On the vectors associated with the roots of max-plus characteristic polynomials

Yuki Nishida; Sennosuke Watanabe; Yoshihide Watanabe

Displaying similar documents to “On the vectors associated with the roots of max-plus characteristic polynomials”

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p, n)$ , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials. ...

Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du, Carlos Martins da Fonseca (2023)

Czechoslovak Mathematical Journal

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We analyse the roots of the polynomial $x^{n} - p x^{n - 1} - q x - 1$ for $p ⩾ q ⩾ 1$ . This is the characteristic polynomial of the recurrence relation $F_{k, p, q} (n) = p F_{k, p, q} (n - 1) + q F_{k, p, q} (n - k + 1) + F_{k, p, q} (n - k)$ for $n ⩾ k$ , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f |}_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

Cycles on algebraic models of smooth manifolds

Wojciech Kucharz (2009)

Journal of the European Mathematical Society

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Every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of $M$ . We study modulo 2 homology classes represented by algebraic subsets of $X$ , as $X$ runs through the class of all algebraic models of $M$ . Our main result concerns the case where $M$ is a spin manifold.

On the distribution of the roots of polynomial $z^{k} - z^{k - 1} - \dots - z - 1$

Carlos A. Gómez, Florian Luca (2021)

Commentationes Mathematicae Universitatis Carolinae

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We consider the polynomial $f_{k} (z) = z^{k} - z^{k - 1} - \dots - z - 1$ for $k \geq 2$ which arises as the characteristic polynomial of the $k$ -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of $f_{k} (z)$ which lie inside the unit disk.

The R₂ measure for totally positive algebraic integers

V. Flammang (2016)

Colloquium Mathematicae

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Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates $α ₁ = α, . . ., α_{d}$ are positive real numbers. We study the set ₂ of the quantities ${(\prod_{i = 1}^{d} {(1 + α ²_{i})}^{1 / 2})}^{1 / d}$ . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.

Polynomial relations amongst algebraic units of low measure

John Garza (2014)

Acta Arithmetica

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For an algebraic number field and a subset $α_{1}, . . ., α_{r} \subseteq_{}$ , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $ℚ [x_{1}, . . ., x_{r}]$ vanishing at the point $(α_{1}, . . ., α_{r})$ .

Rational solutions of certain Diophantine equations involving norms

Maciej Ulas (2014)

Acta Arithmetica

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We present some results concerning the unirationality of the algebraic variety $_{f}$ given by the equation $N_{K / k} (X ₁ + α X ₂ + α ² X ₃) = f (t)$ , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and $_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_{f}$ is k-unirational. A similar result is proved for a broad family of quintic polynomials...

An alternative polynomial Daugavet property

Elisa R. Santos (2014)

Studia Mathematica

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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $m a x_{ω \in} | | I d + ω P | | = 1 + | | P | |$ . We study the stability of the APDP by c₀-, $ℓ_{\infty}$ - and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty} (μ, X)$ and C(K,X), where X has the APDP.

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

Symmetric identity for polynomial sequences satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$

Farid Bencherif, Rachid Boumahdi, Tarek Garici (2021)

Communications in Mathematics

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Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$ with $A_{0} (x)$ a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

Algebraic Superconnections, $S U (2 ∣ 1)$ and Electroweak Interactions

R. Coquereaux (1992)

Recherche Coopérative sur Programme n°25

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Diagonalization and rationalization of algebraic Laurent series

Boris Adamczewski, Jason P. Bell (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^{A}$ and height at most $A p^{A}$ , where $A$ is an effective constant that only...

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an $2 n \times 2 n$ Brualdi-Li tournament matrix $B_{2 n}$ . We show that $B_{2 n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2 n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.