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Multivariate orthogonal Laurent polynomials and integrable systems. (English) Zbl 1490.37090

The article aims to connect the theory of multivariate orthogonal Laurent polynomials on the unit torus, presented in the main part of the text, to the theory of integrable systems. This work is an extension of previous works by the same authors, where the main idea to investigate integrable hierarchies is based on the Gauss-Borel factorization.
The authors introduce the so-called longilex order of Laurent monomials, which allows them to construct the moment matrix associated with a Borel measure. Such a matrix satisfies certain (per)symmetry conditions, as well as, what the authors call the string equations. For some particular cases, it is shown that the moment matrix admits a block Gauss-Borel factorization. This is used to define the multivariate Laurent polynomials in many complex variables, which are shown to form a biorthogonal system.
Then the authors define a Christoffel perturbed measure, induced by multiplication by a Laurent polynomial. This leads to a number of concepts and properties proved in the text, and, in particular, the Jacobi matrices turn out to have an intimate connection with integrable systems, and produce the Lax operators, as well as the zero-curvature condition. In addition, a connection to the Baker-Akhiezer functions is also pointed out.
This is a very technical paper, which is however quite readable and contains many interesting results. There are some explicitly worked-out examples that illustrate the general theory given in the text. A small helpful section devoted to an overview of the previous works of the authors is also included. Finally, the authors provide a number of appendices which contain detailed proofs of the theorems formulated in the main text.

MSC:

37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
15A23 Factorization of matrices
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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