Gröbner Bases and Statistics

Abstract

Abstract This survey describes how to use methods of Algebraic Geometry and Commutative Algebra to study some problems arising in Design of Experiments, a branch of Statistics. Introduction Whenever two branches of Science meet it is a great time for research. Sometimes such events are unexpected, sometimes they are more natural. In this survey I will try to set up a foundation for a fruitful interaction between Design of Experiments (DoE) on one side and Commutative Algebra, Algebraic Geometry and Computer Algebra on the other. DoE is a branch of Statistics, and in its history algebraic methods are not rare, so this kind of interaction falls into the category of natural events. Furthermore, it is now clear how strong was, and still is, the influence of Computer Algebra methods in Commutative Algebra and Algebraic Geometry: for instance Grobner basis theory, which originated with the fundamental work by Buchberger (see Buchberger (1965), Buchberger (1970) and Buchberger (1985)), is nowadays recognized to be a basic tool applicable in several sectors of Science. From now on we can say that this influence extends into the realm of DoE. Let us have a look at the main concepts in DoE by discussing an example. Suppose that a firm wants to introduce a new product, say a portable telephone, into the market. They need to obtain useful a priori information from the potential customers.