  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 $f(d) \in \mathcal{O}(d)$.
