Multivariate Orthogonal Laurent Polynomials and Integrable Systems

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