Computation of moderate-degree fully-symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving

A novel method is presented for expressing the moment equations involved in computing fully symmetric cubature rules on the triangle, by using symmetric polynomials to represent the two kinds of invariance inherent in these rules. This method results in a system of polynomial equations that is amenable to solution using algebraic solving techniques; using Gröbner bases, rules of degree up to 15 are computed and presented, some of them new and with all their points inside the triangle.Since all solutions to the polynomial system are computed, it is for the first time possible to prove whether a given rule type results in specific rules of a given quality; it is thus proved that for degrees up to 14 there are no non-fortuitous rules that can improve on the presented results. For degree 10, an example is also provided showing how the proposed method can be used to exclude the existence of better fortuitous rules as well.