For hybrid finite elements based, for example, on complementary energy functional, polynomial trial functions of stress fields are required. Systematic schemes are given for the 2D and 3D elasticity, respectively. For 2D problems, the paper shows that there are maximal four independent polynomials for the nth order homogeneous polynomial Airy stress functions: two for the first order, three for the second order, and four for the nth order (n not less than 3). For 3D problems, there are 3(2n + 1) independent polynomials for the nth order homogeneous polynomial displacement. The corresponding stress field can be given directly. As examples, explicit expressions of the corresponding independent polynomials are listed: up to the tenth order for the 2D case and up to the third order for the 3D case.