Seminars & Symposium

[10/31] The 5th WPI-AIMR Joint Seminar in FY2014

The 5th WPI-AIMR Joint Seminar in FY2014

Title

Gröbner bases of toric ideals and their application

Speaker

Prof. Hidefumi Ohsugi
(Department of Mathematical Sciences, Kwansei Gakuin University)

Date

October 31 (Fri.), 2014 16:00-17:00

Venue

Seminar Room, 2nd floor, WPI-AIMR Main Bldg.

Abstract

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., [4]. 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 [1]);
• Correspondence between regular triangulations [3] of integral convex polytopes and Gröbner bases of toric ideals (see [5]);
• 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 [2]).

In this talk, starting with introduction to Gröbner bases and toric ideals, we study some topics related with breakthroughs above.

References

• 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.

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