Quantum Lorentz Degrees of Polynomials and a Pólya Theorem for Polynomials Positive on q-Lattices

Rachid Ait-Haddou (1) , Ron Goldman (2) , Marie-Laurence Mazure (3)

1 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals
2 Department of computer Science [Houston]
3 CVGI - Calcul des Variations, Géométrie, Image

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 1/q-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.

Mots clés: quantum Lorentz degrees; quantum Pólya's theorems; q-lattices; q-Bernstein basis; q-positivity; q-blossom; degree elevation

Domaines: Mathématiques [math]

Rachid Ait-Haddou, Ron Goldman, Marie-Laurence Mazure. Quantum Lorentz Degrees of Polynomials and a Pólya Theorem for Polynomials Positive on q-Lattices. Applied Numerical Mathematics, 2021, 165, pp.553-577. ⟨10.1016/j.apnum.2021.03.009⟩. ⟨hal-03040560⟩