Łojasiewicz Exponent of Overdetermined Mappings

Stanisław Spodzieja; Anna Szlachcińska

Displaying similar documents to “Łojasiewicz Exponent of Overdetermined Mappings”

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

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

Jacek Chadzyński, Tadeusz Krasiński (1988)

Banach Center Publications

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A characterization of complex $L_{1}$ -preduals via a complex barycentric mapping

Petr Petráček, Jiří Spurný (2016)

Commentationes Mathematicae Universitatis Carolinae

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We provide a complex version of a theorem due to Bednar and Lacey characterizing real $L_{1}$ -preduals. Hence we prove a characterization of complex $L_{1}$ -preduals via a complex barycentric mapping.

Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, $W \subset k^{n + 1}$ , and $F : k ⁿ \to k^{n + 1}$ such that ${F |}_{V} = f$ and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, $W \subset k^{m}$ with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

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

The Brouwer Fixed Point Theorem for Some Set Mappings

Dariusz Miklaszewski (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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For some classes $X \subset 2^{ₙ}$ of closed subsets of the disc ₙ ⊂ ℝⁿ we prove that every Hausdorff-continuous mapping f: X → X has a fixed point A ∈ X in the sense that the intersection A ∩ f(A) is nonempty.

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...

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 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.

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 $R (x)$ .

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₊.

Diagonals of separately continuous functions of $n$ variables with values in strongly $σ$ -metrizable spaces

Olena Karlova, Volodymyr Mykhaylyuk, Oleksandr Sobchuk (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove the result on Baire classification of mappings $f : X \times Y \to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $P P$ -space, $Y$ is a topological space and $Z$ is a strongly $σ$ -metrizable space with additional properties. We show that for any topological space $X$ , special equiconnected space $Z$ and a mapping $g : X \to Z$ of the $(n - 1)$ -th Baire class there exists a strongly separately continuous mapping $f : X^{n} \to Z$ with the diagonal $g$ . For wide classes...

$F_{σ}$ -mappings and the invariance of absolute Borel classes

Petr Holický, Jiří Spurný (2004)

Fundamenta Mathematicae

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It is proved that $F_{σ}$ -mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any $F_{σ}$ -mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable $F_{σ}$ -set $X_{\infty} \subset X$ satisfying $f (X_{\infty}) = Y$ .

On a result by Clunie and Sheil-Small

Dariusz Partyka, Ken-ichi Sakan (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $𝔻$ , if $F (𝔻)$ is a convex domain, then the inequality $| G (z_{2}) - G (z_{1}) | < | H (z_{2}) - H (z_{1}) |$ holds for all distinct points $z_{1}, z_{2} \in 𝔻$ . Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F = H + \bar{G}$ , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $Ω$ in $ℂ$ and improve it provided $F$ is additionally a quasiconformal mapping...

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) >...

Equidistribution towards the Green current

Vincent Guedj (2003)

Bulletin de la Société Mathématique de France

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Let $f : ℙ^{k} \to ℙ^{k}$ be a dominating rational mapping of first algebraic degree $λ \geq 2$ . If $S$ is a positive closed current of bidegree $(1, 1)$ on $ℙ^{k}$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks $λ^{- n} {(f^{n})}^{*} S$ converge to the Green current $T_{f}$ . For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$ -invariant currents.

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 $_{q ²}$ .