Stably computing the multiplicity of known roots given leading coefficients

Abstract

SummaryWe 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 . 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.