Grassmann凸性与乘性Sturm理论
Grassmann convexity and multiplicative Sturm theory, revisited
摘要：
本文解决了B.和M. Shapiro提出的Grassmann凸性猜想的一个特例。给出了非共轭线性实时常微分方程任意基本解的连续Wronskian的最大实零点总数的一个解析公式.我们表明，这个公式给出了下界所需的总数为一个任意阶的方程的实零点，并使用我们的结果格拉斯曼凸性，我们证明了上述公式是正确的订单$4$和$5$的方程。
英文摘要：
In this paper we settle a special case of the Grassmann convexity conjecture formulated earlier by B.and M.Shapiro. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders $4$ and $5$.
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