Details:
Title  The circuit ideal of a vector configuration  Author(s)  T. Bogart, Anders N. Jensen, Rekha R. Thomas  Type  Article in Journal  Abstract  Given 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.  Keywords  Toric ideal, Circuit ideal, Initial ideal, Primary decomposition, Associated primes, Fans  ISSN  00218693 
URL 
http://www.sciencedirect.com/science/article/pii/S0021869306004984 
Language  English  Journal  Journal of Algebra  Volume  309  Number  2  Pages  518  542  Year  2007  Note  Computational Algebra  Edition  0  Translation 
No  Refereed 
No 
