A new variant of the Stanley-Reisner ring and the number of bipartitions of finite point sets in real affine space, induced by hyperplanes

In my lecture, I will report on joint work with Hans-Juergen Bandelt, Jack Koolen and Victor Cepoy. It will be shown that given a finite set of points in the affine real space, the number of bipartitions induced on this set by hyperplanes not meeting this set depends only on the simplicial complex of affinely independent subsets of this set.

More precisely, it will be shown that this number coincides with the dimension of the quotient of the exterior algebra generated freely by the given finite set modulo the ideal generated by (a) all (exterior) products over its affinely dependent subsets and (b) the boundaries of those subsets. The obvious generalizations to oriented matroids hold true, too, and will be discussed as well as some further generalizations concerning the number of maps from a finite set into {+1,-1} satisfying some combinatorial sparseness conditions.