The existence of Gaussian cubature formulas

We provide a necessary and sufficient condition for the existence of Gaussian cubature formulas. It consists of checking whether an overdetermined linear system has a solution and so complements Mysovskikh’s theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that the existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables, the worse it gets. And for fixed number of variables, the larger the precision, the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of the existence of a polynomial with very specific properties.