Low degree equations for phylogenetic group-based models

Abstract

Abstract. Motivated by phylogenetics, our aim is to obtain a system of low degree equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group G, we provide an explicit construction of codim X polynomial equations (phylogenetic invariants) of degree at most |G| that define the variety X on a Zariski open set U . The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result confirms [Mic12, Conjecture 7.9] and, on the set U , Conjectures 29 and 30 of [SS05].

Finite phylogenetic complexity ofZpand invariants forZ3Michałek1 2017European Journal of CombinatoricsSelf Cite2010View full textAdd to dashboardCite

“…For the class of G-models that includes all the models introduced in this article, on the level of projective schemes the bounds were obtained in [Mic13]. Finally, for group-based models, but only on Zariski open set, the bound of the degrees by |G| was proved in [CFSM14]. Our second main theorem is as follows.…”

“…Group-based Models polynomials defining: degree ≤ |G| [CFSM14] As one can see the higher the row, the finer algebraic properties are required. On the other hand columns to the right provide bigger and more general groups.…”

Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Z 3 Scheme-Theoretically

“…In the next two sections we assume that L or R contains at least two different pairs, e.g. (1,2) and (a, b). There are a few possibilities.…”

“…If u / ∈ {x, y} or v / ∈ {s, t} then we could use a quadric relation, e.g. (1, x, y) g1 + (u, 2) = (u, x, y) g1 + (1, 2), to get (1,2) or (a, b) in R and delete a pair. Hence, without loss of generality, we consider only the situation where u = x and v = s, that is, R contains (x, 2) and (s, b).…”

“…Hence the triple (1, x, y) g1 cannot be the only element of R containing 1 with g 1 . Assume that 1 appears with g 1 in a pair (1, e) ∈ R. If x = e then we apply the quadric relation (x, 2) + (1, e) = (x, e) + (1, 2) in R and delete (1,2). And if s = e, we can use the relation (s, b) + (1, e) = (1, b) + (s, e) in R and delete (1, b) after swapping elements of pairs in L as in Remark 4.1.…”