Computing Gröbner bases of pure binomial ideals via submodules of \(\mathbb Z^n\) (Q432760)

This paper is about the computation of Gröbner bases of lattice ideals, that is binomial ideals of the form \(\langle x^u - x^v : u-v \in L\rangle\), where \(L\subset \mathbb{Z}^n\) is an integer lattice. It is obvious that the notion of a Gröbner basis can be defined combinatorially, and that computations can be done in \(\mathbb{Z}^n\) [\textit{R. Hemmecke} and \textit{P. N. Malkin}, J. Symb. Comput. 44, No. 10, 1463--1476 (2009; Zbl 1200.13048)].NEWLINENEWLINEInstead of working with the lattice \(L\subset \mathbb{Z}^n\), the present paper suggest to use a presentation \(F: \mathbb{Z}^m \to L \to 0\) and work in the presentation. Their method builds on a sign-consistent decomposition of \(L\) and Hilbert basis computations for the respective cones. There are theoretical reasons for the author's observation that this yields no interesting results or computational speed-ups: Gröbner bases of toric ideals (the argument also works for lattice ideals) have an exponential degree bound [\textit{B. Sturmfels}, Tôhoku Math. J., II. Ser. 43, No. 2, 249--261 (1991; Zbl 0714.14034)], while merely determining the size of a Hilbert basis is \#P-complete and NP-hard [\textit{M. Hermann, L. Juban} and \textit{P. G. Kolaitis}, Lect. Notes Comput. Sci. 1705, 13--32 (1999; Zbl 1044.11631)].NEWLINENEWLINEThe paper also uses notions subtly different from those established in the literature. In particular, in computational algebraic geometry an ideal \(I\subset k[x_1,\dots,x_n]\) is \textit{saturated} if \(I:\langle x_1,\dots,x_n\rangle = I\). The authors use the term for \(\left( I:\prod_{i=1}^n x_i\right ) = I\). A binomial ideal such that \(\left( I:\prod_{i=1}^n x_i\right ) = I\) is usually called a \textit{lattice ideal} and although the title suggests much more, the methods in the paper are restricted to this case.NEWLINENEWLINEAnybody interested in the computation of Gröbner bases of lattice ideals should read Hemmecke/Malkin, [loc. cit.] instead.