Confluent Vandermonde matrix and related topics

Abstract: In this note, we explore the connections between the confluent Vandermonde matrix and several mathematical topics, including interpolating polynomials, Hasse derivatives, LU factorization, companion matrices and their Jordan forms, and the Chinese remainder theorem. Using a unified approach based on polynomial evaluations and derivative computations at selected points, we provide accessible proofs that not only clarify key results but also offer insights for both experienced researchers and those new to the subject.