ApCoCoA-1:BBSGen.JacobiLin: Difference between revisions

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{{Version|1}}
<command>
<command>
   <title>BBSGen.TraceSyzStep</title>
   <title>BBSGen.JacobiLin</title>
   <short_description>:  This function computes  the K[c]-linear polynomial  entries of the Jacobi identity   [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] where m,k,l is from {1...n}.
   <short_description>This function computes  the K[c]-linear polynomial  entries of the Jacobi identity matrix [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] where m,k,l is from {1,...,N}.
</short_description>
</short_description>
    
    
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</syntax>
</syntax>
   <description>
   <description>
Let R=K[x_1,...,x_n] and A_k be the generic multiplication matrix associated to x_k. Let  Tau^kl_ij be the polynomial in the (i,j) position of the  [A_k,A_l] where k,l \in {1,..,n}.  
Let R:=K[x_1,...,x_N] and A_k be the generic multiplication matrix associated to x_k. Let OO be the order ideal and BO its order. Let Mu:=Len(OO) and Nu:= Len(BO). Let  tau^kl_ij be the polynomial in the (i,j) position of the  [A_k,A_l] where k,l in {1,..,N}. Let m,k,l in {1,...,N}.  
 
<par/>
Let m,k,l \in {1,...,n}. This function computes the polynomial entries of the Jacobi identity  J^{mkl}= [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] that has constant coeffiecients. During the computation  entries of the commutators Tau^kl_ij will be  considered as indeterminates  t[k,l,i,j]\in XX. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring CoeffRing[t[1..N,1..N,1..Mu,1..Mu]].
This function computes the polynomial entries of the Jacobi identity  J^mkl= [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k]] that has constant coeffiecients. During the computation  entries of the commutators Tau^kl_ij will be  considered as indeterminates  t[k,l,i,j] in K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring K[t[1..N,1..N,1..Mu,1..Mu]].


Please note that this function does not work for the case, where n=2.  
Please note that this function does not work for the case, where N=2.  




<itemize>
<itemize>
   <item>@param  Order ideal OO, border BO, the number of Indeterminates of the Polynomial.(see <commandref>BB.Border</commandref> in package borderbasis)
   <item>@param  Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].
 
</item>
</item>
   <item>@return  The K[c]-linear entries of the Jacobi Identity J^{ikl}.
   <item>@return  The K[c]-linear entries of the Jacobi Identity J^ikl.
.</item>
.</item>
</itemize>
</itemize>
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OO:=[1,x[1]];
OO:=[1,x[1]];
BO:=BB.Border(OO);
BO:=$apcocoa/borderbasis.Border(OO);
Mu:=Len(OO);
Mu:=Len(OO);
Nu:=Len(BO);
Nu:=Len(BO);
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Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  


BBSGen.BoxJacobiLin(OO,BO,N);
BBSGen.JacobiLin(OO,BO,N);




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   </description>
   </description>
   <types>
   <types>
     <type>borderbasis</type>
     <type>bbsmingensys</type>
     <type>ideal</type>
     <type>poly</type>
     <type>apcocoaserver</type>
     <type>apcocoaserver</type>
   </types>
   </types>
<see>BB.Border</see>
  <see>BB.Box</see>
<see>BBSGen.JacobiFull</see>
<see>BBSGen.JacobiStep</see>


  <key>Wmat</key>
<see>ApCoCoA-1:BBSGen.JacobiFull|BBSGen.JacobiFull</see>
   <key>BBSGen.Wmat</key>
<see>ApCoCoA-1:BBSGen.JacobiStep|BBSGen.JacobiStep</see>
   <key>bbsmingensys.Wmat</key>
 
   <wiki-category>Package_bbsmingensys</wiki-category>
   <key>BBSGen.JacobiLin</key>
   <key>JacobiLin</key>
 
 
   <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category>
</command>
</command>

Latest revision as of 09:50, 7 October 2020

This article is about a function from ApCoCoA-1.

BBSGen.JacobiLin

This function computes the K[c]-linear polynomial entries of the Jacobi identity matrix [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] where m,k,l is from {1,...,N}.

Syntax

BBSGen.JacobiLin(OO,BO,N);
BBSGen.JacobiLin(OO:LIST,BO:LIST,N:INTEGER):MATRIX

Description

Let R:=K[x_1,...,x_N] and A_k be the generic multiplication matrix associated to x_k. Let OO be the order ideal and BO its order. Let Mu:=Len(OO) and Nu:= Len(BO). Let tau^kl_ij be the polynomial in the (i,j) position of the [A_k,A_l] where k,l in {1,..,N}. Let m,k,l in {1,...,N}.

This function computes the polynomial entries of the Jacobi identity J^mkl= [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k]] that has constant coeffiecients. During the computation entries of the commutators Tau^kl_ij will be considered as indeterminates t[k,l,i,j] in K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring K[t[1..N,1..N,1..Mu,1..Mu]].

Please note that this function does not work for the case, where N=2.


  • @param Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].

  • @return The K[c]-linear entries of the Jacobi Identity J^ikl. .


Example

Use R::=QQ[x[1..3]];

OO:=[1,x[1]];
BO:=$apcocoa/borderbasis.Border(OO);
Mu:=Len(OO);
Nu:=Len(BO);
N:=Len(Indets());
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; 

BBSGen.JacobiLin(OO,BO,N);


[[   [    -t[2,3,1,2],0],
    [ t[2,3,1,1] - t[2,3,2,2], t[2,3,1,2]]]]


BBSGen.JacobiFull

BBSGen.JacobiStep




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