Some remarks about Descartes’ rule of signs

Albouy, Alain; Fu, Yanning

doi:10.4171/em/262

Cited by 25 publications

(79 citation statements)

References 8 publications

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“…The crucial role of sign vectors in the characterization of existence and uniqueness of positive solutions to parametrized polynomial equations suggests a comparison with Descartes' rule of signs for univariate (generalized) polynomials [47,35,29]. A sharp rule [1] states that a univariate polynomial with given sign sequence has exactly one positive solution for all (positive) coefficients if and only if there is exactly one sign change. Indeed, this statement follows from our main result which can be seen as a multivariate generalization of the sharp Descartes' rule for exactly one positive solution.…”

Section: Introductionmentioning

confidence: 99%

On the Bijectivity of Families of Exponential/Generalized Polynomial Maps

Müller¹,

Hofbauer²,

Regensburger³

2019

SIAM J. Appl. Algebra Geometry

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We start from a parametrized system of d generalized polynomial equations (with real exponents) for d positive variables, involving n generalized monomials with n positive parameters. Existence and uniqueness of a solution for all parameters and for all right-hand sides is equivalent to the bijectivity of (every element of) a family of generalized polynomial/exponential maps. We characterize the bijectivity of the family of exponential maps in terms of two linear subspaces arising from the coefficient and exponent matrices, respectively. In particular, we obtain conditions in terms of sign vectors of the two subspaces and a nondegeneracy condition involving the exponent subspace itself. Thereby, all criteria can be checked effectively. Moreover, we characterize when the existence of a unique solution is robust with respect to small perturbations of the exponents or/and the coefficients. In particular, we obtain conditions in terms of sign vectors of the linear subspaces or, alternatively, in terms of maximal minors of the coefficient and exponent matrices. Finally, we present applications to chemical reaction networks with (generalized) mass-action kinetics.

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Section: Introductionmentioning

confidence: 99%

On the Bijectivity of Families of Exponential/Generalized Polynomial Maps

Müller¹,

Hofbauer²,

Regensburger³

2019

SIAM J. Appl. Algebra Geometry

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“…(3) For real, but not necessarily hyperbolic degree d polynomials, one can ask the question: It seems that the question has been explicitly formulated for the first time in [2]. The answer to it is not trivial and the exhaustive one is known for d ≤ 8, see [7], [1], [5], [8] and [9]. The proof of the realizability of certain cases is often done by means of a concatenation lemma, see Lemma 2 in Section 7.…”

Section: Remarksmentioning

confidence: 99%

Descartes' rule of signs and moduli of roots

Kostov¹

2020

Publ. Math. Debrecen

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A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly c positive and exactly p negative roots counted with multiplicity, where c and p are the numbers of sign changes and sign preservations in the sequence of its coefficients. For c = 1 and 2, we discuss the question: When the moduli of all the roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its positive roots depending on the positions of the sign changes in the sequence of coefficients?

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“…It is clear that if Theorem 1 is true, then one should not be able to deduce with the help of Lemma 2 the realizability of the sign pattern σ 0 with the admissible pair (1,8). Now we show that this is indeed impossible.…”

Section: Commentsmentioning

confidence: 99%

“…In 1828 Carl Friedrich Gauss (1777-1855) has shown that if the roots are counted with multiplicity, then the number of positive roots has the same parity as c. When applied to P (−x), these results give an upper bound on the number of negative roots of P . It is proved in [1] that all possible cases (i.e. of c, c − 2, c − 4, .…”

Section: Introductionmentioning

confidence: 99%

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Polynomials, sign patterns and Descartes' rule of signs

Kostov¹

2018

Math.Boh.

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By Descartes' rule of signs, a real degree d polynomial P with all nonvanishing coefficients, with c sign changes and p sign preservations in the sequence of its coefficients (c + p = d) has pos ≤ c positive and neg ≤ p negative roots, where pos ≡ c( mod 2) and neg ≡ p( mod 2). For 1 ≤ d ≤ 3, for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair (pos, neg) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly neg negative roots (all of them simple). For d ≥ 4 this is not so. It was observed that for 4 ≤ d ≤ 10, in all nonrealizable cases either pos = 0 or neg = 0. It was conjectured that this is the case for any d ≥ 4. We show a counterexample to this conjecture for d = 11. Namely, we prove that for the sign pattern (+, −, −, −, −, −, +, +, +, +, +, −) and the pair (1, 8) there exists no polynomial with 1 positive, 8 negative simple roots and a complex conjugate pair.

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Some remarks about Descartes’ rule of signs

Cited by 25 publications

References 8 publications

On the Bijectivity of Families of Exponential/Generalized Polynomial Maps

On the Bijectivity of Families of Exponential/Generalized Polynomial Maps

Descartes' rule of signs and moduli of roots

Polynomials, sign patterns and Descartes' rule of signs

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