Counting dope matrices

Noga Alon, Noah Kravitz*, Kevin O'Bryant

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

For a polynomial P of degree n and an m-tuple Λ=(λ1,…,λm) of distinct complex numbers, the dope matrix of P with respect to Λ is DP(Λ)=(δij)i∈[1,m],j∈[0,n], where δij=1 if P(j)i)=0, and δij=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 languageEnglish
Pages (from-to)502-518
Number of pages17
JournalJournal of Algebra
Volume620
DOIs
StatePublished - 15 Apr 2023

Keywords

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

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