Gale duality bounds for roots of polynomials with nonnegative coefficients
Gale duality bounds for roots of polynomials with nonnegative coefficients
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非负系数多项式根的 Gale 对偶界
DOI：
10.1016/j.jcta.2009.10.009
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
J. Pfeifle
中科院分区：
文献类型：
--
作者：
J. Pfeifle
我们给出了具有非负系数的多项式的根关于不超过d次多项式的向量空间的一个固定但任意基的位置.为此，我们将基多项式解释为实平面中的向量场，并在平面上的每一点分析了Gale对偶向量构型的组合.这种方法允许我们以统一的方式合并系数之间的任意线性方程和不等式，以获得关于根的位置的更精确的界。我们应用我们的技巧来确定Ehrhart多项式和色多项式的根的位置。最后，我们对随机多项式的根图中出现的聚集现象进行了解释。
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.