Analytical Study of Cubature Formulas on a Sphere in Computer Algebra Systems

Abstract

The problem of finding the weights and nodes of cubature formulas of a given order on a unit sphere that are invariant under the icosahedral rotation groups (A.S. Popov’s problem) is studied analytically in computer algebra systems. Popov’s algorithm for reducing the problem to a system of nonlinear equations is implemented in the Sage computer algebra system. It is shown that, in Sage, difficulties with studying the resulting system of nonlinear algebraic equations arise starting from the order of approximation of 23. It is also shown that Popov’s problem of this order leads to a polynomial ideal whose Gröbner basis contains polynomials with extremely large integer coefficients, which makes it quite difficult to explore with the standard tools implemented in Sage. This basis was found in our computer algebra system GInv, the new version of which was made public by one of the authors of this article in 2021. This made it possible to fully describe the set of solutions of Popov’s problem in Sage. The exact solutions found in the article are compared with the solutions found numerically by Popov. The potential of using Popov’s problem as a test problem for systems specializing in computing the Gröbner basis is discussed.