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

TL;DR

This paper explores generalized Pell's equations linked to Christoffel functions and equilibrium measures for compact sets, extending classical results to higher dimensions and complex geometric shapes through algebraic and optimization techniques.

Contribution

It introduces a new interpretation of Pell's equations via Positivstellensatz, extending to arbitrary semi-algebraic sets and connecting orthogonal polynomials with real algebraic geometry.

Findings

Pell's equations are satisfied by Christoffel functions for various sets.

The approach applies to classical shapes like the simplex, ball, and box.

Connections established between orthogonal polynomials, sum-of-squares, and convex optimization.

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)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.