Explicit consistency conditions for fully symmetric cubature on the tetrahedron

Abstract

A novel fully symmetric basis is derived for the S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_4$$\\end{document}-invariant polynomial 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 corresponding linear integer programming problem, optimal and quasi-optimal rule structures are derived. Explicit formulas to calculate the estimated lower bounds in the number of integration points are also given. Additionally, the new basis is of practical interest in calculating specific 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.