Computational Commutative Algebra and Combinatorics
The study of algebraic combinatorics on convex polytopes as well as of computational commutative algebra for monomials and binomials has been developed mainly by using Gröbner bases as a basic tool. In a word, a Gröbner basis is a finite set of polynomials with remarkable algebraic properties. The theory and computation of Gröbner bases have greatly contributed toward the research on computational aspect of commutative algebra and algebraic geometry, and have promoted the development of a wide variety of software syatems. Moreover, beyond the scope of pure mathematics, it turns out that Gröbner bases bring a profound impact on the application to statistics.
A convex polytope is a high-dimensional version of building blocks, such as the triangular prism, the triangular pyramid, or the cube. The algebraic combinatorics on convex polytopes is a field of research of counting the number of faces of a convex polytope and that of lattice points belonging to a convex polytope by using abstract algebra, in particular the modern theory of commutative algebra. An expository book on this field is my book "Commutative Algebra and Combinatorics" (in Japanese), Springer Tokyo, 1995. When we employ Gröbner bases in algebraic combinatorics, the exploration of ideals generated by monomials and ideals generated by binomials is essential. The recent development of monomial ideals is discussed in detail in the textbook by J. Herzog and T. Hibi, "Monomial Ideals," GTM 260, Springer, 2011.
- 1981 Graduate, Faculty of Science, Nagoya University
- 1985 Assistant, Faculty of Science, Nagoya University
- 1990 Assistant Professor, Faculty of Science, Hokkaido University
- 1991 Associate professor, Faculty of Science, Hokkaido University
- 1995 Professor, Faculty of Science, Osaka University
- 1996 Professor, Graduate School of Science, Osaka University
- 2002 Professor, Graduate School of Information Science and Technology, Osaka University
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