### Abstract

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated d × d-minors in a d × n totally-positive matrix is O(nd-dd+1). For the case d = 2 we also show that our bound is optimal. We consider some special families of totally positive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of n points in the plane? This problem seems to be interesting in its own right especially since it seems to have a flavor of additive combinatorics and relates to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of O(n43) and provide a lower bound of n1+1O(loglogn).

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

Pages (from-to) | 149-161 |

Number of pages | 13 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 128 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

### Fingerprint

### Keywords

- Discrete geometry
- Geometric incidences
- Totally positive

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series A*,

*128*(1), 149-161. https://doi.org/10.1016/j.jcta.2014.08.004

**On totally positive matrices and geometric incidences.** / Farber, Miriam; Ray, Saurabh; Smorodinsky, Shakhar.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series A*, vol. 128, no. 1, pp. 149-161. https://doi.org/10.1016/j.jcta.2014.08.004

}

TY - JOUR

T1 - On totally positive matrices and geometric incidences

AU - Farber, Miriam

AU - Ray, Saurabh

AU - Smorodinsky, Shakhar

PY - 2014/1/1

Y1 - 2014/1/1

N2 - A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated d × d-minors in a d × n totally-positive matrix is O(nd-dd+1). For the case d = 2 we also show that our bound is optimal. We consider some special families of totally positive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of n points in the plane? This problem seems to be interesting in its own right especially since it seems to have a flavor of additive combinatorics and relates to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of O(n43) and provide a lower bound of n1+1O(loglogn).

AB - A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated d × d-minors in a d × n totally-positive matrix is O(nd-dd+1). For the case d = 2 we also show that our bound is optimal. We consider some special families of totally positive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of n points in the plane? This problem seems to be interesting in its own right especially since it seems to have a flavor of additive combinatorics and relates to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of O(n43) and provide a lower bound of n1+1O(loglogn).

KW - Discrete geometry

KW - Geometric incidences

KW - Totally positive

UR - http://www.scopus.com/inward/record.url?scp=84906684180&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906684180&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2014.08.004

DO - 10.1016/j.jcta.2014.08.004

M3 - Article

AN - SCOPUS:84906684180

VL - 128

SP - 149

EP - 161

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -