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 .
T. Maćkowiak (1976)
Colloquium Mathematicae
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Didier D'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 .
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...
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.
Stanisław Spodzieja, Anna Szlachcińska (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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A mapping 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 can be reduced to the case m = n.
Romain Tessera (2007)
Bulletin de la Société Mathématique de France
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Let be a compactly generated locally compact group and let be a compact generating set. We prove that if has polynomial growth, then is a Følner sequence and we give a polynomial estimate of the rate of decay of 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...
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.
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₊.
M. K. Sen (1971)
Annales Polonici Mathematici
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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 .
M. K. Aouf (1988)
Matematički Vesnik
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K. Orlov (1981)
Matematički Vesnik
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Swami Jnanananda (1936)
Časopis pro pěstování matematiky a fysiky
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P. Srivastava, K. K. Azad (1981)
Matematički Vesnik
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Ralph McKenzie (1971)
Colloquium Mathematicae
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Janusz Matkowski (1989)
Annales Polonici Mathematici
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Stephan Baier (2004)
Acta Arithmetica
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A. Szymański (1977)
Colloquium Mathematicae
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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, , and such that 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, with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.
Stephan Baier (2005)
Acta Arithmetica
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А.М. Вершик (1972)
Zapiski naucnych seminarov Leningradskogo
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