Quantum Lorentz degrees of polynomials and a Pólya theorem for polynomials positive on q-lattices

We establish the uniform convergence of the control polygons generated by repeated degree elevation of q-Bézier curves (i.e., polynomial curves represented in the q-Bernstein bases of increasing degrees) on [0,1], q>1, to a piecewise linear curve with vertices on the original curve. A similar result is proved for q<1, but surprisingly the limit vertices are not on the original curve, but on the q⁻¹-Bézier curve with control polygon taken in the reverse order. We introduce a q-deformation (quantum Lorentz degree) of the classical notion of Lorentz degree for polynomials and we study its properties. As an application of our convergence results, we introduce a notion of q-positivity which guarantees that the q-Lorentz degree is finite. We also obtain upper bounds for the quantum Lorentz degrees. Finally, as a by-product we provide a generalization to polynomials positive on q-lattices of the univariate Pólya theorem concerning polynomials positive on the non-negative axis.