Diffusion-orthogonal polynomial systems of maximal weighted degree
A diffusion-orthogonal polynomial system is a bounded domain Ω in R d endowed with the measure μ and the second-order elliptic differential operator L , self adjoint w.r.t L 2 (Ω ,μ ) , preserving the space of polynomials of degree 6 n for any n . This notion was initially defined in , and 2 -dimensional models were classified. It turns out that the boundary of Ω is always an algebraic hypersurface of degree 6 2 d . It was pointed out in  that in dimension 2 , when the degree is maximal (so, equals 4 ), the symbol of L (denoted by g ij ) is a cometric of constant curvature. We present the self-contained classification-free proof of this property, and its multidimensional generalisation.