Algebraic statistics of codings for fractional factorial designs 1 Giovanni Pistone Department of Mathematics - Politecnico di Torino, Italy Maria-Piera Rogantin ∗ Department of Mathematics - Universit` a di Genova, Italy Abstract Key words: Algebraic Statistics, Complex coding, Mixed designs, Regular fractions, Orthogonal Arrays 1 Introduction We discuss here the properties of a fractional factorial design with m fac- tors X 1 ,...,X m , where the set of treatments is described by the solutions of a system of polynomial equations with rational coefficients. The solutions themselves are not restricted to be rational. In fact they could be real or com- plex algebraic numbers. We discuss both qualitative and quantitative factors. The approach has been introduced in Pistone and Wynn (1996) and Pistone et al. (2001). Through a number of research papers, the original ideas have developed into a technology based on symbolic computation software, which we wish to illustrate with reference to original works and typical examples. A design is a list or array of treatments. We do not consider replication of treatments. As a consequence, a design is just a set of treatments. In our ∗ Corresponding author. Email addresses: giovanni.pistone@polito.it (Giovanni Pistone), rogantin@dima.unige.it (Maria-Piera Rogantin). 1 Partially supported by Italian grant PRIN03 coordinated by G. Consonni (Uni- versity of Pavia, Italy) 1