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)(λi)=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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Dope matrices
- Geometry of polynomials
- Hermite-Birkhoff interpolation
- Zero patterns of polynomials