Complex linear models for general fractional factorial designs Modelli lineari complessi per piani fattoriali frazionari generali Giovanni Pistone Maria Piera Rogantin 1 DIMAT Dima Politecnico di Torino Universit` a di Genova pistone@calvino.polito.it rogantin@dima.unige.it Riassunto: I livelli di un fattore possono essere rappresentati dalle radici n-esime dell’unit` a. In questo lavoro mostriamo come definire modelli lineari adatti a questa codifica e con- frontiamo tali modelli con quelli relativi alla codifica tradizionale 0, 1,...,n. Keywords: factorial design, complex design, polynomial ring, Gr¨ obner basis. 1. Introduction Confounding designs is a field in factorial where algebraic methods are widely used. However, the use of finite field algebra, introduced in Bose (1947), has not received much consideration in recent books. The classical monograph by Raktoe et al. (1981) and the recent one by Wu and Hamada (2000) rely more on linear algebra. The use of a different algebraic method, called commutative algebra or polynomial ring algebra, has been firstly advocated in Pistone and Wynn (1996) and developed in Pistone et al. (2001). The specific application to factorial designs has been the object of a series of papers: Fontana et al. (2000) on the binary case with the coding -1, +1, and Robbiano (1998), Robbiano and Rogantin (1998) on general designs. In the present paper, we use a complex coding of levels, where the k-th level of a n-level factor is represented by the complex number exp ( i 2π n k ) , i = √ -1. Complex linear models on group-generated designs were discussed in Kobilinsky (1990), while results on non-binary factorial designs are in Galetto et al. (2001). A basic dictionary of commutative algebra is given below. Main references are Cox et al. (1997) and Kreuzer and Robbiano (2000). Given a numerical coding of factor levels in a number field k, we identify a design D with the set of all solutions of a suitable system of polynomial equations. The set of all polynomials which are zero on the design points is called the design ideal, Ideal (D). Any set of polynomial equations defining (with multiplicity one) the same design D is called a set of generators of Ideal (D).A monomial ordering is a total ordering of monomials which is compatible with the product. Many such orderings exist. Given one of them τ , in each polynomial the maximal monomial is called the leading term. There exist special sets of generators, called Gr¨ obner bases, with the following prop- erty: the set of monomials which are not divided by any of the leading terms form a linear basis for the k-vector space of functions on the design. Theses bases are denoted by Est τ (D) because depend on τ only. By construction such a monomial basis is hierarchi- 1 Il lavoro ` e stato svolto con il contribuito del Politecnico di Torino (Ricerca di Base) e del CNR (PCS “Algoritmi algebrici per la pianificazione statistica degli esperimenti simulati”) – 707 –