Unimodular triangulations of simplicial cones by short vectors

TL;DR

This paper presents a new exponential bound on the length of vectors used in unimodular triangulations of simplicial cones, improving previous bounds by leveraging prime number theory and reduction techniques.

Contribution

It introduces a significantly improved exponential bound on vector lengths in unimodular triangulations, based on prime divisor reduction and number theory methods.

Findings

Bound is exponential in the square of the logarithm of multiplicity

Uses prime number theorem to control vector lengths

Significantly improves previous bounds

Abstract

We establish a bound for the length of vectors involved in a unimodular triangulation of simplicial cones. The bound is exponential in the square of the logarithm of the multiplicity, and improves previous bounds significantly. The proof is based on a successive reduction of the highest prime divisor of the multiplicity and uses the prime number theorem to control the length of the subdividing vectors.