Polynomial-size vectors are enough for the unimodular triangulation of simplicial cones

In a recent paper, Bruns and von Thaden established 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 authors mentioned that the next goal would be a bound that is polynomial in the multiplicity but not knowing if such a bound exists. In this paper we will prove that such a bound, which is polynomial in the multiplicity $\mu$, indeed exists. In detail, the bound is of the type $\mu^{f(d)}$ with $f(d) \in \mathcal{O}(d)$.