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. [ABSTRACT FROM AUTHOR]