Characteristic Polynomial Criteria in Optimal Experimental Design L´ opez-Fidalgo J. Rodr´ ıguez-D´ ıaz J.M. Department of Pure and Applied Mathematics, University of Salamanca, E-37008 Spain ABSTRACT: Many optimality criteria have been used in the literature of experimental design. Two of the most common are A-optimality and D- optimality. They are the first and the last coefficients of the characteristic polynomial of the inverse information matrix. In this paper, criteria from the remaining coefficients are considered, and some properties are stud- ied. While A-optimality focuses on the average of the estimate variances and D-optimality focuses on all of the covariances, these criteria take into consideration the covariances considered in groups of two, three, four,... KEYWORDS: Characteristic Polynomial, Experimental Design, Informa- tion matrix, Optimality Criteria. 1 Introduction The criteria of A-optimality and D-optimality have been greatly used in optimal design theory. They come from the first and the last coefficients of the characteristic polynomial of the inverse of the information matrix. This paper is an introduction to the study of the remaining coefficients. All of them are invariant by linear transformations. They are also differentiable. While A-optimality centers on the average of the variances, D-optimality considers all of the variances and covariances together. The second coef- ficient, for instance, focuses on pairs of regression coefficients. Something similar can be said for the other. Let us introduce some notation. Let X be the set of all observable points and y(x) the observation at the point x. We assume the linear regression model y(x)= α t f (x)+ ε x , where E[ε x ]=0; var[y(x)] ≡ σ 2 (x);