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

Jean B. Lasserre

doi:10.5802/crmath.465

Statistiques

Pell’s equation, sum-of-squares and equilibrium measures on a compact set
[Équation de Pell, sommes de carrés et mesure d’équilibre d’ensembles compacts]

Jean B. Lasserre ¹

¹ LAAS-CNRS and Institute of Mathematics, BP 54200, 7 Avenue du Colonel Roche, 31031 Toulouse cedex 4, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 935-952.

Résumé
Abstract

Nous fournissons d’abord une interprétation particulière de l’équation polynomiale de Pell satisfaite par les polynômes de Chebyshev. Pour chaque degré $t$ , il en découle une équation similaire satisfaite par les fonctions de Christoffel de la mesure d’équilibre $μ$ de l’intervalle $[- 1, 1]$ et de la mesure $(1 - x^{2}) d μ$ . Nous généralisons ensuite ce point de vue à des ensembles semi-algébriques compacts, et vérifions le résultat pour $t = 1, 2, 3$ sur la boule unité Euclidienne, la boite unité, et le simplex en dimension $2$ . Cette interprétation met en lumière une connection plutôt inattendue entre d’un coté, polynômes orthogonaux, fonctions de Christoffel et mesure d’équilibre, et de l’autre, optimisation convexe et certificats de positivité en géométrie algébrique réelle.

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” $2 t$ , associated with the equilibrium measure $μ$ of the interval $[- 1, 1]$ and the measure $(1 - x^{2}) d μ$ . 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” $2 t$ associated with (i) the equilibrium measure $μ$ of $S$ and (ii), measures $g d μ$ 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 $2 D$ -simplex, the $2 D$ -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.

Reçu le : 2022-10-05
Révisé le : 2022-12-08
Accepté le : 2023-01-12
Publié le : 2023-07-18

DOI : 10.5802/crmath.465

Classification : 42C05, 47B32, 33C47, 90C23, 90C46

Affiliations des auteurs :

Jean B. Lasserre ¹

¹ LAAS-CNRS and Institute of Mathematics, BP 54200, 7 Avenue du Colonel Roche, 31031 Toulouse cedex 4, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jean B. Lasserre},
     title = {Pell{\textquoteright}s equation, sum-of-squares and equilibrium measures on a compact set},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {935--952},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.465},
     language = {en},
}

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Jean B. Lasserre. Pell’s equation, sum-of-squares and equilibrium measures on a compact set. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 935-952. doi : 10.5802/crmath.465. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.465/

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