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 the Łojasiewicz exponent of F is attained on the set .
Jacek Chądzyński, Tadeusz Krasiński (1997)
Annales Polonici Mathematici
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We show that for a polynomial mapping the Łojasiewicz exponent of F is attained on the set .
Rachid Boumahdi, Jesse Larone (2018)
Archivum Mathematicum
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Let be a polynomial with integral coefficients. Shapiro showed that if the values of at infinitely many blocks of consecutive integers are of the form , where is a polynomial with integral coefficients, then for some polynomial . In this paper, we show that if the values of at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form where is an integer greater than 1, then for some polynomial .
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 be the mth coefficient of the square f(x)² of...
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 in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than with .
Carlos A. Gómez, Florian Luca (2021)
Commentationes Mathematicae Universitatis Carolinae
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We consider the polynomial for which arises as the characteristic polynomial of the -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of which lie inside the unit disk.
Bartłomiej Bzdęga (2016)
Acta Arithmetica
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We prove that for every ε > 0 and every nonnegative integer w there exist primes such that for the height of the cyclotomic polynomial is at least , where and is a constant depending only on w; furthermore . In our construction we can have for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.
Innocent Ndikubwayo (2020)
Czechoslovak Mathematical Journal
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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence generated by a three-term recurrence relation with the standard initial conditions where and are arbitrary real polynomials.
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 : There does not exist a sequence of complex singular points of F such that the imaginary parts tend to (0,0), the real parts tend to ∞ and . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition and if, in addition, the restriction of F to every real level set is proper for values of |c| large enough.
Mohamed Amine Hachani (2017)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be a polynomial of degree having no zeros in , , and let . It was shown by Govil that if and are attained at the same point of the unit circle , then The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.
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 to all of . The compact subsets of on which every piecewise polynomial function is extensible to can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of . 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...
Xiang-dong Hou (2014)
Acta Arithmetica
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Let . We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of . This result allows us to solve a related problem: Let (n ≥ 0, ) be the polynomial defined by the functional equation . We determine all n of the form , α > β ≥ 0, for which is a permutation polynomial of .
Leokadia Białas-Cież (2011)
Banach Center Publications
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Let E be a compact set in the complex plane, be the Green function of the unbounded component of with pole at infinity and where the supremum is taken over all polynomials of degree at most n, and . The paper deals with recent results concerning a connection between the smoothness of (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence . Some additional conditions are given for special classes of sets.
(2016)
Acta Arithmetica
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For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≢ 0 of the form , , , such that divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that . We find the size of and for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...
Vladimir Petrov Kostov (2019)
Mathematica Bohemica
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By Descartes’ rule of signs, a real degree polynomial with all nonvanishing coefficients with sign changes and sign preservations in the sequence of its coefficients () has positive and negative roots, where and . For , for every possible choice of the sequence of signs of coefficients of (called sign pattern) and for every pair satisfying these conditions there exists a polynomial with exactly positive and exactly negative roots (all of them simple). For ...
Peter Borwein, Tamás Erdélyi, Géza Kós (2013)
Acta Arithmetica
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For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≠ 0 of the form , 1/paj ∈ ℂsuch that divides P(x). For n ∈ ℕ and L > 0 let be the largest possible value of k for which there is a polynomial P ≠ 0 of the form , , , such that divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...
Wojciech Kozłowski (2007)
Annales Polonici Mathematici
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In the space of polynomial p-forms in ℝⁿ we introduce some special inner product. Let be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that splits as the direct sum , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.
Wentang Kuo, Shuntaro Yamagishi (2021)
Czechoslovak Mathematical Journal
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Let denote the polynomial ring over , the finite field of elements. Suppose the characteristic of is not or . We prove that there exist infinitely many such that the set contains a Sidon set which is an additive basis of order .
Hamid Ben Yakkou, Jalal Didi (2024)
Mathematica Bohemica
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Let be a pure number field generated by a complex root of a monic irreducible polynomial , where , , are three positive natural integers. The purpose of this paper is to study the monogenity of . Our results are illustrated by some examples.
Christophe Debry (2014)
Journal de Théorie des Nombres de Bordeaux
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Let be a cubic, monic and separable polynomial over a field of characteristic and let be the elliptic curve given by . In this paper we prove that the coefficient at in the –th division polynomial of equals the coefficient at in . For elliptic curves over a finite field of characteristic , the first coefficient is zero if and only if is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...