Overview

Empirical polynomials, i.e. polynomials with some coefficients of limited accuracy, have been introduced in Chapter 3; in the multivariate setting, they have been further considered in section 7.2. Everything said there now refers to the individual empirical polynomials ( p _?, e _?) of a system ( P, E) of such polynomials. For easy reference, we reformulate the definitions thus obtained:

Definition 9.1

(Compare Definitions 3.3 3.5 and Definition 7.1) A system ( P, E) = {( p _?, e _?). ? = 1(1) n} of empirical polynomials in s > 1 variables defines a family of neighborhoods N _?( P, E) of the system :

where N _? ( p _?, e _?) is defined by (7.11).

Note that we employ one common validity parameter ? for the n empirical polynomials in the system ( P, E). This makes it desirable that the individual tolerance vectors e _? are compatible in the following sense: For a fixed ? > 0, the neighborhoods N _?( p _?, e _?), ? = 1(1) n, include instances of polynomials of (approximately) the same validity.

Remember than an s-variate empirical polynomial ( p _?, e _?) has an empirical support which contains the subscript vectors j of those M _? coefficients which are empirical;