Exponent of growth of polynomial mappings of $C^2$ into $C^2$

Jacek Chadzyński; Tadeusz Krasiński

Displaying similar documents to “Exponent of growth of polynomial mappings of $C^{2}$ into $C^{2}$ ”

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 $z \in ℂ^{n} : f ₁ (z) \cdot . . . \cdot f ₘ (z) = 0$ .

On sets of confluent and related mappings in the space $Y^{X}$

T. Maćkowiak (1976)

Colloquium Mathematicae

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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 $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

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.

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...

Entire functions of exponential type not vanishing in the half-plane $ℑ z > k$ , where $k > 0$

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $P (z)$ be a polynomial of degree $n$ having no zeros in $| z | < k$ , $k \leq 1$ , and let $Q (z) : = z^{n} \bar{P (1 / \bar{z})}$ . It was shown by Govil that if ${max}_{| z | = 1} | P^{'} (z) |$ and ${max}_{| z | = 1} | Q^{'} (z) |$ are attained at the same point of the unit circle $| z | = 1$ , then $max_{| z | = 1} | P^{'} (z) | \leq \frac{n}{1 + k^{n}} max_{| z | = 1} | P (z) | .$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

Polynomials with values which are powers of integers

Rachid Boumahdi, Jesse Larone (2018)

Archivum Mathematicum

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Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q (m)$ , where $Q$ is a polynomial with integral coefficients, then $P (x) = Q (R (x))$ for some polynomial $R$ . In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^{q}$ where $q$ is an integer greater than 1, then $P (x) = {(R (x))}^{q}$ for some polynomial $R (x)$ .

Łojasiewicz Exponent of Overdetermined Mappings

Stanisław Spodzieja, Anna Szlachcińska (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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A mapping $F : ℝ ⁿ \to ℝ^{m}$ is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping $F : ℝ ⁿ \to ℝ^{m}$ can be reduced to the case m = n.

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.

Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

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Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then ${(U^{n})}_{n \in ℕ}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $\frac{μ (U^{n + 1} ∖ U^{n})}{μ (U^{n})} .$ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain...

On nonsingular polynomial maps of ℝ²

Nguyen Van Chau, Carlos Gutierrez (2006)

Annales Polonici Mathematici

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We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $(J_{\infty})$ : There does not exist a sequence $(p_{k}, q_{k}) \subset ℂ ²$ of complex singular points of F such that the imaginary parts $(ℑ (p_{k}), ℑ (q_{k}))$ tend to (0,0), the real parts $(ℜ (p_{k}), ℜ (q_{k}))$ tend to ∞ and $F (ℜ (p_{k}), ℜ (q_{k}))) \to a \in ℝ ²$ . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $(J_{\infty})$ and if, in addition, the restriction of F to every real level set $P^{- 1} (c)$ is proper for values of |c| large enough.

On a generalization of the Beiter Conjecture

Bartłomiej Bzdęga (2016)

Acta Arithmetica

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We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_{1}, . . ., p_{w}$ such that for $n = p_{1} . . . p_{w}$ the height of the cyclotomic polynomial $Φ_{n}$ is at least $(1 - ε) c_{w} M_{n}$ , where $M_{n} = \prod_{i = 1}^{w - 2} p_{i}^{2^{w - 1 - i} - 1}$ and $c_{w}$ is a constant depending only on w; furthermore $l i m_{w \to \infty} c_{w}^{2^{- w}} \approx 0.71$ . In our construction we can have $p_{i} > h (p_{1} . . . p_{i - 1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Degrees of compatible $L$ -subsets and compatible mappings

Fu Gui Shi, Yan Sun (2024)

Kybernetika

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Based on a completely distributive lattice $L$ , degrees of compatible $L$ -subsets and compatible mappings are introduced in an $L$ -approximation space and their characterizations are given by four kinds of cut sets of $L$ -subsets and $L$ -equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible $L$ -subsets and compatible degrees of $L$ -subsets. Finally, the notion of complete $L$ -sublattices is introduced and it is shown...

On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

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Let $f = a x + b x^{q} + x^{2 q - 1} \in_{q} [x]$ . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of $_{q ²}$ . This result allows us to solve a related problem: Let $g_{n, q} \in_{p} [x]$ (n ≥ 0, $p = c h a r_{q}$ ) be the polynomial defined by the functional equation $\sum_{c \in_{q}} {(x + c)}^{n} = g_{n, q} (x^{q} - x)$ . We determine all n of the form $n = q^{α} - q^{β} - 1$ , α > β ≥ 0, for which $g_{n, q}$ is a permutation polynomial of $_{q ²}$ .