Gröbner bases of toric ideals and their application
大杉 英史 教授
Gröbner bases has a lot of application in many research areas, and is implemented in various mathematical software. The most basic application is an elimination of variables from a system of polynomial equations. See, e.g., . In this talk, we discuss basic and recent developments in the theory of Gröbner bases of toric ideals. In 1990's, several breakthroughs on toric ideals were done:
- Conti--Traverso algorithm for integer programming using Gröbner bases of toric ideals (see );
- Correspondence between regular triangulations  of integral convex polytopes and Gröbner bases of toric ideals (see );
- Diaconis--Sturmfels algorithm for Markov chain Monte Carlo method in the examination of a statistical model using a set of generators of toric ideals (see ).
In this talk, starting with introduction to Gröbner bases and toric ideals, we study some topics related with breakthroughs above.
- P. Conti and C. Traverso, Buchberger algorithm and integer programming, In Proceedings of AAECC-9 (New Orleans), pp.130--139. Springer LNCS 539, 1991.
- P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, The Annals of Statistics, 26 (1998) 363--397.
- I.M. Gel'fand, A.V. Zelevinskii, and M.M. Kapranov, Hypergeometric functions and toral manifolds, Functional Analysis and Its Applications, 23 (1989) 94--106.
- T. Hibi (ed.), ``Gröbner Bases: Statistics and Software Systems,'' Springer, 2013.
- B. Sturmfels, Gröbner bases of toric varieties, Tohoku Math. J., 43 (1991) 249--261.