Leading Coefficients and the Multiplicity of Known Roots

We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over $\mathbb{C}$, and then apply it to obtain new characteristic polynomials of hypergraphs.