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

Jean-Bernard Lasserre (1) , Yuan Xu (2)

1 LAAS-POP - Equipe Polynomial OPtimization
2 Department of Mathematics, University of Oregon [Eugene]

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.

Keywords: MSC: 44A60 14Q30 42C05 90C25 90C46 32U15 14P10

Domains: Probability [math.PR]; Optimization and Control [math.OC]

Jean-Bernard Lasserre, Yuan Xu. A Generalized Pell's equation for a class of multivariate orthogonal polynomials. 2023. ⟨hal-04163153v1⟩