IJPAM: Volume 39, No. 2 (2007)

BIRKHOFF INTERPOLATION OVER A FINITE FIELD

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Fix integers $n \ge m-1 \ge 0$, a Birkhoff interpolation problem $\mathcal {B}$ (interpolation of a polynomial and certain of its derivatives of order $\le n$ at $m$ points of a field) induced by a matrix $E = [e_{i,k}]$, $1 \le i \le m$, $0 \le k \le n$, $e_{i,k}\in \{0,1\}$, a prime $p > n$ and a $p$-power $q$. Here we prove the regularity of $\mathcal {B}$ at $(t_1,\dots ,t_m)\in \mathbb {F}_q^m$ if it is regular at $(t_1^{q/p},\dots ,t_1^{q/p})\in \mathbb {F}_p^m$. The regularity over $\mathbb {F}_p$ was recently studied by T. Tassa to solve a cryptographic model (hierarchical thresold secret sharing).

Received: March 19, 2007

AMS Subject Classification: 14N05

Key Words and Phrases: Hermite interpolation, Birkhoff interpolation, finite field

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 39
Issue: 2