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TitleThe circuit ideal of a vector configuration
Author(s) T. Bogart, Anders N. Jensen, Rekha R. Thomas
TypeArticle in Journal
AbstractGiven a configuration A = a 1 , , a n ⊂ Z d , a basis ideal of A is an ideal J B = 〈 x u + − x u − : u ∈ B 〉 ⊂ k [ x 1 , , x n ] where B spans the lattice L A = u ∈ Z n : ∑ a i u i = 0 . Our main interest is to understand when the toric ideal, I A , of A equals a basis ideal J B with radical I A . The circuit ideal, J C A , of A is an example of such a basis ideal. We study such a J B in relation to I A from various algebraic and combinatorial perspectives with a special focus on J C A . We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of J B and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that the embedded primes of J B are indexed by certain faces of the cone spanned by A . We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to I A and J C A . The Gröbner fan of J C A is shown to refine that of I A when the codimension of the ideals is at most two.
KeywordsToric ideal, Circuit ideal, Initial ideal, Primary decomposition, Associated primes, Fans
URL http://www.sciencedirect.com/science/article/pii/S0021869306004984
JournalJournal of Algebra
Pages518 - 542
NoteComputational Algebra
Translation No
Refereed No