Voronoi Diagram Properties in Polynomials with Polynomiography Applications and Extensions

We have previously established delicate connections between the Voronoi diagram of polynomial roots and their basins of attraction with respect to the Basic Family of iteration functions. We have also previously defined polynomiography, visualization techniques in solving a polynomial equation, resulting in a medium with multidisciplinary applications. Here we describe several novel results and survey our recent work on polynomials and polynomiography and their extensions and applications. These include, (i) extension of root-finding methods to analytic functions and the analysis of infinite Voronoi diagram and basins of attraction of their roots, (ii) an application of polynomiography in the visualization of least square polynomial approximations, (iii) a Voronoi diagram property of roots of a cubic equation in connection to its critical points and its algorithmic application, (iv) association of polynomiography to special matrices such as permutation matrices, Latin Squares and Sudoku solutions, and alternating sign matrices. We will consider the application of the above and present corresponding polynomiographies.