Abstract
We study phylogenetic invariants of general group-based models of evolution with group of symmetries \({\mathbb{Z}_3}\). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal \({I'}\) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that \({I = I'}\).
Highlights
We study phylogenetic invariants of general group-based models of evolution with group of symmetries Z3
We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal I generated by elements of degree at most 3 are the same
One of the most important questions in phylogenetic algebraic geometry, motivated by applications, is to determine the ideal of phylogenetic invariants, i.e., the ideal of polynomials vanishing on an algebraic variety corresponding to a model of evolution
Summary
One of the most important questions in phylogenetic algebraic geometry, motivated by applications, is to determine the ideal of phylogenetic invariants, i.e., the ideal of polynomials vanishing on an algebraic variety corresponding to a model of evolution. 29] is true, it implies that I = I Since comparing these two ideals is a difficult task, there have been a few attempts to compare geometric objects defined by them: projective schemes, sets of zeroes, or even sets of zeroes in the open orbit of a toric model. The saturation of I with respect to the irrelevant ideal is equal to I Note that this result implies the set-theoretic one: to check whether a point lies in the set of zeroes of the ideal of phylogenetic invariants for Z3 it is sufficient to see if the invariants of degree at most 3 vanish. Several examples illustrating these ideas are provided.
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