Title: On universal sign patterns Show Chinese title

Title: 关于通用符号模式

Abstract: We consider polynomials $Q:=\sum _{j=0}^da_jx^j$, $a_j\in \mathbb{R}^*$, with all roots real. When the {\em sign pattern} $\sigma (Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm sgn}(a_0))$ has $\tilde{c}$ sign changes, the polynomial $Q$ has $\tilde{c}$ positive and $d-\tilde{c}$ negative roots. We suppose the moduli of these roots distinct. The {\em order} of these moduli is defined when in their string as points of the positive half-axis one marks the places of the moduli of negative roots. A sign pattern $\sigma^0$ is {\em universal} when for any possible order of the moduli there exists a polynomial $Q$ with $\sigma (Q)=\sigma^0$ and with this order of the moduli of its roots. We show that when the polynomial $P_{m,n}:=(x-1)^m(x+1)^n$ has no vanishing coefficients, the sign pattern $\sigma (P_{m,n})$ is universal. We also study the question when $P_{m,n}$ can have vanishing coefficients. Show Chinese abstract

Abstract: 我们考虑多项式$Q:=\sum _{j=0}^da_jx^j$，$a_j\in \mathbb{R}^*$，它们的所有根都是实数。 当 {\em 符号模式} $\sigma (Q):=({\rm sgn}(a_d),{\rm sgn}(a_{d-1})$ , $\ldots$, ${\rm sgn}(a_0))$具有 $\tilde{c}$符号变化时，多项式 $Q$具有 $\tilde{c}$个正根和 $d-\tilde{c}$个负根。 我们假设这些根的模是不同的。 这些模的{\em 顺序}在它们作为正半轴上的点的字符串中被定义，当标记负根的模的位置时。 符号模式$\sigma^0$是{\em 通用}的，当对于任何可能的模的顺序，都存在一个多项式$Q$具有$\sigma (Q)=\sigma^0$且具有其根的模的这个顺序。 我们证明当多项式$P_{m,n}:=(x-1)^m(x+1)^n$没有零系数时，符号模式$\sigma (P_{m,n})$是普遍的。 我们还研究了$P_{m,n}$在什么情况下可以有零系数。