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

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 <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

 [A_{k},A_{l}]=A_{k}A_{l}-A_{l}A_{k}  

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

  I(B_O)=\langle \tau \rangle.

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

    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 ideal and ring. Functions <commandref>TraceSyzFull</commandref>, <commandref>JacobiFull</commandref>, <commandref>LinIndep</commandref> and <commandref>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.