Question: Does anyone know of interesting applications of [Gröbner bases](http://en.wikipedia.org/wiki/Gr%C3%B6bner_basis) to theoretical computer science?

Gröbner bases are used to solve multi-variate polynomial equations, an NP-hard problem in general. I was wondering whether some tractable special cases were used to provide efficient algorithms/constructions/proofs in TCS or TCS-related areas (combinatorics, coding theory).

Answer: 
Alekseyev and Pevzner use them in [this paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.142.3719&rep=rep1&type=pdf) to compute the $k$-break distance between two genomes in linear time. Loosely speaking, that distance is defined as the minimum number of times you need to cut a given genome into $k$ pieces and rearrange them in order to transform that genome into another given one.
Following <http://arxiv.org/pdf/1502.05912.pdf> sometimes grobner basis are used to decide isomorphism (when graphs are encoded by systems of equations). But this joins the use of grobner basis in refutating CNFS.