非线性嵌入：应用于分析、分形和多项式求根
Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding
摘要：
我们引入$\mathcal{B}_κ$ -嵌入，非线性数学结构，通过$κ$参数化的光滑路径，连接在$κ=0$的有限或可数的对象集（例如，数字，函数，向量，生成函数的系数.）到它们在$κ\to \infty$的普通总和。我们表明，$\mathcal{B}_κ$ -嵌入可以用来设计非线性不可逆过程，通过这种连接。一些越来越复杂的例子来说明这个概念所揭示的可能性。这些不仅包括光滑的功能，但也分形的实线和复平面上。作为一个应用，我们使用$\mathcal{B}_κ$ -嵌入制定一个强大的方法，找到所有的根的一元多项式，而不因式分解或紧缩的多项式。我们说明这种方法通过寻找所有根的多项式的19次。
英文摘要：
We introduce $\mathcal{B}_κ$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $κ$, a finite or denumerable set of objects at $κ=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $κ\to \infty$. We show that $\mathcal{B}_κ$-embeddings can be used to design nonlinear irreversible processes through this connection. A number of examples of increasing complexity are worked out to illustrate the possibilities uncovered by this concept. These include not only smooth functions but also fractals on the real line and on the complex plane. As an application, we use $\mathcal{B}_κ$-embeddings to formulate a robust method for finding all roots of a univariate polynomial without factorizing or deflating the polynomial. We illustrate this method by finding all roots of a polynomial of 19th degree.
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