A Degroebnerization Approach to Algebraic Statistics

Abstract

In this paper, we describe a new variation of the interpolation algorithm by Möller, proposed in a way that completely avoids Gröbner bases and does not need a term order, but only a well order on terms. This algorithm takes a set of functionals describing a Macaulay chain, namely, roughly speaking, the functionals are chosen and ordered in such a way that the first functional defines a zero-dimensional ideal and all the sets one gets by adding the functionals one after the other define zero-dimensional ideals as well. Starting from this set, the algorithm describes the zero-dimensional ideals of the Macaulay chain via a basis of the quotient algebra and Auzinger-Stetter matrices. Our algorithm shows how Degroebnerization can give symmetric representations to design ideals, a crucial feature for Algebraic Statistics, showing also that such feature can always be attained without using Gröbner bases and Buchberger reduction. The paper further investigates the potential applications of our new algorithm to describe design ideals into non-commutative algebraic settings.