论文标题
标志模式和刚性模量订单
Sign patterns and rigid moduli orders
论文作者
论文摘要
我们考虑一组元度$ d $真实的单变量多项式$ q_d = x^d+\ sum_ {j = 0}^{d-1} a_jx^j $及其{\ em hyprobolicity域} $ y} $ pollotity域} $ photial $ a_j $ a_j $ a_j $ q op a polyn的$ q os $ a_j $ q opy $ q os a_j $ q是$ a_j $ q nece。子集$ e_d \subsetπ_d$是$ q_d $的负根模量等于$ q_d $的正词。在某个时刻，$ q_d $具有$ d $不同的根源，正好$ s $（$ 1 \ leq s \ leq [d/2] $）正源根和负根的模量之间的平等性，集合$ e_d $在本地是$ s $ spooth Smooth Hypersurfaces的横向交叉点。在某个时刻，$ q_d $具有两个双重的根，并且根部之间没有其他平等性，$ e_d $在本地是$ \ mathbb {r}^{d-3} $的直接产品，而在$ \ mathbb {r}^3 $中具有Whitney Singulare umbrella umbrella umbrella Singularity。对于$ d \ leq 4 $，我们绘制集合$π_d$和〜$ e_d $的图片。
We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\sum_{j=0}^{d-1}a_jx^j$ and its {\em hyperbolicity domain} $Π_d$, i.e. the subset of values of the coefficients $a_j$ for which the polynomial $Q_d$ has all roots real. The subset $E_d\subset Π_d$ is the one on which a modulus of a negative root of $Q_d$ is equal to a positive root of $Q_d$. At a point, where $Q_d$ has $d$ distinct roots with exactly $s$ ($1\leq s\leq [d/2]$) equalities between positive roots and moduli of negative roots, the set $E_d$ is locally the transversal intersection of $s$ smooth hypersurfaces. At a point, where $Q_d$ has two double opposite roots and no other equalities between moduli of roots, the set $E_d$ is locally the direct product of $\mathbb{R}^{d-3}$ and a hypersurface in $\mathbb{R}^3$ having a Whitney umbrella singularity. For $d\leq 4$, we draw pictures of the sets $Π_d$ and~$E_d$.
