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arXiv · 1905.07517

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

Yihui Liang
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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.

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Yihui Liang. 2019-05-18. Degree bounds for Gr\"{o}bner bases of modules. https://doi.org/10.1016/j.jsc.2021.11.003

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