Degree bounds for Gröbner bases of modules

Abstract

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.

There is an error in the proof of lemma 49 in version 2. To replace it, we quote a similar lemma with a larger bound (lemma 39 in this version)