Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky, CMC)

The unique minimal generating set for a semigroup of lattice points in an n-dimensional polyhedral cone is called its Hilbert basis. In the case when the cone is spanned by lattice points, this Hilbert basis is finite, however its cardinality can be much larger than n. Nevertheless, a conjecture of Sebö states that each point in this semigroup can be spanned by no more than n elements of the Hilbert basis. We will brief discuss this conjecture and its current status, and then shift to the situation where the cone in question is the positive orthant in R^n. In this case the Hilbert basis consists of the so-called indecomposable elements, notation coming from the theory of universal quadratic forms over algebraic number fields. We will classify lattices that have finite versus infinite Hilbert basis of indecomposables. We will then discuss an analogue of Sebö’s conjecture for indecomposables in planar lattices and finish by presenting a counting estimate on the number of indecomposables of bounded norm in this 2-dimensional case. Joint work with Filiana Kostopoulou.