Department of
Applied
Mathematics

Grobner bases

Gabor Hegedus

Abstract

The main theme of our work is the description of the Gr\"obner bases and the standard monomials of finite set of points of combinatorial significance. This line of research has its roots in linear algebra methods of combinatorics. My research related to incidence matrices and the polynomial subspace method, which have been developed and applied by Alon, Babai, Blokhuis, Frankl, Lov\'asz, Wilson, and many others. On the algebraic side we build on the general theory of Gr\"obner bases.

We found applications of Gr\"obner basis techniques (in particular, standard monomials and Gr\"obner reduction) to problems of extremal combinatorics. A conjecture of L. Babai and P. Frankl was solved this way. Our methods offer a new approach to the rank computation of inclusion matrices.