a degree bound for the graver basis of non-saturated lattices

let $l$ be a lattice in $zz^n$ of dimension $m$. we prove that there exist integer constants $d$ and $m$ which are basis-independent such that the total degree of any graver element of $l$ is not greater than $m(n-m+1)md$. the case $m=1$ occurs precisely when $l$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. as a corollary, we show that the castelnuovo-mumford regularity of the corresponding lattice ideal $i_l$ is not greater than $rac{1}{2}m(n-1)(n-m+1)md$.