**Supplementary Materials for "Identifiability of Two Tree Mixtures for Group-Based Models"**

This webpage contains supplementary material for the paper Identifiability of two tree mixtures for group-based models by Elizabeth Allman, Sonja Petrovic, John Rhodes, and Seth Sullivant. In particular, this page provides code in the computer algebra package Singular that verifies the claims in Theorem 22, Proposition* 24, Proposition* 25, and Example 26. Computations can be verified by downloading Singular and the files below, and running the command "Singular < filename" at the terminal prompt. We provide brief descriptions of what each file computes below, but refer the reader to the paper for the details on phylogenetic varieties, mixture models, etc. All computations were performed on a 2.5 GHz Intel MacBook Pro with 2 GB of memory, except for the computations for the K3P model, which were performed on a MacPro with 16 Intel Xeon 2.67GHz cores and 32GB of real memory with the new Intel ``Nehalem'' quadcore memory bus running Ubuntu Linux 2.6.28-13.44. Thanks to Erich Kaltofen for the use of the second computer, which was purchased by NSF grant DMS-0532140. All timings are in seconds. Note that your computation times will vary depending on your system and the precise values of the random numbers that are used. |

** Theorem 22 **

Description | File | Timing |

This file verifies the containment of phylogenetic varieties claimed in Theorem 22. | theorem22-varietycontainment.sing | 107 |

This file computes the primary decomposition described in the paragraphs following Theorem 22. | theorem22-stochEG.sing | 173 |

** Proposition* 24 **

Description | File | Timing |

This file verifies identifiability for the Jukes-Cantor model for a two-tree mixture on two different four leaf trees with randomly choosen parameters. | prop24-jc.sing | 74 |

This file would verify identifiability for the Kimura 2-parameter model for a two-tree mixture on two different four leaf trees. However, the computation did not terminate after running for 3 weeks. | prop24-k2p.sing | >1814400 |

This file would verify identifiability for the Kimura 3-parameter model for a two-tree mixture on two different four leaf trees. However, the computation did not terminate after running for 3 weeks. | prop24-k3p.sing | >1814400 |

** Proposition* 25 **

Description | File | Timing |

This file verifies that the parametrizing map is generically 12-to-1 for the Jukes-Cantor model for a two-tree mixture on the same four-leaf tree with randomly choosen parameters. | prop25-jc.sing | 3 |

This file verifies that the parametrizing map is generically 32-to-1 for the Kimura 2-parameter model for a two-tree mixture on the same four-leaf tree with randomly choosen parameters. Note however, that the model is identifiable up to label swapping, the extra preimages arising because of the high symmetry of the model in Fourier coordinates. | prop25-k2p.sing | 1 |

This file verifies that the parametrizing map is generically 128-to-1 for the Kimura 3-parameter model for a two-tree mixture on the same four-leaf tree with randomly choosen parameters. Note however, that the model is identifiable up to label swapping, the extra preimages arising because of the high symmetry of the model in Fourier coordinates. | prop25-k3p.sing | 42420 |

** Example 26 **

Description | File | Timing |

Example parameter values with 2 biologically relevant preimages. | example26-2stocRel.sing | 3 |

Example parameter values with 4 biologically relevant preimages. | example26-4stocRel.sing | 3 |

Example parameter values with 8 biologically relevant preimages. | example26-8stocRel.sing | 3 |