# Degree bounds for Gr\"{o}bner bases of modules

## TL;DR

This paper establishes explicit degree bounds for Gr"obner bases of submodules of free modules over polynomial rings, generalizing previous bounds for ideals and considering both graded and non-graded cases.

## Contribution

It provides new, explicit degree bounds for Gr"obner bases of modules, extending prior results from ideals to modules and covering graded and non-graded scenarios.

## Key findings

- Degree bounds depend on module parameters and are exponential in the module dimension.
- Bounds are tighter for graded modules compared to non-graded modules.
- Generalizes classical bounds for ideals to modules in polynomial rings.

## 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 Gr\"{o}bner 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 Dub\'{e}(1990) and Mayr-Ritscher(2013)'s bounds for ideals in a polynomial ring.