On the phenomena of constant curvature in the diffusion-orthogonal polynomials

Abstract

We consider the systems of diffusion-orthogonal polynomials, defined in the work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why these systems with boundary of maximal possible degree should always come from the group, generated by reflections. Our proof works for the dimensions 2 (on which this phenomena was discovered) and 3, and fails in the dimensions 4 and higher, leaving the possibility of existence of diffusion-orthogonal systems related to the Einstein metrics.The methods of our proof are algebraic / complex analytic in nature and based mainly on the consideration of the double covering of C d , branched in the boundary divisor.Author wants to thank Stepan Orevkov, Misha Verbitsky and Dmitry Korb for useful discussions.

Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree

“…So, we say that (g, Γ) is a solution of the (1, ∞)-AlgDOP Problem, if it is a solution of the (1, w)-AlgDOP Problem for some w > 2, and a coordinate change (5) with an arbitrary polynomial p(x) will be called (1, ∞)-admissible.…”

“…In [3] the following problem is studied (see also [1], [2], [5], [6]): describe all triples (Ω, L, µ) where Ω is a domain in R d , L is an elliptic second order operator of the form…”

“…curvature is constant and non-negative for all solutions with det(g) of maximal degree. For maximal degree solutions, Lev Soukhanov [5], [6] proved the constancy of curvature (but not its non-negativity) not using the classification. He also proved a generalization of this result to an arbitrary dimension.…”