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Stably computing the multiplicity of known roots given leading coefficients
Author(s) -
Clark Gregory J.,
Cooper Joshua N.
Publication year - 2020
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2275
Subject(s) - mathematics , monic polynomial , multiplicity (mathematics) , algebraically closed field , univariate , homogeneous , polynomial , characteristic polynomial , zero (linguistics) , field (mathematics) , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , multivariate statistics , linguistics , statistics , philosophy
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 C . 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.