Finite phylogenetic complexity and combinatorics of tables

Michałek, Mateusz; Ventura, Emanuele

doi:10.2140/ant.2017.11.235

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Toric Varieties and Phylogenetics2

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Cited by 4 publications

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“…The results of [MV17] imply that for finite abelian group G the function φ(G, ·) is eventually constant. The ensuing results would be a desired strengthening.…”

Section: Toric Varieties and Phylogeneticsmentioning

confidence: 99%

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Selected topics on toric varieties

Michałek

Advanced Studies in Pure Mathematics

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“…The results of [MV17] imply that for finite abelian group G the function φ(G, ·) is eventually constant. The ensuing results would be a desired strengthening.…”

Section: Toric Varieties and Phylogeneticsmentioning

confidence: 99%

“…Definition 11.2 (Polytope P G,n , [MV17], [SS05]). Consider the lattice M ∼ = Z |G| with a basis corresponding to elements of G. Consider M n with the basis e (i,g) indexed by pairs (i, g) ∈ [n]×G.…”

Section: Toric Varieties and Phylogeneticsmentioning

confidence: 99%

Selected topics on toric varieties

Michałek

Advanced Studies in Pure Mathematics

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When is a Polynomial Ideal Binomial After an Ambient Automorphism?

2018

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Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials x a − λx b with λ ∈ k, or by unital binomials (i.e., with λ = 0 or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions about a family I of ideals over an arbitrary base B. The main results in this general setting are algorithms to find the locus of points in B over which the fiber of I • is contained in the fiber of a second family I ′ of ideals over B;• defines a variety of dimension at least d;• is generated by binomials; or • is generated by unital binomials. A faster containment algorithm is also presented when the fibers of I are prime. The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. Applications include the setting where a second group T acts on affine space, in addition to G, in which case algorithms compute the set of G-translates of I• whose stabilizer subgroups in T have maximal dimension; or • that admit a faithful multigrading by Z r of maximal rank r. Even with no ambient group action given, the final application is an algorithm to• decide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible; in some cases they are closed.

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