  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.
