We provide a rigorous treatment of continuous limits for various generalizations of the pentagram map on polygons in ${\mathbb{RP}}^d$ by means of quantum calculus. Describing this limit in detail for the case of the short-diagonal pentagram map, we verify that this construction yields the $(2,d+1)$-KdV equation, and moreover, the Lax form of the pentagram map in the limit is proved to become the Lax representation of the corresponding KdV system. More generally, we introduce the χ-pentagram map, a geometric construction defining curve evolutions by directly taking intersections of subspaces through specified points. We show that its different configurations yield certain other KdV equations and provide an argument towards disproving the conjecture that any KdV-type equation can be discretized through pentagram-type maps.

我们为多边形上的五角星图的各种概括提供了对连续限制的严格处理 ${\mathbb{RP}}^d$通过量子演算。针对短对角五角星图的情况详细描述此限制，我们验证此构造产生$(2,d+1)$-KdV方程，而且证明极限中五角星图的Lax形式成为相应KdV系统的Lax表示。更一般地说，我们介绍了χ -五角星图，这是一种几何构造，通过直接通过指定点取子空间的交点来定义曲线演化。我们展示了它的不同配置产生了某些其他 KdV 方程，并提供了一个论据来反驳任何 KdV 型方程都可以通过五角星型映射离散化的猜想。