Weakly Invariant Designs, Rotatable Designs and Polynomial Designs

In this article, we state a polynomial characterization of weakly invariant designs and show how to derive the construction of weakly invariant designs from the construction of rotatable designs using a new moment generating function introduced to that purpose. We apply the methods of computational commutative algebra to provide existence results for spherical rotatable polynomial designs in R-3. These designs can be of use in various settings such as response surface methodology or three-dimensional shape analysis.