TY - JOUR

T1 - Counting dope matrices

AU - Alon, Noga

AU - Kravitz, Noah

AU - O'Bryant, Kevin

N1 - Funding Information:
The first author is supported in part by NSF grant DMS–1855464 and BSF grant 2018267 . The second author is supported by an NSF Graduate Research Fellowship (grant DGE–2039656 ). The third author is supported by a PSC–CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York .
Publisher Copyright:
© 2023 Elsevier Inc.

PY - 2023/4/15

Y1 - 2023/4/15

N2 - 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.

AB - 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.

KW - Dope matrices

KW - Geometry of polynomials

KW - Hermite-Birkhoff interpolation

KW - Zero patterns of polynomials

UR - http://www.scopus.com/inward/record.url?scp=85146443723&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85146443723&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2023.01.006

DO - 10.1016/j.jalgebra.2023.01.006

M3 - Article

AN - SCOPUS:85146443723

SN - 0021-8693

VL - 620

SP - 502

EP - 518

JO - Journal of Algebra

JF - Journal of Algebra

ER -