非负系数多项式根的Gale对偶界
Gale duality bounds for roots of polynomials with nonnegative coefficients
摘要：
我们绑定的多项式的根的位置，有非负系数的多项式的次数至多$d$的向量空间的一个固定的，但任意的基础。为此，我们将基多项式解释为实平面上的向量场，并在平面上的每一点上分析盖尔对偶向量配置的组合学。这种方法使我们能够将任意的线性方程和不等式之间的系数在一个统一的方式，以获得更精确的边界上的位置的根。我们应用我们的技术来界定的Ehrhart和色多项式的根的位置。最后，我们给出了一个解释的聚类图中看到的随机多项式的根。
英文摘要：
We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most $d$. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.
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