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Border bases for lattice ideals

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 Z n). 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 Z 2 .
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Submitted on : Tuesday, August 2, 2016 - 9:27:20 AM
Last modification on : Monday, August 29, 2016 - 1:48:38 PM
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  • HAL Id : hal-01350887, version 1
  • ARXIV : 1608.08004



Giandomenico Boffi, Alessandro Logar. Border bases for lattice ideals. MEGA'2015 (Special Issue), Jun 2015, Trento, Italy. ⟨hal-01350887⟩



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