Łojasiewicz exponent of the gradient near the fiber

Ha Huy Vui; Nguyen Hong Duc

Displaying similar documents to “Łojasiewicz exponent of the gradient near the fiber”

On asymptotic critical values and the Rabier Theorem

Zbigniew Jelonek (2004)

Banach Center Publications

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Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f ₁, . . ., f_{m}) : X \to k^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with...

On gradient at infinity of semialgebraic functions

Didier D&#039;Acunto, Vincent Grandjean (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number $ϱ_{c} \leq 1$ such that |x|·|∇f| and ${| f (x) - c |}^{ϱ_{c}}$ are separated at infinity. If c is a regular value and $ϱ_{c} < 1$ , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

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We study the leading order behaviour of positive solutions of the equation $- Δ u + ϵ u - {| u |}^{p - 2} u + {| u |}^{q - 2} u = 0, x \in ℝ^{N}$ , where $N \geq 3$ , $q > p > 2$ and when $ϵ > 0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$ , $q$ and $N$ . The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*} = \frac{2 N}{N - 2}$ . For $p < 2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p > 2^{*}$ the solution asymptotically...

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...

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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We show that for a polynomial mapping $F = (f ₁, . . ., f ₘ) : ℂ^{n} \to ℂ^{m}$ the Łojasiewicz exponent $_{\infty} (F)$ of F is attained on the set $z \in ℂ^{n} : f ₁ (z) \cdot . . . \cdot f ₘ (z) = 0$ .

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .

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) >...

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

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Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_{m}$ be the mth coefficient of the square f(x)² of...

Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević (2002)

Colloquium Mathematicae

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We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f (x, y) = p x + (1 - p) y - \sum_{s = m}^{\infty} s (x, y)$ uniformly in a neighborhood of the origin, where m > 1, $_{s} (x, y) = \sum_{i = 0}^{s} a_{i, s} x^{s - i} y^{i}$ ; (c) $ₘ (1, 1) = \sum_{i = 0}^{m} a_{i, m} > 0$ . Let x₀,x₁ ∈ (0,α) and $x_{n + 1} = f (x ₙ, x_{n - 1})$ , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $x ₙ \sim {((2 - p) / ((m - 1) \sum_{i = 0}^{m} a_{i, m}))}^{1 / (m - 1)} 1 / \sqrt[m - 1]{n}$ .

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $𝔻$ denote the unit disk ${z : | z | < 1}$ in the complex plane $ℂ$ . In this paper, we study a family of polynomials $P$ with only one zero lying outside $\bar{𝔻}$ . We establish criteria for $P$ to satisfy implying that each of $P$ and $P^{'}$ has exactly one critical point outside $\bar{𝔻}$ .

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Convergence of minimax structures and continuation of critical points for singularly perturbed systems

Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)

Journal of the European Mathematical Society

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In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $- Δ u + u^{3} + β u v^{2} = λ u, - Δ v + v^{3} + β u^{2} v = μ v, u, v \in H_{0}^{1} (Ω), u, v > 0$ , as the interspecies scattering length $β$ goes to $+ \infty$ . For this system we consider the associated energy functionals $J_{β}, β \in (0, + \infty)$ , with $L^{2}$ -mass constraints, which limit $J_{\infty}$ (as $β \to + \infty$ ) is strongly irregular. For such functionals, we construct multiple critical points...

On the value set of small families of polynomials over a finite field, II

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in $_{q} [T]$ of degree d for which s consecutive coefficients $a = (a_{d - 1}, . . ., a_{d - s})$ are fixed. Our estimate asserts that $(d, s, a) = μ_{d} q + (q^{1 / 2})$ , where $μ_{d} : = \sum_{r = 1}^{d} ({(- 1)}^{r - 1}) / (r!)$ . We also prove that $₂ (d, s, a) = μ ²_{d} q ² + (q^{3 / 2})$ , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of $_{q} [T]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂ (d, 0) = μ ²_{d} q ² + (q)$ , where ₂(d,0) denotes the average second moment for...

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino, Monica Musso, Frank Pacard (2010)

Journal of the European Mathematical Society

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The role of the second critical exponent $p = (n + 1) / (n - 3)$ , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $Δ u + u^{p} = 0$ , $u > 0$ under zero Dirichlet boundary conditions, in a domain $Ω$ in $ℝ^{n}$ with bounded, smooth boundary. Given $Γ$ , a geodesic of the boundary with negative inner normal curvature we find that for $p = (n + 1) / (n - 3 - ε)$ , there exists a solution $u_{ε}$ such that $| \nabla u_{ε} |^{2}$ converges weakly to a Dirac measure on $Γ$ as $ε \to 0^{+}$ , provided that $Γ$ is nondegenerate in the sense of second...

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.

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study the extensibility of piecewise polynomial functions defined on closed subsets of $ℝ^{2}$ to all of $ℝ^{2}$ . The compact subsets of $ℝ^{2}$ on which every piecewise polynomial function is extensible to $ℝ^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$ . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

On the irreducible factors of a polynomial over a valued field

Anuj Jakhar (2024)

Czechoslovak Mathematical Journal

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We explicitly provide numbers $d$ , $e$ such that each irreducible factor of a polynomial $f (x)$ with integer coefficients has a degree greater than or equal to $d$ and $f (x)$ can have at most $e$ irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field.

Weak polynomial identities and their applications

Vesselin Drensky (2021)

Communications in Mathematics

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Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$ . The polynomial $f (x_{1}, ..., x_{n})$ of the free associative algebra $K 〈 x_{1}, x_{2}, ... 〉$ is a weak polynomial identity for the pair $(R, V)$ if it vanishes in $R$ when evaluated on $V$ . We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of...

On a class of $(p, q)$ -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi (2017)

Communications in Mathematics

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Let $Ω \subset ℝ^{n}$ be a bounded starshaped domain and consider the $(p, q)$ -Laplacian problem $\begin{matrix} - Δ_{p} u - Δ_{q} u = λ (𝐱) {| u |}^{p^{☆} - 2} u + μ {| u |}^{r - 2} u \end{matrix}$ where $μ$ is a positive parameter, $1 < q \leq p < n$ , $r \geq p^{☆}$ and $p^{☆} : = \frac{n p}{n - p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$ -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

Rigidity of critical circle mappings I

Edson de Faria, Welington de Melo (1999)

Journal of the European Mathematical Society

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We prove that two $C^{3}$ critical circle maps with the same rotation number in a special set $𝔸$ are $C^{1 + α}$ conjugate for some $α > 0$ provided their successive renormalizations converge together at an exponential rate in the $C^{0}$ sense. The set $𝔸$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of $C^{\infty}$ critical circle maps with the same rotation number that are not $C^{1 + β}$ conjugate for any $β > 0$ . The class of rotation numbers for which such examples exist...