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 the total degree of any Graver element of $L$ is not greater than $m(n-m+1)MD$‎, ‎where the integer $M$ is defined by the set of circuits of $L$‎, ‎and the integer $D$ is defined by the saturation of $L$‎. ‎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 $\frac{1}{2}m(n-1)(n-m+1)MD$‎.