Parity binomial edge ideals
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Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gröbner bases and are radical if and only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.
KeywordsBinomial ideals Primary decomposition Mesoprimary decomposition Binomial edge ideals Markov bases
Mathematics Subject ClassificationPrimary 05E40 Secondary 13P10 05C38
The authors would like to thank Rafael Villareal for posting the question of radicality of parity binomial edge ideals. We thank Fatemeh Mohammadi for pointing us at . The authors appreciate the many comments and suggestions by Issac Burke and Mourtadha Badiane. T.K. and C.S. are supported by the Center for Dynamical Systems (CDS) at Otto-von-Guericke University Magdeburg. T.W. is supported by the German National Academic Foundation and TopMath, a graduate program of the Elite Network of Bavaria.
- 4.Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics, Oberwolfach Seminars, A Birkhäuser book, vol. 39, Springer, Berlin (2009)Google Scholar
- 6.Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
- 15.Miller, E.: Theory and applications of lattice point methods for binomial ideals. In: Proceedings of the Abel Symposium held at Voss, Norway, 2009, pp. 99–154. Springer (2011)Google Scholar
- 17.Pérez Millán, M., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74, 1027–1065 (2012)Google Scholar
- 20.Windisch, T.: BinomialEdgeIdeals, a Macaulay2 package for (parity) binomial edge ideals.https://github.com/windisch/BinomialEdgeIdeals