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

@article{CRMATH_2023__361_G5_935_0,
     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},
}

TY  - JOUR
AU  - Jean B. Lasserre
TI  - Pell’s equation, sum-of-squares and equilibrium measures on a compact set
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 935
EP  - 952
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.465
LA  - en
ID  - CRMATH_2023__361_G5_935_0
ER  -

%0 Journal Article
%A Jean B. Lasserre
%T Pell’s equation, sum-of-squares and equilibrium measures on a compact set
%J Comptes Rendus. Mathématique
%D 2023
%P 935-952
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.465
%G en
%F CRMATH_2023__361_G5_935_0

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/

Bibliographie
Cité par

[1] Eric Bedford; Bert A. Taylor The complex equilibrium measure of a symmetric convex set in $ℝ^{n}$ , Trans. Am. Math. Soc., Volume 294 (1986) no. 2, pp. 705-717 | Zbl

[2] Iain Dunning; Joey Huchette; Miles Lubin JuMP: A Modeling Language for Mathematical Optimization, SIAM Rev., Volume 59 (2017) no. 2, pp. 295-320 | DOI | Zbl

[3] Michael Grant; Stephen Boyd CVX: Matlab Software for Disciplined Convex Programming, version 2.1, http://cvxr.com/cvx, 2014

[4] Jean B. Lasserre Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2015 | DOI | Zbl

[5] Jean B. Lasserre A disintegration of the Christoffel function, C. R. Math. Acad. Sci. Paris, Volume 360 (2022), pp. 1071-1079 | Zbl

[6] Jean B. Lasserre; Edouard Pauwels; Mihai Putinar The Christoffel–Darboux Kernel for Data Analysis, Cambridge Monographs on Applied and Computational Mathematics, 38, Cambridge University Press, 2022 | DOI

[7] James Mc Laughlin Multivariable-polynomial solutions to Pell’s equation and fundamental units in real quadratic fields, Pac. J. Math., Volume 210 (2002) no. 2, pp. 335-348 | DOI

[8] Yurii Nesterov Squared functional systems and optimization problems, High Performance Optimization (Hans Frenk; Kees Roos; Tamás Terlaky; Shuzhong Zhang, eds.) (Applied Optimization), Volume 33, Springer, 2000, pp. 405-440 | DOI | Zbl

[9] Ngoc Hoang Anh Mai; Jean B. Lasserre; Victor Magron; Jie Wang Exploiting constant trace property in large scale polynomial optimization (2020) (https://arxiv.org/abs/2012.08873, to appear in ACM Trans. Math. Softw.)

[10] Mihai Putinar Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., Volume 42 (1993) no. 3, pp. 969-984 | DOI | Zbl

[11] William A. Webb; Hisashi Yokota Polynomial Pell’s equation, Proc. Am. Math. Soc., Volume 131 (2003) no. 4, pp. 993-1006 | DOI | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

A disintegration of the Christoffel function

Jean B. Lasserre

C. R. Math (2022)

Équation de Pell–Abel et applications

Quentin Gendron

C. R. Math (2022)

On the Christoffel function and classification in data analysis

Jean B. Lasserre

C. R. Math (2022)