Third Degree Integration Formulas with Four Real Points and Positive Weights in Two Dimensions

A theorem of Stroud and Mysovskikh ascertains that we can construct a two-dimensional integration formula of degree $N = 2m - 1$ with the $m^2 $ distinct, finite common zeros of two orthogonal polynomials $P1(x,y)$ and $P2(x,y)$ of degree m as points.This paper gives the pairs of orthogonal polynomials of second degree, dependent on a characteristic number introduced by Mysovskikh, whose zeros give rise to a third degree integration formula with four real points and positive weights.