具有稀疏指数的多项式的根结构
Root structures of polynomials with sparse exponents
摘要：
对于实多项式的（稀疏）指数在一些固定的设置，\[ Ψ(t)=x+y_1t^{k_1}+\ldots +y_L t^{k_L}, \]我们分析的类型的根结构，可能会出现的系数变化。我们首先建立一个分层的根到层，每个层包含的根的大小相当。然后，我们表明，存在一个合适的小参数$ε>0$，这样，对于任何根$w\in \mathbb{C}$，$B(w,ε|w|)$包含最多$L$根，计数的多重性。我们的分析建议考虑原始多项式的粗糙因子分解，并且我们建立了相应的根结构的封闭性：存在由球覆盖的根，其中a）每个球包含原始多项式及其粗糙因子分解的相同数量的根，并且b）球是强分离的。
英文摘要：
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.
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