2 - 5x[1]y[1] + 3y[1]^2 + 4x[1] + y[1] - 5, x[1]y[1]^9 + 6x[1]y[1]^8 + 2y[1]^9 + 6x[1]^2y[1]^6 + 4x[1]y[1]^7 - 6y[1]^8 - 3x[1]^2y[1]^5 + 4x[1]y[1]^6 - 3x[1]^2y[1]^4 - 4x[1]y[1]^5 + 6y[1]^6 - 6x[1]y[1]^4 + 3y[1]^5 + x[1]^2y[1]^2 - 6x[1]y[1]^3 + y[1]^4 - 2x[1]^2y[1] + 6x[1]y[1]^2 - y[1]^3 + 3x[1]^2 - 2x[1]y[1] + 6y[1]^2 - 5x[1] + 2y[1] + 5, x[1]^3 + x[1]y[1]^2 + y[1] - 2] ------------------------------- 4 -- which is now size of reduced GB of the ideal I ------------------------------- -- Done. -------------------------------
See also
Introduction to Groebner Basis in CoCoA
