有限捕获与受限多项式根的闭包
Semialgebraic interior layers in root closures of bounded-height integer polynomials
AI总结：
本文研究了受限多项式根集的闭包与有限捕获集的关系，通过构造陷阱和围栏定义了有限捕获集，并证明了其连通性领域与非实参数的包含关系。
摘要：
我们研究如何一个可数代数根集转化为分形连通性领域。令$D_n=\{-n+1,-n+2,\ldots,n-1\}$，令$R_n$为所有非主系数位于$D_n$中的单项式多项式的根集。我们研究$\overline{R_n}\setminus\overline{\mathbb{D}}$。在闭合单位盘外，这个集合等于一个共线仿射迭代函数系统的连通性领域$M_n$，或者等价于递归幂级数$1+\sum_{k\ge1} d_k c^{-k}$的零集，其中$d_k\in D_n$。对于非实参数在透镜$X_n=\{\,c\in\mathbb{C}\setminus\overline{\mathbb{D}}:\\ |c\pm1|<\sqrt{2n}\\,\}$中，我们构造了一个标准陷阱和围栏用于关联的差吸引子，并用它们定义有限捕获集$Θ_k(n)$对于标记点$2c$。我们的主要结果是统一包含$\overline{Θ_k(n)}\cap(X_n\setminus\mathbb{R})\subsetΘ_{k+2}(n)$对于每一个$k\ge0$。因此，$(M_n\cap X_n)\setminus\mathbb{R}$恰好是有限捕获领域的闭包。本文结合了显式陷阱几何与认证反向搜索。此外，$M_n\setminus\mathbb{R}\subset X_n$对于每一个$n\ge20$，并且对于$2\le n\le19$来说这是尖锐的。因此，对于$n\ge20$，$\overline{R_n}\setminus\overline{\mathbb{D}}$的非实部分恰好是有限捕获领域的闭包。
英文摘要：
Finite-degree root plots of bounded-height monic integer polynomials can exhibit gaps around interior points of their root closures. Outside the closed unit disk, these closures coincide with Mandelbrot sets for affine iterated function systems with collinear digits. We construct explicit semialgebraic interior layers, certified when finite inverse orbits enter canonical self-covering parallelograms. Such parallelograms exist exactly in the non-real lens, which contains the non-real locus precisely from twenty maps onward. Within the non-real lens, limits of each capture level are captured within two additional steps, optimally, and the layers recover the locus by closure. The layers expose persistent interior deficits: explicit quadratic parameters evade every finite level; sufficiently close distinct parameters have logarithmically growing capture depths; the underlying lattice obstruction yields degree-dependent root-free radii, explaining such gaps. For non-real parameters of modulus at least the square root of the number of maps, exact enclosure trees have sharp, alphabet-independent branching bounds.
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