Betti numbers of polynomial hierarchical models for experimental designs

Abstract

Polynomial models, in statistics, interpolation and other fields, relate an output ? to a set of input variables (factors), x?=?(x 1,..., x d ), via a polynomial ?(x 1,...,x d ). The monomials terms in ?(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 ?(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".