Castelnuovo-Mumford regularity

[w-bruns]

Add Castelnuovo-Mumford regularity to computation goals (independently of Hilbert series).
More general: "minimal" point in interior of cone. Use optimization method similar to computation of subdivision point.

[mahrud]

How is regularity defined for a cone?

[w-bruns]

I think one cannot define CM regularity for a cone in general. Classically it is defined for a K-algebra R where K is a field, R is graded with R = K[R_1]. We can generalize this to the case in which R is a finitely generated module over K[R_1].

In the world of cones and polytopes this means that one can speak of CM regularity if P is a lattice polytope and C is the cone over it. In this case the monoid algebra K[M] where M is the monoid of lattice points in C is the type of algebra considered above.

For every cone with a grading one can compute the least degree of a lattice point in the interior. Let it be g and d = dim C. The CM regularity, if fined, then is d - g.

You can compute g by Normaliz via the Ehrhart series or, perhaps more efficiently, via the generators of the "ideal" of interior points. But both approaches are kind of an overkill, since you get much more information than asked for. I wrote this issue to remind me that another method should be introduced. I have this in mind, but I am not sure when it will come.