Betti numbers of polynomial hierarchical models for experimental designs
Polynomial models, in statistics, interpolation and other fields, relate an output eta to a set of input variables (factors), x = (x (1),..., x (d) ), via a polynomial eta(x (1),...,x (d) ). The monomials terms in eta(x) are sometimes referred to as "main effect" terms such as x (1), x (2), ..., or "interactions" such as x (1) x (2), x (1) x (3), ... Two theories are related in this paper. First, when the models are hierarchical, in a well-defined sense, there is an associated monomial ideal generated by monomials not in the model. Second, the so-called "algebraic method in experimental design" generates hierarchical models which are identifiable when observations are interpolated with eta(x) based at a finite set of points: the design. We study conditions under which ideals associated with hierarchical polynomial models have maximal Betti numbers in the sense of Bigatti (Commun Algebra 21(7):2317-2334, 1993). This can be achieved for certain models which also have minimal average degree in the design theory, namely "corner cut models".
betti numbers experimental design hilbert function grobner fan