2006/02/26 by Batyrev, Victor · 1 citation
#13H10 #14M25 #52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0602593
Let Δ⊂ \Rn be an n-dimensional lattice polytope. It is well-known that hΔ^*(t) := (1-t)n+1 ∑k ≥ 0 |kΔ∩ \Zn| tk is a polynomial of degree d ≤ n with nonnegative integral coefficients. Let AGL(n, \Z) be the group of invertible affine integral transformations which naturally acts on \Rn. For a given polynomial h^* ∈ \Z[t], we denote by Ch^*(n) the number AGL(n, \Z)-equivalence classes of n-dimensional lattice polytopes such that h^* = hΔ^*(t). In this paper we show that \Ch^*(n) \n ≥ 1 is a monotone increasing sequence which eventually becomes constant. This statement follows from a more general combinatorial result whose proof uses methods of commutative algebra.