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Irreducible decomposition of binomial ideals

Published online by Cambridge University Press:  01 April 2016

Thomas Kahle
Faculty of Mathematics, Otto-von-Guericke Universität, Magdeburg, Germany email
Ezra Miller
Mathematics Department, Duke University, Durham, NC 27708, USA email
Christopher O’Neill
Mathematics Department, Texas A&M University, TX 77843, USA email
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Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

Research Article
© The Authors 2016 


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