### Abstract

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.

Original language | English (US) |
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Title of host publication | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |

Pages | 32-40 |

Number of pages | 9 |

DOIs | |

State | Published - Oct 3 2012 |

Event | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 - Piscataway, NJ, United States Duration: Jun 27 2012 → Jun 29 2012 |

### Publication series

Name | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
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### Other

Other | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
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Country | United States |

City | Piscataway, NJ |

Period | 6/27/12 → 6/29/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Keywords

- Basins of attraction
- Complex polynomials
- Dynamical systems
- Iteration functions
- Newton's method
- Voronoi diagrams
- Zeros

### Cite this

*Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012*(pp. 32-40). [6257654] (Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012). https://doi.org/10.1109/ISVD.2012.10

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*Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012.*, 6257654, Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012, pp. 32-40, 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012, Piscataway, NJ, United States, 6/27/12. https://doi.org/10.1109/ISVD.2012.10

**Voronoi diagram properties in polynomials with polynomiography applications and extensions.** / Kalantari, Bahman.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Voronoi diagram properties in polynomials with polynomiography applications and extensions

AU - Kalantari, Bahman

PY - 2012/10/3

Y1 - 2012/10/3

N2 - 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.

AB - 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.

KW - Basins of attraction

KW - Complex polynomials

KW - Dynamical systems

KW - Iteration functions

KW - Newton's method

KW - Voronoi diagrams

KW - Zeros

UR - http://www.scopus.com/inward/record.url?scp=84866785613&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866785613&partnerID=8YFLogxK

U2 - 10.1109/ISVD.2012.10

DO - 10.1109/ISVD.2012.10

M3 - Conference contribution

AN - SCOPUS:84866785613

SN - 9780769547244

T3 - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012

SP - 32

EP - 40

BT - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012

ER -