### Abstract

To each permutation matrix we associate a complex permutation polynomial with roots at lattice points corresponding to the position of the ones. More generally, to an alternating sign matrix (ASM) we associate a complex alternating sign polynomial. On the one hand visualization of these polynomials through polynomiography, in a combinatorial fashion, provides for a rich source of algorithmic art-making, interdisciplinary teaching, and even leads to games. On the other hand, this combines a variety of concepts such as symmetry, counting and combinatorics, iteration functions and dynamical systems, giving rise to a source of research topics. More generally, we assign classes of polynomials to matrices in the Birkhoff and ASM polytopes. From the characterization of vertices of these polytopes, and by proving a symmetry-preserving property, we argue that polynomiography of ASMs form building blocks for approximate polynomiography for polynomials corresponding to any given member of these polytopes. To this end we offer an algorithm to express any member of the ASM polytope as a convex of combination of ASMs. In particular, we can give exact or approximate polynomiography for any Latin Square or Sudoku solution. We exhibit some images.

Original language | English (US) |
---|---|

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 2 |

State | Published - Nov 10 2011 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

### Keywords

- Alternating sign matrices
- Doubly stochastic matrices
- Latin squares
- Linear programming
- Newton's method
- Polynomial roots
- Polynomiography
- Voronoi diagram

### Cite this

*Electronic Journal of Combinatorics*,

*18*(2), 1-22.

}

*Electronic Journal of Combinatorics*, vol. 18, no. 2, pp. 1-22.

**Alternating sign matrices and polynomiography.** / Kalantari, Bahman.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Alternating sign matrices and polynomiography

AU - Kalantari, Bahman

PY - 2011/11/10

Y1 - 2011/11/10

N2 - To each permutation matrix we associate a complex permutation polynomial with roots at lattice points corresponding to the position of the ones. More generally, to an alternating sign matrix (ASM) we associate a complex alternating sign polynomial. On the one hand visualization of these polynomials through polynomiography, in a combinatorial fashion, provides for a rich source of algorithmic art-making, interdisciplinary teaching, and even leads to games. On the other hand, this combines a variety of concepts such as symmetry, counting and combinatorics, iteration functions and dynamical systems, giving rise to a source of research topics. More generally, we assign classes of polynomials to matrices in the Birkhoff and ASM polytopes. From the characterization of vertices of these polytopes, and by proving a symmetry-preserving property, we argue that polynomiography of ASMs form building blocks for approximate polynomiography for polynomials corresponding to any given member of these polytopes. To this end we offer an algorithm to express any member of the ASM polytope as a convex of combination of ASMs. In particular, we can give exact or approximate polynomiography for any Latin Square or Sudoku solution. We exhibit some images.

AB - To each permutation matrix we associate a complex permutation polynomial with roots at lattice points corresponding to the position of the ones. More generally, to an alternating sign matrix (ASM) we associate a complex alternating sign polynomial. On the one hand visualization of these polynomials through polynomiography, in a combinatorial fashion, provides for a rich source of algorithmic art-making, interdisciplinary teaching, and even leads to games. On the other hand, this combines a variety of concepts such as symmetry, counting and combinatorics, iteration functions and dynamical systems, giving rise to a source of research topics. More generally, we assign classes of polynomials to matrices in the Birkhoff and ASM polytopes. From the characterization of vertices of these polytopes, and by proving a symmetry-preserving property, we argue that polynomiography of ASMs form building blocks for approximate polynomiography for polynomials corresponding to any given member of these polytopes. To this end we offer an algorithm to express any member of the ASM polytope as a convex of combination of ASMs. In particular, we can give exact or approximate polynomiography for any Latin Square or Sudoku solution. We exhibit some images.

KW - Alternating sign matrices

KW - Doubly stochastic matrices

KW - Latin squares

KW - Linear programming

KW - Newton's method

KW - Polynomial roots

KW - Polynomiography

KW - Voronoi diagram

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UR - http://www.scopus.com/inward/citedby.url?scp=80555154213&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:80555154213

VL - 18

SP - 1

EP - 22

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

ER -