Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding

TL;DR

This paper introduces $\\mathcal{B}_{\kappa}$-embeddings, nonlinear structures connecting objects via smooth paths, and demonstrates their applications in analysis, fractals, and a novel polynomial root-finding method.

Contribution

It presents the concept of $\mathcal{B}_{\kappa}$-embeddings and applies them to develop a new robust polynomial root-finding technique without factorization.

Findings

Successfully applied to find roots of a 19th-degree polynomial.

Illustrated the use of embeddings with fractals and smooth functions.

Provided a new framework for nonlinear irreversible processes.

Abstract

We introduce $\mathcal{B}_{\kappa}$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $\kappa$, a finite or denumerable set of objects at $\kappa=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $\kappa \to \infty$. We show that $\mathcal{B}_{\kappa}$-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 $\mathcal{B}_{\kappa}$-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.