431 research outputs found
Geometry of Higher-Order Markov Chains
We determine an explicit Gr\"obner basis, consisting of linear forms and
determinantal quadrics, for the prime ideal of Raftery's mixture transition
distribution model for Markov chains. When the states are binary, the
corresponding projective variety is a linear space, the model itself consists
of two simplices in a cross-polytope, and the likelihood function typically has
two local maxima. In the general non-binary case, the model corresponds to a
cone over a Segre variety.Comment: 9 page
A sagbi basis for the quantum Grassmannian
The maximal minors of a p by (m + p) matrix of univariate polynomials of
degree n with indeterminate coefficients are themselves polynomials of degree
np. The subalgebra generated by their coefficients is the coordinate ring of
the quantum Grassmannian, a singular compactification of the space of rational
curves of degree np in the Grassmannian of p-planes in (m + p)-space. These
subalgebra generators are shown to form a sagbi basis. The resulting flat
deformation from the quantum Grassmannian to a toric variety gives a new
`Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum
Schubert calculus. The coordinate ring of the quantum Grassmannian is an
algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and
Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner
basis. This holds more generally for skew quantum Schubert varieties. These
results are well-known for the classical Schubert varieties (n=0). We also show
that the row-consecutive p by p-minors of a generic matrix form a sagbi basis
and we give a quadratic Gr\"obner basis for their algebraic relations.Comment: 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of
Gian-Carlo Rot
Elimination Theory in Codimension Two
New formulas are given for Chow forms, discriminants and resultants arising
from (not necessarily normal) toric varieties of codimension 2. Exact
descriptions are also given for the secondary polygon and for the Newton
polygon of the discriminant.Comment: 20 pages, Late
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