## Algebraic models

In this chapter we introduce the basic algebraic machinery: rings of polynomials, ideals, varieties and Gro¨bner bases. The eﬀort is motivated by a general deﬁnition of a model to be given in Section 2.1. In our deﬁnition of model, factors or inputs are denoted by the letter x,

responses or outputs are denoted by y, parametric functions denoted by θ or functions of θ. These are related by polynomial algebraic relations, possibly implicit. Another feature of the deﬁnition is that constraints of polynomial type may be included in the speciﬁcation of the model. Implicit models and the introduction of constraints can lead to the use of dummy variables. All this requires complex polynomial computations to be tackled with advanced tools from polynomial ring theory together with their computer implementation. In this algebraic framework, the parameters of the model as interpreted in

statistics are functions of any form with the restriction that they belong to a speciﬁed ﬁeld. For example, Q(θ1, . . . , θp) is the set of all rational functions in θ1, . . . , θp with rational coeﬃcients. Another example is Q(eθ1 , . . . , eθp), the set of all exponential rational functions. Parameters are treated as unknown quantities and in most of the cases they appear in linear form. The process of actual estimation of parameter values, given observed

values of factors and responses, will be formalised in Chapter 6. For the purpose of the present chapter it suﬃces to represent the statistical error (deviation from the model) by dummy variables. Our working algebraic space is k[x1, . . . , xs], the commutative ring of

all polynomials in the indeterminates x1, . . . , xs and with coeﬃcients in k, where k is a ﬁeld. Most of the time k will be the rational numbers Q, or a ﬁeld extension of Q. There is often no ordering on the indeterminates x1, . . . , xs, in particular

no ordering is necessary when talking of ideals in k[x1, . . . , xs] and varieties in ks (see Deﬁnition 5). Nevertheless in some algebraic and statistical situations it is necessary to order indeterminates.