Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus (C∗)D, inspired by the Cantero–Moral– Velázquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus T. The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second kind functions are found as well as Plemej type formulæ. Spectral matrices give string equations for the moment matrix which model the three-term relations as well as the Christoffel–Darboux formulæ. Christoffel type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used for the finding of a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for nice Laurent polynomials which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of nice Laurent polynomial perturbation and poised sets is given; full column rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime nice Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi type matrices and its quasi-determinants alow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted τ-functions in the 1D scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial differencedifferential equations that involve only one site in the integrable lattice behaving as a Kadomstev–Petviashvili type system. 1991 Mathematics Subject Classification. 15A23,33C45,37K10,37L60,42C05,46L55.