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The LC method described in this work seeks to approximate the roots of polynomial equations in one variable. This book allows you to explore the LC method, which uses geometric structures of Lines L and Circumferences C in the plane of complex numbers, based on polynomial coefficients. These structures depend on the inclination angle of a line with fixed point that seeks to contain one of the roots; they are associated with an error measure that indicates the degree of proximity to that root, without knowing a priori its location. Using a computer with parallel processing capabilities, it is feasible to construct several of these geometric structures at the same time, varying the inclination angle of the lines with fixed point, in order to obtain an error measure map, with which it is possible to identify, approximately, the location of all polynomial roots. To show how the LC method works, this book includes numerical examples for quadratic, cubic, and quartic polynomials, and also for polynomials of degree greater than or equal to 5; this book also includes R programs that allow you to reproduce the results of the examples on a typical personal computer; these R programs use vectorization of operations instead of loops, which can be seen as a basic and accessible form of parallel processing. This book, in the end, invites us to explore beyond the basic ideas and concepts described here, motivating the development of a more efficient and complete computational implementation of the LC method., Comment: PDF document with 270 pages

This paper describes a geometrical method for finding the roots $r_1$, $r_2$ of a quadratic equation in one complex variable of the form $x^2+c_1 x+c_2=0$, by means of a Line $L$ and a Circumference $C$ in the complex plane, constructed from known coefficients $c_1$, $c_2$. This Line-Circumference (LC) geometric structure contains the sought roots $r_1$, $r_2$ at the intersections of its component elements $L$ and $C$. Line $L$ is mapped onto Circumference $C$ by a Mobius transformation. The location and inclination angle of $L$ can be computed directly from coefficients $c_1$, $c_2$, while $C$ is constructed by dividing the constant term $c_2$ by each point from $L$. This paper describes the technical details for the quadratic LC method, and then shows how the quadratic LC method works through a numerical example. The quadratic LC method described here, although more elaborate than the traditional quadratic formula, can be extended to find initial approximations to the roots of polynomials in one variable of degree $n \geq 3$. As an additional feature, this paper also studies an interesting property of the rectilinear segments connecting key points in a quadratic LC structure., Comment: 7 pages, 2 figures