A Generalized Pell's equation for a class of multivariate orthogonal polynomials

TL;DR

This paper generalizes Pell's equation for multivariate orthogonal polynomials on various domains, linking it to positivity certificates, Christoffel functions, and extremal properties of orthonormal polynomials.

Contribution

It extends Pell's equation to multivariate orthogonal polynomials on general domains, connecting it to positivity certificates and extremal polynomial properties.

Findings

Establishes a multivariate Pell's equation for orthogonal polynomials.

Links the equation to positivity certificates in real algebraic geometry.

Shows the solution reflects extremal properties of orthonormal polynomials.

Abstract

We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary dimension. Moreover, we show that such an equation is strongly connected (i) to a certificate of positivity (from real algebraic geometry) on the domain, as well as (ii) to the Christoffel functions of the equilibrium measure on the domain. In addition, the solution to Pell's equation reflects an extremal property of orthonormal polynomials associated with an entropy-like criterion.