论文标题
表面多项式和脐带的根
Roots of Polynomials and Umbilics of Surfaces
论文作者
论文摘要
对于某些多项式，我们将单位圆内的根数与欧几里得3空间中真实分析表面上的非脱位分离的脐带的索引联系起来。 特别是，对于$ n> 0 $，我们证明，对于某个（$ n+2 $） - 真正的多项式多项式$ n $的实际维度家族，单元圆内的根数小于或等于$ 1+n/2 $。该界限的确定如下。 从多项式中，我们构建一个凸出的凸真分析表面，该分析表面包含一个孤立的脐带，从而使脐点的索引由位于单位圆内的多项式根部的根数确定。然后，根部的绑定是从汉堡的界面上绑定在凸真分析表面上孤立的脐点的索引上的。 出现的多项式类别是具有自我转入二阶导数的那些类别。因此，事实证明，单位圆内的根数是具有自发性第二个导数的多项式的界限。
For certain polynomials we relate the number of roots inside the unit circle with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In particular, for $N>0$ we prove that for a certain ($N+2$)-real dimensional family of complex polynomials of degree $N$, the number of roots inside the unit circle is less than or equal to $1+N/2$. This bound is established as follows. From the polynomial we construct a convex real analytic surface containing an isolated umbilic point, such that the index of the umbilic point is determined by the number of roots of the polynomial that lie inside the unit circle. The bound on the number of roots then follows from Hamburger's bound on the index of an isolated umbilic point on a convex real analytic surface. The class of polynomials that arise are those with self-inversive second derivative. Thus the number of roots inside the unit circle is proven to be bounded for a polynomial with self-inversive second derivative.
