Abstract
The main ingredient to construct an O-border basis of an ideal I⊆K[x1,…,xn] is the order ideal O, which is a basis of the K-vector space K[x1,…,xn]/I. In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal IM (where M is a lattice of Zn). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Gröbner bases. Finally, we give a complete and explicit description of all the border bases for ideals IM in case M is a 2-dimensional lattice contained in Z2.
Highlights
Let I be a zero-dimensional ideal in the polynomial ring K[x1, . . . , xn], a border basis of I is composed by a finite set O of monomials closed under division, which is a basis of the K-vector space K[x1, . . . , xn]/I and a set of polynomials f1, . . . , fm ∈ I, such that fi = bi − j aijtj, where tj ∈ O, aij ∈ K and bi are elements in the borderLogar). 1 Partially supported by the UNINT grant “Metodi relativi allo studio degli ideali polinomiali.” 2 Partially supported by the FRA 2013 grant “Geometria e topologia delle varieta”, Universita di Trieste, and by the PRIN 2010-2011 grant “Geometria delle varieta algebriche”.of O
In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal IM
For lattice ideals of positive dimension we show that, a border basis is infinite, it can be defined in finite terms
Summary
We do not focus our attention on the problem of determining zeros of systems of polynomials, but we consider a different question: we want to find all the border bases of a given lattice ideal. The construction we propose determines a finite graph whose maximal cliques allow to recover the required order ideals In this way we see that there are only finitely many order ideals and the ideal IM has only finitely many border bases. We show that in this case every border basis comes from a Grobner basis and we see that the results obtained in the previous sections allow a complete and explicit description of all the border (Grobner) bases of IM
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