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We first interpret Pell's equation satisfied by Chebyshev polynomials foreach degree t, as a certain Positivstellensatz, which then yields for eachinteger t, what we call a generalized Pell's equation, satisfied by reciprocalsof Christoffel functions of ''degree'' 2t, associated with the equilibriummeasure $\mu$ of the interval [--1, 1] and the measure (1 -- x 2)d$\mu$. Wenext 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 theinterval [--1, 1]). Under some conditions, for each t the equation is satisfiedby reciprocals of Christoffel functions of ''degree'' 2t associated with (i)the equilibrium measure $\mu$ of S and (ii), measures gd$\mu$ for anappropriate set of generators g of S. These equations depend on the particularchoice 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 forthe equilibrium measures of the 2D-simplex, the 2D-Euclidean unit ball and unitbox. Interestingly, this view point connects orthogonal polynomials,Christoffel functions and equilibrium measures on one side, withsum-of-squares, convex optimization and certificates of positivity in realalgebraic geometry on another side.

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