论文标题
多项式的根结构，稀疏指数
Root structures of polynomials with sparse exponents
论文作者
论文摘要
对于带有（稀疏）指数的真实多项式，\ [ψ（t）= x+y_1t^{k_1}+\ ldots+y_l t^{k_l}，\]，我们分析可能以系数为vary的根结构的类型。我们首先将根分层建立为层，每个层都包含相当大小的根。然后，我们证明存在一个合适的小参数$ε> 0 $，这样，对于任何root $ w \ in \ mathbb {c} $，$ b（w，ε| w |）$最多包含$ l $ roots，以多种为数。我们的分析表明，考虑原始多项式的粗略分解，我们建立了相应的根结构的接近性：存在球对根的覆盖，其中a）a）每个球都包含原始多项式的根数及其粗糙分解的根和b）球是强烈分离的。
For real polynomials with (sparse) exponents in some fixed set, \[ Ψ(t)=x+y_1t^{k_1}+\ldots +y_L t^{k_L}, \] we analyse the types of root structures that might occur as the coefficients vary. We first establish a stratification of roots into tiers, each containing roots of comparable sizes. We then show that there exists a suitable small parameter $ε>0$ such that, for any root $w\in \mathbb{C}$, $B(w,ε|w|)$ contains at most $L$ roots, counted with multiplicity. Our analysis suggests the consideration of a rough factorisation of the original polynomial and we establish the closeness of the corresponding root structures: there exists a covering of the roots by balls wherein a) each ball contains the same number of roots of the original polynomial and of its rough factorisation and b) the balls are strongly separated.
