Constructive degree bounds for group-based models
Constructive degree bounds for group-based models
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基于组的模型的构造度界限
DOI：
10.1016/j.jcta.2013.06.003
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
M. Michałek
中科院分区：
文献类型：
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作者：
M. Michałek
基于群体的模型出现在代数统计学中，同时研究进化过程。它们由嵌入的环面代数簇表示。从理论和应用的观点来看，人们对确定定义簇的理想感兴趣。Sturmfels和Sullivant（2005）[25，Conjectures 29，30]给出了这些理想生成的度的猜测界。证明了对于3-Kimura模型，对应于群G= Z2 × Z2，投射概型可以由4次生成的理想定义.特别是，考虑4度系统发育不变量就足以测试给定点是否属于该变种。我们还研究了G-模型，一个推广的阿贝尔群为基础的模型。对于任意的G-模型，我们证明了存在一个常数d，使得对于任意的树，相关的投射方案可以由度至多为d的理想所定义。
Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant (2005)[25, Conjectures 29, 30]. We prove that for the 3-Kimura model, corresponding to the group G= Z 2× Z 2, the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant d, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d.