Gröbner Fan

The Gröbner fan of a polynomial ideal is a polyhedral fan describing how its initial ideal changes with a real weight vector. For over a field , weight vectors and are equivalent iff

Following Fukuda et al. (2007), take the closures of the equivalence classes that contain a vector with all coordinates positive, together with every nonempty face of these polyhedral cones. The full-dimensional cones correspond to the distinct monomial ideals obtained as initial ideals for monomial orders, or equivalently to the reduced Gröbner bases with their leading terms specified. For a homogeneous ideal, the fan is complete. For a general ideal, intersecting these cones with the nonnegative coordinate region and its faces gives the restricted Gröbner fan.

For example, let in . Using greatest weight for initial forms, its initial ideals are when , when , and when , where . Its Gröbner fan consists of the two closed half-planes and their common boundary line.