29 research outputs found
Constructive degree bounds for group-based models
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. We prove that
for the 3-Kimura model, corresponding to the group G=Z2xZ2, 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 , such that for any tree, the associated projective scheme can be
defined by an ideal generated in degree at most d.Comment: Boundedness results for equations defining the projective scheme were
extended to G-models (including 2-Kimura and all JC
Toric geometry of the 3-Kimura model for any tree
In this paper we present geometric features of group based models. We focus
on the 3-Kimura model. We present a precise geometric description of the
variety associated to any tree on a Zariski open set. In particular this set
contains all biologically meaningful points. Our motivation is a conjecture of
Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura
model is generated
Obstructions to combinatorial formulas for plethysm
Motivated by questions of Mulmuley and Stanley we investigate
quasi-polynomials arising in formulas for plethysm. We demonstrate, on the
examples of and , that these need not be counting
functions of inhomogeneous polytopes of dimension equal to the degree of the
quasi-polynomial. It follows that these functions are not, in general, counting
functions of lattice points in any scaled convex bodies, even when restricted
to single rays. Our results also apply to special rectangular Kronecker
coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples;
v3: final version as in Electronic Journal of Combinatoric
Derived category of toric varieties with Picard number three
We construct a full, strongly exceptional collection of line bundles on the
variety X that is the blow up of the projectivization of the vector bundle
O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b
is a non-negative integer
Secant cumulants and toric geometry
We study the secant line variety of the Segre product of projective spaces
using special cumulant coordinates adapted for secant varieties. We show that
the secant variety is covered by open normal toric varieties. We prove that in
cumulant coordinates its ideal is generated by binomial quadrics. We present
new results on the local structure of the secant variety. In particular, we
show that it has rational singularities and we give a description of the
singular locus. We also classify all secant varieties that are Gorenstein.
Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous
results for the tangential variety.Comment: Some improvements to previous results, with other minor changes.
Updated reference
Local equations for equivariant evolutionary models
Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open “salmon conjecture”, see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states ¿, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied (“d-claw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.Peer ReviewedPostprint (author's final draft
Polynomial systems admitting a simultaneous solution
We provide a complete description of the ideal that serves as the resultant
ideal for n univariate polynomials of degree d. We in particular describe a set
of generators of this resultant ideal arising as maximal minors of a set of
cascading matrices formed from the coefficients of the polynomials,
generalising the classical Sylvester resultant of two polynomials.Comment: 9 page