Pell’s equation, sum-of-squares and equilibrium measures on a compact set

Abstract

We first interpret Pell’s equation satisfied by Chebyshev polynomials for each degree $t$, as a certain Positivstellensatz, which then yields for each integer $t$, what we call a generalized Pell’s equation, satisfied by reciprocals of Christoffel functions of “degree” $2t$, associated with the equilibrium measure $\mu $ of the interval $[-1,1]$ and the measure $(1-x^2)\mathrm{d}\mu $. We next extend this point of view to arbitrary compact basic semi-algebraic set $S\subset \mathbb{R}^n$ and obtain a generalized Pell’s equation (by analogy with the interval $[-1,1]$). Under some conditions, for each $t$ the equation is satisfied by reciprocals of Christoffel functions of “degree” $2t$ associated with (i) the equilibrium measure $\mu $ of $S$ and (ii), measures $g\mathrm{d}\mu $ for an appropriate set of generators $g$ of $S$. These equations depend on the particular choice of generators that define the set $S$. In addition to the interval $[-1,1]$, we show that for $t=1,2,3$, the equations are indeed also satisfied for the equilibrium measures of the $2D$-simplex, the $2D$-Euclidean unit ball and unit box. Interestingly, this view point connects orthogonal polynomials, Christoffel functions and equilibrium measures on one side, with sum-of-squares, convex optimization and certificates of positivity in real algebraic geometry on another side.