We first interpret Pell’s equation satisfied by Chebyshev polynomials for each degree , as a certain Positivstellensatz, which then yields for each integer , what we call a generalized Pell’s equation, satisfied by reciprocals of Christoffel functions of “degree” , associated with the equilibrium measure of the interval and the measure . We next extend this point of view to arbitrary compact basic semi-algebraic set and obtain a generalized Pell’s equation (by analogy with the interval ). Under some conditions, for each the equation is satisfied by reciprocals of Christoffel functions of “degree” associated with (i) the equilibrium measure of and (ii), measures for an appropriate set of generators of . These equations depend on the particular choice of generators that define the set . In addition to the interval , we show that for , the equations are indeed also satisfied for the equilibrium measures of the -simplex, the -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.
Nous fournissons d’abord une interprétation particulière de l’équation polynomiale de Pell satisfaite par les polynômes de Chebyshev. Pour chaque degré , il en découle une équation similaire satisfaite par les fonctions de Christoffel de la mesure d’équilibre de l’intervalle et de la mesure . Nous généralisons ensuite ce point de vue à des ensembles semi-algébriques compacts, et vérifions le résultat pour sur la boule unité Euclidienne, la boite unité, et le simplex en dimension . 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.
Revised:
Accepted:
Published online:
Jean B. Lasserre 1
@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}, }
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/
[1] The complex equilibrium measure of a symmetric convex set in , Trans. Am. Math. Soc., Volume 294 (1986) no. 2, pp. 705-717 | Zbl
[2] JuMP: A Modeling Language for Mathematical Optimization, SIAM Rev., Volume 59 (2017) no. 2, pp. 295-320 | DOI | Zbl
[3] CVX: Matlab Software for Disciplined Convex Programming, version 2.1, http://cvxr.com/cvx, 2014
[4] Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2015 | DOI | Zbl
[5] A disintegration of the Christoffel function, C. R. Math. Acad. Sci. Paris, Volume 360 (2022), pp. 1071-1079 | Zbl
[6] The Christoffel–Darboux Kernel for Data Analysis, Cambridge Monographs on Applied and Computational Mathematics, 38, Cambridge University Press, 2022 | DOI
[7] 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] 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] Exploiting constant trace property in large scale polynomial optimization (2020) (https://arxiv.org/abs/2012.08873, to appear in ACM Trans. Math. Softw.)
[10] Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., Volume 42 (1993) no. 3, pp. 969-984 | DOI | Zbl
[11] Polynomial Pell’s equation, Proc. Am. Math. Soc., Volume 131 (2003) no. 4, pp. 993-1006 | DOI | Zbl
Cited by Sources:
Comments - Policy