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Betti Numbers of Cut Ideals of Trees

Journal Name:

  • Journal of Algebraic Statistics

Publication Year:

  • 2013

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Abstract (Original Language): 
Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions of cut ideals of tree graphs. By employing basic methods from combinatorial topology, we obtain upper bounds for the Betti numbers of this type of ideals. These take the form of simple formulas on the number of vertices, which arise from the enumeration of induced subgraphs of certain incomparability graphs associated to the edge sets of trees.
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REFERENCES

References: 

[1] 4ti2 team. 4ti2 { a software package for algebraic, geometric and combinatorial problems
on linear spaces. Available at http://www.4ti2.de.
[2] A. Dochtermann and A. Engstrom. Algebraic properties of edge ideals via combinatorial
topology. Electron. J. Combin., 16(2), 2009. 24 pp.
[3] A. Engstrom. Cut ideals of K4-minor free graphs are generated by quadrics. Michigan
Math. J., 60(3):705{714, 2011.
[4] D. R. Grayson and M. E. Stillman. Macaulay2, a software system for research in
algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
[5] E. Miller and B. Sturmfels. Combinatorial Commutative Algebra, volume 227 of Graduate
Texts in Mathematics. Springer-Verlag, New York, 2005. 420 pp.
[6] U. Nagel and S. Petrovic. Properties of cut ideals associated to ring graphs. J. Commut.
Algebra, 1(3):547{565, 2009.
[7] S. Potka and C. Sarmiento. A python script for computing the number of labeled
occurrences of an induced subgraph in the incomparability graph of a boolean lattice
on n elements. Available at http://personal-homepages.mis.mpg.de/sarmient/.
[8] B. Sturmfels. Grobner Bases and Convex Polytopes, volume 8 of University Lecture
Series. American Mathematical Society, Providence, R.I., 1996. 162 pp.
[9] B. Sturmfels and S. Sullivant. Toric geometry of cuts and splits. Michigan Math. J.,
57:689{709, 2008.

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