Stably computing the multiplicity of known roots given leading coefficients

Abstract

Summary We show that a monic univariate polynomial over a field of characteristic zero, with k distinct nonzero known roots, is determined by precisely k of its proper leading coefficients. Furthermore, we give an explicit, numerically stable algorithm for computing the exact multiplicities of each root over double-struckC. 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. Then, we demonstrate how these results can be used to obtain the full homogeneous spectra of symmetric tensors—in particular, complete characteristic polynomials of hypergraphs.

Applications of the Harary-Sachs theorem for hypergraphs

“…To demonstrate Theorem 3 we have computed the first sixteen coefficients of the characteristic polynomial of the Fano Plane with two edges removed (called the Rowling hypergraph in [3]), the Fano Plane with one edge removed, and the Fano Plane in Fig. 2.…”

“…2. Aside from the novelty of performing these computations, the coefficients can be used to determine the characteristic polynomial of a given hypergraph [3]. Consider the Rowling hypergraph (see Fig.…”