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## On the zeros of polynomials: an extension of the Enestrom-Kakeya theorem

Fonte: Academic Press Inc Elsevier Science; Amsterdan
Publicador: Academic Press Inc Elsevier Science; Amsterdan

Tipo: Artigo de Revista Científica

Português

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#ENESTROM-KAKEYA THEOREM#ZEROS OF PERTURBED POLYNOMIALS#STABILITY OF BROWN (K, L) METHODS#JELTSCH CONJECTURE#MULTISTEP MULTIDERIVATIVE METHODS#POWER-SERIES#STABILITY#MECÂNICA DOS FLUÍDOS COMPUTACIONAL#MATHEMATICS, APPLIED#MATHEMATICS

This paper presents an extension of the Enestrom-Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown (K, L) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994)[6]. (C) 2011 Elsevier Inc. All rights reserved.

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## On the zeros of polynomials: An extension of the Enestrom-Kakeya theorem

Fonte: Academic Press Inc. Elsevier B.V.
Publicador: Academic Press Inc. Elsevier B.V.

Tipo: Artigo de Revista Científica
Formato: 1151-1161

Português

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#Enestrom-Kakeya theorem#Zeros of perturbed polynomials#Stability of Brown (K, L) methods#Jeltsch conjecture

This paper presents an extension of the Enestrom-Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown (K, L) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994)[6]. (C) 2011 Elsevier B.V. All rights reserved.

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## Zeros de polinômios ortogonais de variável discreta; Zeros of orthogonal polynomials of discrete variable

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 15/03/2012
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Neste trabalho estudamos o comportamento de zeros de polinômios ortogonais clássicos de variável discreta. Provamos que certas funções que envolvem os zeros dos polinômios de Charlier, Meixner, Kravchuck e Hahn são funções monótonas dos parâmetros dos quais os correspondentes polinômios dependem. Com esse resultado obtemos novos limitantes extremamente precisos para os zeros dessas famílias de polinômios em função dos zeros dos polinômios ortogonais clássicos, que são mais estudados. Analisamos quais são os melhores limitantes explícitos para os zeros desses polinômios e aplicamos aos nossos resultados, obtendo assim limitantes explícitos para os zeros dos polinômios de Charlier, Meixner, Kravchuck e Hahn. São feitas comparações entre os nossos resultados e os melhores resultados encontrados na literatura para os zeros desses polinômios e verifica-se que nossos limitantes são, em uma grande parte, melhores. Devido à sua grande aplicabilidade, um estudo ainda mais minucioso foi feito para os zeros dos polinômios de Gram, um caso particular de Hahn, que resultou em limitantes para os zeros dos polinômios de Gram. Experimentos numéricos comprovam a qualidade dos resultados.; In this thesis we study the behavior of zeros of classical orthogonal polynomials of discrete variable. We prove that certain functions which involve the zeros of polynomials of Charlier...

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## Cauchy-Mirimanoff and related polynomials

Fonte: Universidade Nacional da Austrália
Publicador: Universidade Nacional da Austrália

Tipo: Thesis (PhD); Doctor of Philosophy (PhD)

Português

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This thesis investigates the Cauchy-Mirimanoff polynomials En and their close relatives Rn, Sn and Tn, with an emphasis on irreducibility. The Cauchy-Mirimanoff polynomials were first identified and studied by Cauchy and Liouville in 1839 in relation to Fermat's Last Theorem, but Mirimanoff in 1903 first proposed their irreducibility over Q for n a prime number. None of the standard irreducibility criteria apply directly, for example Helou showed En is always reducible modulo any prime for all odd n>=9. Computing irreducibility is problematic as the largest coefficients grow rapidly with n. The difficulty of the problem is apparent since it remains unresolved after more than 100 years. Helou, Filaseta and Beukers in 1997, Tzermias in 2007, Irick in 2010 and Lynch in 2012 have progressed the area using a range of methods, but this thesis describes an alternative original method and it is used to generalize some of the earlier results. In essence the method uses proven properties of the polynomials to reveal an inconsistency in the 2-adic or 3-adic valuation of their coefficients, depending on the polynomial under consideration.
After proving several properties of the polynomials the new method is used to prove that Rm, Sm, Tm are irreducible over Q for odd m>=3...

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## New bounds for roots of polynomials based on Fiedler companion matrices

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Publicado em 15/06/2014
Português

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Several matrix norms of the classical Frobenius companion matrices of a monic polynomial p(z) have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots lambda of p(z). Recently, M. Fiedler (2003) [9] has introduced a new family of companion matrices of p(z) that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on vertical bar lambda vertical bar. The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1- and infinity-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on vertical bar lambda vertical bar is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials.; This work has been supported by the Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542.

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## Backward stability of polynomial root-finding using Fiedler companion matrices

Fonte: Oxford University Press
Publicador: Oxford University Press

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article; info:eu-repo/semantics/conferenceObject

Publicado em 08/09/2014
Português

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#roots of polynomials#eigenvalues#characteristic polynomial#fiedler companion matrices#backward stability#conditioning#Matemáticas

Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices. In other words, given a polynomial p(z), the question is to determine whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm for the standard eigenvalue problem, are the set of roots of a nearby polynomial or not. We show that, if the coefficients of p(z) are bounded in absolute value by a moderate number, then algorithms for polynomial root-finding using Fiedler matrices are backward stable, and Fiedler matrices are as good as the Frobenius companion matrices. This allows us to use Fiedler companion matrices with favorable structures in the polynomial root-finding problem. However, when some of the coefficients of the polynomial are large, Fiedler companion matrices may produce larger backward errors than Frobenius companion matrices, although in this case neither Frobenius nor Fiedler matrices lead to backward stable computations. To prove this we obtain explicit expressions for the change...

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## Gale duality bounds for roots of polynomials with nonnegative coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We bound the location of roots of polynomials that have nonnegative
coefficients with respect to a fixed but arbitrary basis of the vector space of
polynomials of degree at most $d$. For this, we interpret the basis polynomials
as vector fields in the real plane, and at each point in the plane analyze the
combinatorics of the Gale dual vector configuration. This approach permits us
to incorporate arbitrary linear equations and inequalities among the
coefficients in a unified manner to obtain more precise bounds on the location
of roots. We apply our technique to bound the location of roots of Ehrhart and
chromatic polynomials. Finally, we give an explanation for the clustering seen
in plots of roots of random polynomials.; Comment: 25 pages, 10 figures. Final version incorporating referees' comments,
to appear in J. Comb. Theory, Ser. A

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## Symbolic computation of the roots of any polynomial with integer coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/01/2000
Português

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The roots of any polynomial of degree m with integer coefficients, can be
computed by manipulation of sequences made from 2m distinct symbols and
counting the different symbols in the sequences. This method requires only
'primitive' operations like replacement of sequences and counting of symbols.
No calculations using 'advanced' operations like multiplication, division,
logarithms etc. are needed. The method can be implemented as a geometric
construction of roots of polynomials to arbitrary accuracy using only a
straight edge, a compass, and pencils of 2m different colors. In particular,
the ancient problem of the "doubling of cube" is soluble asymptotically by the
above-mentioned construction. This method, by which a cube can be doubled,
albeit, in infinite steps, is probably the closest to the original problem of
construction using only a straight edge and compass in a finite number of
steps.
Moreover, to every polynomial of degree m over the field of rationals, can be
associated an m-term recurrence relation for generating integer sequences. A
set of m such sequences, which together exhibit interesting properties related
to the roots of the polynomial, can be obtained if the m initial terms of each
of these m sequences is chosen in a special way using a matrix associated with
the polynomial. Only two of these integer sequences need to be computed to
obtain the real root having the largest absolute value. Since this method
involves only integers...

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## Choosing roots of polynomials smoothly

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/01/1998
Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#Mathematics - Spectral Theory#26C10, 47A55

We clarify the question whether for a smooth curve of polynomials one can
choose the roots smoothly and related questions. Applications to perturbation
theory of operators are given.; Comment: to appear Israel J. Math

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## Optimal regularity of roots of polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/06/2015
Português

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We study the regularity of the roots of complex univariate polynomials whose
coefficients depend smoothly on parameters. We show that any continuous choice
of the roots of a $C^{n-1,1}$-curve of monic polynomials of degree $n$ is
locally absolutely continuous with locally $p$-integrable derivatives for every
$1 \le p < n/(n-1)$, uniformly with respect to the coefficients. This result is
optimal: in general, the derivatives of the roots of a smooth curve of monic
polynomials of degree $n$ are not locally $n/(n-1)$-integrable, and the roots
may have locally unbounded variation if the coefficients are only of class
$C^{n-1,\alpha}$ for $\alpha <1$. We also give a generalization of Ghisi and
Gobbino's higher order Glaeser inequalities.; Comment: 28 pages

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## Geometry of the locus of polynomials of degree 4 with iterative roots

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/11/2010
Português

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We study polynomial iterative roots of polynomials and describe the locus of
complex polynomials of degree 4 admitting a polynomial iterative square root.; Comment: 7 pages, accepted by Central European Journal of Mathematics

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## A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We specify a small set, consisting of $O(d(\log\log d)^2)$ points, that
intersects the basins under Newton's method of \emph{all} roots of \emph{all}
(suitably normalized) complex polynomials of fixed degrees $d$, with
arbitrarily high probability. This set is an efficient and universal
\emph{probabilistic} set of starting points to find all roots of polynomials of
degree $d$ using Newton's method; the best known \emph{deterministic} set of
starting points consists of $\lceil 1.1d(\log d)^2\rceil$ points.; Comment: 17 pages, 6 figures. Minor update and slight improvements upon
referee recommendations

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## Choosing roots of polynomials with symmetries smoothly

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/03/2006
Português

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The roots of a smooth curve of hyperbolic polynomials may not in general be
parameterized smoothly, even not $C^{1,\alpha}$ for any $\alpha > 0$. A
sufficient condition for the existence of a smooth parameterization is that no
two of the increasingly ordered continuous roots meet of infinite order. We
give refined sufficient conditions for smooth solvability if the polynomials
have certain symmetries. In general a $C^{3n}$ curve of hyperbolic polynomials
of degree $n$ admits twice differentiable parameterizations of its roots. If
the polynomials have certain symmetries we are able to weaken the assumptions
in that statement.; Comment: 19 pages, 2 figures, LaTeX

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## On a Generalisation of Obreshkoff-Ehrlich Method for Simultaneous Extraction of All Roots of Polynomials Over an Arbitrary Chebyshev System

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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New modifications of the methods for simultaneous extraction of all roots of
polynomials over an arbitrary Chebyshev system are elaborated. A cubic
convergence of iterations is proved. The method presented is a generalisation
of the classical methods of Obreshkoff and Ehrlich for simultaneous seeking of
all roots of algebraic equations. Numerical examples are provided.; Comment: 4 pages

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## On a stratification defined by real roots of polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We consider the family of polynomials $P(x,a)=x^n+a_1x^{n-1}+... +a_n$,
$x,a_i\in {\bf R}$, and the stratification of ${\bf R}^n\cong \{(a_1,...
,a_n)|a_i\in {\bf R}\}$ defined by the multiplicity vector of the real roots of
$P$. We prove smoothness of the strata and a transversality property of their
tangent spaces.

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## Choosing roots of polynomials smoothly, II

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We show that the roots of any smooth curve of polynomials with only real
roots can be parametrized twice differentiably (but not better).; Comment: Amstex, 5 pages; some misprints corrected

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## Regularity of roots of polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Algebraic Geometry#26C10, 26A46, 30C15, 32S45

We show that smooth curves of monic complex polynomials $P_a
(Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset
\mathbb R$ a compact interval, have absolutely continuous roots in a uniform
way. More precisely, there exists a positive integer $k$ and a rational number
$p >1$, both depending only on the degree $n$, such that if $a_j \in C^{k}$
then any continuous choice of roots of $P_a$ is absolutely continuous with
derivatives in $L^q$ for all $1 \le q < p$, in a uniform way with respect to
$\max_j\|a_j\|_{C^k}$. The uniformity allows us to deduce also a multiparameter
version of this result. The proof is based on formulas for the roots of the
universal polynomial $P_a$ in terms of its coefficients $a_j$ which we derive
using resolution of singularities. For cubic polynomials we compute the
formulas as well as bounds for $k$ and $p$ explicitly.; Comment: 32 pages, 2 figures; minor changes; accepted for publication in Ann.
Sc. Norm. Super. Pisa Cl. Sci. (5)

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## Finding roots of polynomials over finite fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/06/2006
Português

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We propose an improved algorithm for finding roots of polynomials over finite
fields. This makes possible significant speedup of the decoding process of
Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and some other error-correcting
codes.; Comment: 6 pages. IEEE Transactions on Communications

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## Newton's method in practice: finding all roots of polynomials of degree one million efficiently

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/08/2015
Português

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#Mathematics - Numerical Analysis#Computer Science - Numerical Analysis#Mathematics - Dynamical Systems#49M15, 65H04, 37F10, 37N30, 37-04, 65-04

We use Newton's method to find all roots of several polynomials in one
complex variable of degree up to and exceeding one million and show that the
method, applied to appropriately chosen starting points, can be turned into an
algorithm that can be applied routinely to find all roots without deflation and
with the inherent numerical stability of Newton's method.
We specify an algorithm that provably terminates and finds all roots of any
polynomial of arbitrary degree, provided all roots are distinct and exact
computation is available. It is known that Newton's method is inherently
stable, so computing errors do not accumulate; we provide an exact bound on how
much numerical precision is sufficient.; Comment: 32 pages, 21 figures

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## Additive Complexity and the Roots of Polynomials Over Number Fields and p-adic Fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Consider any nonzero univariate polynomial with rational coefficients,
presented as an elementary algebraic expression (using only integer exponents).
Letting sigma(f) denotes the additive complexity of f, we show that the number
of rational roots of f is no more than 15 + sigma(f)^2 (24.01)^{sigma(f)}
sigma(f)!. This provides a sharper arithmetic analogue of earlier results of
Dima Grigoriev and Jean-Jacques Risler, which gave a bound of C^{sigma(f)^2}
for the number of real roots of f, for some constant C with 1

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