The associated primes of initial ideals of lattice ideals

This paper concerns the associated primes and primary decompositions of the monomial initial ideals of lattice ideals. For a fixed initial ideal, we show that the multiplicities of its associated primes and its arithmetic degree are the cardinalities of sets of polytopes in which the origin is the unique lattice point. The associated primes are shown to exhibit a rare connectivity property: each embedded prime contains an associated prime of one higher dimension. The length of any such chain of associated primes can be bounded above by a function that depends only on the codimension of the lattice ideal. We express the unique irredundant irreducible decomposition of an initial ideal of a lattice ideal using maximal lattice point free polytopes defined by the lattice and the cost vector inducing the initial ideal.

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Published 1 January 1999