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.

我们建立由q- Bézier曲线（即，以递增的q -Bernstein基数表示的多项式曲线）的重复次数升高生成的控制多边形的一致收敛。$[0，\mathrm{1个}]$， $q>\mathrm{1个}$，到在原始曲线上具有顶点的分段线性曲线。证明了类似的结果$q<\mathrm{1个}$，但令人惊讶的是，极限顶点不在原始曲线上，而是在 $q^{-\mathrm{1个}}$-Bézier曲线，控制多边形的取反顺序。我们介绍了多项式的Lorentz度经典概念的q形变（量子Lorentz度），并研究了其性质。作为我们收敛结果的一种应用，我们引入了q-正性的概念，该概念保证q -Lorentz度是有限的。我们还获得了量子洛伦兹度的上限。最后，作为副产品，我们对单变量Pólya定理的q格上的正多项式与在非负轴上的正多项式进行了推广。