Abstract
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.
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References
Bayer, D., Popescu, S., Sturmfels, B.: Syzygies of unimodular Lawrence ideals. Journal für die Reine und Angewandte Mathematik 534, 169–186 (2001)
Conradi, C., Kahle, T.: Detecting binomiality. Adv. Appl. Math. 71, 52–67 (2015)
De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and Geometric Ideas in the Theory of Discrete Optimization. MPS-SIAM Series on Optimization, SIAM, Cambridge (2013)
Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics, Oberwolfach Seminars, A Birkhäuser book, vol. 39, Springer, Berlin (2009)
Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84(1), 1–45 (1996)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Grossman, J.W., Kulkarni, D.M., Schochetman, I.E.: On the minors of an incidence matrix and its Smith normal form. Linear Algebra Appl. 218(1995), 213–224 (1995)
Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45(3), 317–333 (2010)
Herzog, J., Macchia, A., Madani, S.S., Welker, V.: On the ideal of orthogonal representations of a graph. Adv. Appl. Math. 71, 146–173 (2015)
Hoşten, S., Shapiro, J.: Primary decomposition of lattice basis ideals. J. Symb. Comput. 29(4–5), 625–639 (2000)
Hoşten, S., Sullivant, S.: Ideals of adjacent minors. J. Algebra 277(2), 615–642 (2004)
Kahle, T.: Decompositions of binomial ideals. J. Softw. Algebra Geom. 4, 1–5 (2012)
Kahle, T., Miller, E.: Decompositions of commutative monoid congruences and binomial ideals. Algebra Number Theory 8(6), 1297–1364 (2014)
Kahle, T., Rauh, J., Sullivant, S.: Positive margins and primary decomposition. J. Commut. Algebra 6(2), 173–208 (2014)
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)
Miller, E.: Affine stratifications from finite misère quotients. J. Algebraic Comb. 37(1), 1–9 (2013)
Pérez Millán, M., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74, 1027–1065 (2012)
Rauh, J., Sullivant, S.: Lifting Markov bases and higher codimension toric fiber products. J. Symb. Comput. 74, 276–307 (2016). doi:10.1016/j.jsc.2015.07.003
Swanson, I.: On the embedded primes of the Mayr–Meyer ideals. J. Algebra 275, 143–190 (2004)
Windisch, T.: BinomialEdgeIdeals, a Macaulay2 package for (parity) binomial edge ideals.https://github.com/windisch/BinomialEdgeIdeals
Acknowledgments
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 [9]. 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.
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Kahle, T., Sarmiento, C. & Windisch, T. Parity binomial edge ideals. J Algebr Comb 44, 99–117 (2016). https://doi.org/10.1007/s10801-015-0657-3
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DOI: https://doi.org/10.1007/s10801-015-0657-3