Explicit consistency conditions for fully-symmetric cubature on the tetrahedron

A novel fully symmetric basis is derived for the S4-invariant poly-nomial space, by using symmetric polynomials and invariant theory. This new basis enables deriving explicitly the consistency conditions for non-overdeterminedness of moment equations in the case of fully symmetric cubature rules on the tetrahedron. Solving the correspond-ing linear integer programming problem, optimal and quasi-optimal rule structures are derived. Explicit formulas to calculate the esti-mated lower bounds in the number of integration points are also given. Additionally, the new basis is of practical interest in calculating spe-cific cubature rules, since it allows decomposing the moment equations into a series of successively independent smaller subsystems, which can be exploited in designing more efficient solution methods. Solving the moment equations analytically we obtain several interesting new results.