Degree bounds for Gröbner bases of modules

Let $F$ be a non-negatively graded free module over a polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. Let $M$ be a submodule of $F$ generated by elements in $F$ with degrees bounded by $D$ and dim $F/M$=$r$. We prove that if $M$ is graded, the degree of the reduced Gröbner basis of $M$ for any term order is bounded by $2\left[1/2((Dm)^{n-r}m+D) \right]^{2^{r-1}}$. If $M$ is not graded, the bound is $2\left[1/2((Dm)^{(n-r)^2}m+D) \right]^{2^{r}}$. This is a generalization of Dubé(1990) and Mayr-Ritscher(2013)'s bounds for ideals in a polynomial ring.

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