Difference between revisions of "Category:ApCoCoA-1:Package bbsmingensys"

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Let \tau be the set of  defining equations of a border basis scheme.  
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Let Tau be the set of  defining equations of a border basis scheme.  
 
The package bbsmingensys contains programs, which focus on this generating set.  
 
The package bbsmingensys contains programs, which focus on this generating set.  
  
  
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Let O be an order ideal and let Len(O)=Mu. Let BO be its border and Len(BO)=Nu.(see <commandref>BB.Border</commandref> in the package border basis) Let k,l\in {1,...,N} and k\neq l. We denote a polynomial entry of a commutator operation   
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Let O be an order ideal and let Len(O)=Mu. Let BO be its border and Len(BO)=Nu.(see the functions BB. Border and BB.Box in the package borderbasis) Let k,l in {1,...,N} and k is not equal to l. We denote a polynomial entry of a commutator operation   
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  [A_{k},A_{l}]=A_{k}A_{l}-A_{l}A_{k}  
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    [A_k,A_l]=A_k*A_l-A_l*A_k  
  
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in the position (p,q)  by \tau_{p\,q}^{k\,l} where p,q\in {1,...,Mu}.  These entries generate the vanishing ideal of the border basis scheme.We denote the set of such polynomials by  \tau, thus we have  
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in the position (p,q)  by tau_pq^kl where p,q in {1,...,Mu}.  These entries generate the vanishing ideal of the border basis scheme.We denote the set of such polynomials by  Tau, thus we have  
  
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  I(B_O)=\langle \tau \rangle.
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    I(B_O)= < Tau >.
  
  
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Further we shall consider the I(B_O) as a K[c]-module, where the entries of the commutator operation [A_{k},A_{l}] are a Mu x Mu -tuple ( \tau_{p\,q}^{k\,l})_{1<= p,q<= Mu)  from  I(B_O). Therefore, throughout this package  we will represent every element from  ( \tau_{p\,q}^{k\,l})_{1<= p,q<= Mu) as an indeterminate t[k,l,p,q]. And we construct the ring   
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Further we will represent every element from Tau ( tau_pq^kl_{1<= p,q<= Mu,1<=k<l<=N} ) as an indeterminate t[k,l,p,q]. Therefore, we construct the ring   
 
     XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]  
 
     XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]  
 
with respect to the order ideal and its border defined on R=K[x_1,..,x_N]. Throughout this package, defining this ring exactly  as given in the examples is crucial.  
 
with respect to the order ideal and its border defined on R=K[x_1,..,x_N]. Throughout this package, defining this ring exactly  as given in the examples is crucial.  
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*M.Kreuzer and L.Robbiano, Deformations of Border Basis, Coll Math. 59 (2008),275-297.
 
*M.Kreuzer and L.Robbiano, Deformations of Border Basis, Coll Math. 59 (2008),275-297.
  
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[[Category:ApCoCoA_Manual]]
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[[Category:ApCoCoA-1 Manual]]

Latest revision as of 15:17, 2 October 2020

Let Tau be the set of defining equations of a border basis scheme. The package bbsmingensys contains programs, which focus on this generating set.


Let O be an order ideal and let Len(O)=Mu. Let BO be its border and Len(BO)=Nu.(see the functions BB. Border and BB.Box in the package borderbasis) Let k,l in {1,...,N} and k is not equal to l. We denote a polynomial entry of a commutator operation 

   [A_k,A_l]=A_k*A_l-A_l*A_k

in the position (p,q) by tau_pq^kl where p,q in {1,...,Mu}. These entries generate the vanishing ideal of the border basis scheme.We denote the set of such polynomials by Tau, thus we have 

    I(B_O)= < Tau >.


Further we will represent every element from Tau ( tau_pq^kl_{1<= p,q<= Mu,1<=k<l<=N} ) as an indeterminate t[k,l,p,q]. Therefore, we construct the ring 

    XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]

with respect to the order ideal and its border defined on R=K[x_1,..,x_N]. Throughout this package, defining this ring exactly as given in the examples is crucial.

NOTE: This package is designed for experimenting for some specific shape of order ideals and rings. Functions <commandref>BBSGen.TraceSyzFull</commandref>, <commandref>BBSGen.JacobiFull</commandref>, <commandref>BBSGen.LinIndep</commandref> and <commandref>BBSGen.BBFinder</commandref> may not give results due to the growth of indeterminates in XX or due to the growth of polynomials during matrix multiplications.





The global alias for this package is BBSGen.




References

  • M.Huibregtse, Some Syzygies of the Generators of the Ideal of a Border Basis Scheme,??? (2009),
  • M.Kreuzer and L.Robbiano, Deformations of Border Basis, Coll Math. 59 (2008),275-297.
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