Norms of roots of trinomials
Norms of roots of trinomials
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三项式根的范数
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
T. Wolff
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作者：
T. Theobald;T. Wolff
单变量三项式的根的范数$$z^{s+t} + p z^t + q in mathbb {C}[z]$$ zs+t+pzt+q∈C[z]对于固定支撑$$A = {0,t,s+t} subset mathbb {N}$$ A={0,t，s+t}∧N关于系数$$p,q in mathbb {C}$$ p，q∈C的选择的行为是19世纪末和20世纪初的一个经典问题。虽然P. Bohl在1908年进行了代数表征，但相应系数参数空间的几何和拓扑结构尚未揭示。假设s和t是互素，我们通过用变形虫理论重新解释这个问题，为三项式空间提供了这样一个表征。给定范数的根被参数化为沿着三项式空间$$mathbb {C}$$ c切片的下曲面曲线，该范数的多个根恰好出现在奇点上。作为主要结果，我们证明了具有支持a和相同范数的某些根的所有三项式集及其补集可以变形缩回到环面结$$K(s+t,s)$$ K(s+t,s)，因此是连通的而不是单连通的。一个例外是第t个最小范数与$$(t+1)$$ (t+1)-st个最小范数重合。在这里，补具有不同的拓扑结构，因为它具有基本群$$mathbb {Z}^2$$ Z2。
The behavior of norms of roots of univariate trinomials $$z^{s+t} + p z^t + q in mathbb {C}[z]$$zs+t+pzt+q∈C[z] for fixed support $$A = {0,t,s+t} subset mathbb {N}$$A={0,t,s+t}⊂N with respect to the choice of coefficients $$p,q in mathbb {C}$$p,q∈C is a classical late 19th and early 20th century problem. Although algebraically characterized by P. Bohl in 1908, the geometry and topology of the corresponding parameter space of coefficients had yet to be revealed. Assuming s and t to be coprime we provide such a characterization for the space of trinomials by reinterpreting the problem in terms of amoeba theory. The roots of given norm are parameterized in terms of a hypotrochoid curve along a $$mathbb {C}$$C-slice of the space of trinomials, with multiple roots of this norm appearing exactly on the singularities. As a main result, we show that the set of all trinomials with support A and certain roots of identical norm, as well as its complement can be deformation retracted to the torus knot $$K(s+t,s)$$K(s+t,s), and thus are connected but not simply connected. An exception is the case where the t-th smallest norm coincides with the $$(t+1)$$(t+1)-st smallest norm. Here, the complement has a different topology since it has fundamental group $$mathbb {Z}^2$$Z2.