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

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)d$\mu$. We next extend this point of view to arbitrary compact basic semi-algebraic set S $\subset$ 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 gd$\mu$ 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.