## Abstract

For a polynomial P of degree n and an m-tuple Λ=(λ_{1},…,λ_{m}) of distinct complex numbers, the dope matrix of P with respect to Λ is D_{P}(Λ)=(δ_{ij})_{i∈[1,m],j∈[0,n]}, where δ_{ij}=1 if P^{(j)}(λ_{i})=0, and δ_{ij}=0 otherwise. Our first result is a combinatorial characterization of the 2-row dope matrices (for all pairs Λ); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of m×(n+1) dope matrices, and we show that the number of m×(n+1) dope matrices for a fixed m-tuple Λ is maximized when Λ is generic. Finally, we resolve an “extension” problem of Nathanson and present several open problems.

Original language | English (US) |
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Pages (from-to) | 502-518 |

Number of pages | 17 |

Journal | Journal of Algebra |

Volume | 620 |

DOIs | |

State | Published - Apr 15 2023 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Keywords

- Dope matrices
- Geometry of polynomials
- Hermite-Birkhoff interpolation
- Zero patterns of polynomials