Border bases for lattice ideals (Q321275)

Border bases are a natural generalization of Gröbner bases. For example, the set of irreducible monomials with respect to a Gröbner basis forms an order ideal which is defined as a set of monomials posed under division. There is a natural way to construct a border basis from a given Gröbner basis. However, even for lattice ideals not all border bases come from Gröbner bases.NEWLINENEWLINEFor a \(0\)-dimensional ideal in the polynomial ring, a border basis is composed by an order ideal which is a basis of the coordinate ring of the ideal and a set of elements satisfying certain property. Thus the first step to construct a border basis is to find a monomial basis of the coordinate ring which forms an order ideal.NEWLINENEWLINEGiven a lattice in \(\mathbb{Z}^n\), the authors study the order ideals of the corresponding lattice ideal. They show that there are only finitely many order ideals corresponding to the associated lattice ideal.NEWLINENEWLINEThey give a complete and explicit description of all the border bases for lattice ideals whose corresponding lattice is a \(2\)-dimensional lattice in \(\mathbb{Z}^2\).