Multivariate orthogonal Laurent polynomials and integrable systems

TL;DR

This paper develops a framework for multivariate orthogonal Laurent polynomials on the algebraic torus, linking them to integrable systems, and introduces new formulas for measure perturbations and their relation to Toda lattice hierarchies.

Contribution

It introduces a novel ordering for Laurent polynomials, constructs multivariate biorthogonal Laurent polynomials, and connects measure deformations to Toda type integrable hierarchies.

Findings

Construction of multivariate biorthogonal Laurent polynomials via moment matrix factorization.

Derivation of Christoffel formulas for measure perturbations using sample matrices.

Establishment of Toda lattice equations solved by matrix coefficients of these polynomials.

Abstract

An ordering for Laurent polynomials in the algebraic torus $(\mathbb C^*)^D$, inspired by the Cantero-Moral-Vel\'azquez 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 $\mathbb T^D$. 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. 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. 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.