Gröbner Bases and Convex Polytopes

Lectures on Algebraic Statistics

“…The first is that we need a termination criterion, and the second concerns the combinatorial explosion (which becomes serious for n − rank(L) ≥ 3) of having to look at many fibers until a termination criterion kicks in. The first problem can be addressed by deriving a general bound on the sizes of the coordinates of any element in the Graver basis of L. Such a bound is given in [87,Theorem 4.7,p. 33].…”

“…33]. However, a more conceptual solution for both problems can be given by recasting the Markov basis property in terms of commutative algebra [25,87]. This will be done in Theorem 1.3.6 below.…”

“…The notions of Gröbner bases and Graver bases are also derived from their algebraic analogues. For a detailed account see [87]. In that book, as well as in most statistical applications, the lattice L arises as the kernel of an integer matrix A.…”

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

“…3.3.2] and [10, Exercise 4.8.16]), as well as in connection with algebraic geometry in [8], [9], and [25].…”

GKZ Hypergeometric Structures

“…In Section 3.2 we demonstrate how the classical hypergeometric series of Gauss, Appell and Lauricella appear in the Γ-series format. In Section 3.3 we give estimates for the growth of the coefficients in (23). Formula (23) requires for γ ∈ Z k+1 choices of logarithms for u 1 , .…”