A non-regular Groebner fan

Details A non-regular Groebner fan A non-regular Groebner fan

2005-01-21

arxiv.org/abs/math/0501352v1

Discrete Comput. Geom. 37 (2007), no, 443--453

Mathematics

Combinatorics

11 pages

Scientific paper

10.1007/s00454-006-1289-0

The Groebner fan of an ideal $I\subset k[x_1,...,x_n]$, defined by Mora and Robbiano, is a complex of polyhedral cones in $R^n$. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of $I$ as the term order varies. If $I$ is homogeneous the Groebner fan is complete and is the normal fan of the state polytope of $I$. In general the Groebner fan is not complete and therefore not the normal fan of a polytope. We may ask if the restricted Groebner fan, a subdivision of $R_{>=0}^n$, is regular i.e. the normal fan of a polyhedron. The main result of this paper is an example of an ideal in $Q[x_1,...,x_4]$ whose restricted Groebner fan is not regular.

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