论文标题
实用的多项式和广义的Hermite-Sylvester定理
Real-rooted polynomials and a generalised Hermite-Sylvester theorem
论文作者
论文摘要
如果多项式所有根都是真实的，则将其根深蒂固。对于{\ mathbf r} [t] $中的每个多项式$ f（t）\，Hermite-Sylvester定理将二次形式$φ_2$关联，使得$ f（t）$是真实的，并且只有$φ_2$是正面的。在此注释中，对于每个正整数$ m $，构建200万美元的$ $φ_{2M} $，以至于$ f（t）$仅当$φ_{2M {2M} $才能实现，当时仅是$ m $的$ m $，仅当$ m $ IF $ $; if $φ_{2M}（x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1，x_1， $ m $。
A polynomial is real-rooted if all of its roots are real. For every polynomial $f(t) \in {\mathbf R}[t]$, the Hermite-Sylvester theorem associates a quadratic form $Φ_2$ such that $f(t)$ is real-rooted if and only if $Φ_2$ is positive semidefinite. In this note, for every positive integer $m$, an $2m$-adic form $Φ_{2m}$ is constructed such that $f(t)$ is real-rooted if and only if $Φ_{2m}$ is positive semidefinite for some $m$ if and only if $Φ_{2m}(x_1,\ldots, x_n)$ is positive semidefinite for all $m$.
