On the number of irreducible points in polyhedra

TL;DR

This paper establishes an upper bound on the number of irreducible integer points in high-dimensional polyhedra and applies this to determine the teaching dimension of threshold functions in k-valued logic.

Contribution

It provides a new bound on irreducible points in polyhedra and confirms a hypothesis about the teaching dimension in k-valued logic.

Findings

Bound on irreducible points: O(m^{floor(n/2)} log^{n-1} k)

Teaching dimension of threshold functions is Θ(log^{n-2} k)

Results hold for fixed n and general m, k

Abstract

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in $n$-dimensional polytope with radius $k$ given by a system of $m$ linear inequalities is at most $O(m^{\lfloor\frac{n}{2}\rfloor}\log^{n-1} k)$ if $n$ is fixed. Using this result we prove the hypothesis asserting that the teaching dimension in the class of threshold functions of $k$-valued logic in $n$ variables is $\Theta(\log^{n-2} k)$ for any fixed $n\ge 2$.