Major research activity
The main research subject of our laboratory is “computational commutative algebra” and “discrete and computational geometry.” In particular, we are interested in Gröbner bases of toric ideals.
In 1965, Bruno Buchberger invented “Gröbner bases” of polynomial rings. (Wolfgang Gröbner was his adviser.) Roughly speaking, a Gröbner basis is a “good” set of polynomials. An important keyword is a division algorithm for polynomials. Gröbner bases have a lot of application in many research areas, and are implemented in various mathematical software (Mathematica, Maple, and a lot of free software). The most basic application is an elimination of variables from a system of polynomial equations. In 1990's, several breakthroughs on Gröbner bases of toric ideals were done:
- Conti-Traverso algorithm for integer programming using Gröbner bases of toric ideals;
- Correspondence between regular triangulations of integral convex polytopes and Gröbner bases of toric ideals;
- Diaconis-Sturmfels algorithm for Markov chain Monte Carlo method in the examination of a statistical model using a set of generators of toric ideals.
Major relevant publications
- S. Aoki, T. Hibi and H. Ohsugi, Markov chain Monte Carlo methods for the regular two-level fractional factorial designs and cut ideals, J. Statistical Planning and Inference 143, Issue 10 (2013) 1791-1806.
- T. Hibi, K. Nishiyama, H. Ohsugi and A. Shikama, Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases, J. Algebra 408 (2014), 138-146.
- H. Ohsugi, Normality of cut polytopes of graphs is a minor closed property, Discrete Math. 310 (2010), 1160-1166.
- H. Ohsugi and T. Hibi, Toric rings and ideals of nested configurations, Journal of Commutative Algebra 2 (2010), 187-208.
- H. Ohsugi and K. Shibata, Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part, Discrete and Computational Geometry 47 (2012), 624-628.