A Non-Regular Grobner Fan

The Grobner fan of an ideal \(I\subset k[x_1,\dots,x_n]\), defined by Mora and Robbiano, is a complex of polyhedral cones in \({\Bbb R}^n\). 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 Grobner fan is complete and is the normal fan of the state polytope of I. In general the Grobner fan is not complete and therefore not the normal fan of a polytope. We may ask if the restricted Grobner fan, a subdivision of \({\Bbb R}_{\geq 0}^n\), is regular, i.e. the normal fan of a polyhedron. The main result of this paper is an example of an ideal in n = 4 variables whose restricted Grobner fan is not regular.

Authors and Affiliations

1. Institute for Mathematics and Its Applications, University of Minnesota, 400 Lind Hall, 207 Church Street S.E., Minneapolis, MN 55455-0436, USA

   Anders Nedergaard Jensen

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Correspondence to Anders Nedergaard Jensen.

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