Abstract
We address the problem of studying the toric ideals of phylogenetic invariants for a general group-based model on an arbitrary claw tree. We focus on the group ℤ2 and choose a natural recursive approach that extends to other groups. The study of the lattice associated with each phylogenetic ideal produces a list of circuits that generate the corresponding lattice basis ideal. In addition, we describe explicitly a quadratic lexicographic Gröbner basis of the toric ideal of invariants for the claw tree on an arbitrary number of leaves. Combined with a result of Sturmfels and Sullivant, this implies that the phylogenetic ideal of every tree for the group ℤ2 has a quadratic Gröbner basis. Hence, the coordinate ring of the toric variety is a Koszul algebra.
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References
Allman, E., Rhodes, J.: Phylogenetic ideals and varieties for the general Markov model. Advances in Applied Mathematics, Preprint, arXiv.org:math/0410604 (to appear)
Allman, E., Rhodes, J.: Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites. Preprint, arXiv.org:q-bio/0702050 (2007)
Casanellas, M., Fernandez-Sanchez, J.: Performance of a new invariants method on homogeneous and non-homogeneous quartet trees. Preprint, arXiv.org:q-bio/0610030 (2006)
Conca, A., Rossi, M.E., Valla, G.: Gröbner flags and Gorenstein algebras. Compositio Math. 129 27(1), 95–121 (2001)
Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. In: Eisenbud, D. (ed.) Graduate Texts in Mathematics, vol. 150, Springer, Heidelberg (1995)
Eriksson, N., Ranestad, K., Sturmfels, B., Sullivant, S.: Phylogenetic Algebraic Geometry. In: Projective varieties with unexpected properties, Ciliberto, C., Geramita, A., Harbourne, B., Roig, R.-M., Ranestad, K. (eds.) De Gruyter, Berlin, pp. 237–255 (2005)
Eriksson, N., Yao, Y.: Metric learning for phylogenetic invariants. Preprint, arXiv.org:q-bio/0703034 (2007)
Hemmecke, R., Malkin, P.: Computing generating sets of lattice ideals. Preprint, arXiv.org:math/0508359 (2005)
Pachter, L., Sturmfels, B.: Algebraic statistics for computational biology. Cambridge University Press, New York, NY, USA (2005)
Sturmfels, B.: Gröbner bases and convex polytopes. American Mathematical Society, University Lecture Series 8 (1996)
Sturmfels, B., Sullivant, S.: Toric ideals of phylogenetic invariants. J. Comp. Biol. 12, 204–228 (2005)
Sturmfels, B., Sullivant, S.: Toric geometry of cuts and splits. Preprint, arXiv.org:math.AC/0606683 (2006)
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Chifman, J., Petrović, S. (2007). Toric Ideals of Phylogenetic Invariants for the General Group-Based Model on Claw Trees K 1,n . In: Anai, H., Horimoto, K., Kutsia, T. (eds) Algebraic Biology. AB 2007. Lecture Notes in Computer Science, vol 4545. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73433-8_22
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DOI: https://doi.org/10.1007/978-3-540-73433-8_22
Publisher Name: Springer, Berlin, Heidelberg
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