We consider the embedding function \(c_b(a)\) describing the problem of symplectically embedding an ellipsoid E(1, a) into the smallest scaling of the polydisc P(1, b). Previous work suggests that determining the entirety of \(c_b(a)\) for all b is difficult, as infinite staircases can appear for many sequences of irrational b. In contrast, we show that for every polydisc P(1, b) with \(b>2\), there is an explicit formula for the minimum a such that the embedding problem is determined only by volume. That is, when the ellipsoid is sufficiently stretched, there is a symplectic embedding of E(1, a) fully filling an appropriately scaled polydisc \(P(\lambda ,\lambda b)\). Denoted RF(b), this rigid-flexible (RF) value is piecewise smooth with a discrete set of discontinuities for \(b>2\). At the same time, by exhibiting a sequence of obstructive classes for \(b_n = \frac{n+1}{n}\) at \(a=8\), we show that RF is also discontinuous at \(b=1\).

我们考虑嵌入函数\(c_b(a)\)来描述将椭球体E (1,  a ) 辛嵌入到多圆盘P (1,  b ) 的最小缩放中的问题。之前的工作表明，确定所有b的整个\(c_b(a)\)是很困难的，因为许多无理数b序列可能会出现无限阶梯。相比之下，我们证明对于每个具有\(b>2\)的多圆盘P (1,  b ) ，有一个最小a的显式公式这样嵌入问题仅由体积决定。也就是说，当椭球体被充分拉伸时，存在E (1,  a ) 的辛嵌入，完全填充适当缩放的多圆盘\(P(\lambda ,\lambda b)\)。表示为RF ( b )，该刚性-柔性 ( RF ) 值是分段平滑的，具有一组离散的不连续性（b>2）。同时，通过在\(a=8\)处展示\(b_n = \frac{n+1}{n}\)的一系列阻塞类别，我们表明RF在\(b= 1\) .