Exponent of growth of polynomial mappings of into
Jacek Chadzyński, Tadeusz Krasiński (1988)
Banach Center Publications
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Jacek Chadzyński, Tadeusz Krasiński (1988)
Banach Center Publications
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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.
Gary Hosler-Meisters (1989)
Banach Center Publications
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Elisa R. Santos (2014)
Studia Mathematica
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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies . We study the stability of the APDP by c₀-, - and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely and C(K,X), where X has the APDP.
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 .
Zhibin Du, Carlos Martins da Fonseca (2023)
Czechoslovak Mathematical Journal
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We analyse the roots of the polynomial for . This is the characteristic polynomial of the recurrence relation for , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.
Vladimir Kostov (2018)
Czechoslovak Mathematical Journal
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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree polynomials. ...
Lai-Yi Zhu, Da-Peng Zhou (2017)
Czechoslovak Mathematical Journal
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Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality for a constrained polynomial of degree at most , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval and establish a new asymptotically sharp inequality. ...
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.
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 and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then 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 , where are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...
Vesselin Drensky (2021)
Communications in Mathematics
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Let be an associative algebra over a field generated by a vector subspace . The polynomial of the free associative algebra is a weak polynomial identity for the pair if it vanishes in when evaluated on . 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...
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 .
T. Maćkowiak (1976)
Colloquium Mathematicae
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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...
Dariusz Miklaszewski (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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For some classes 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.
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...
E. Reyes, A. Montesinos, P. M. Gadea (1987)
Colloquium Mathematicae
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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.
Ludwik M. Drużkowski (1997)
Annales Polonici Mathematici
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It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form , where are homogeneous polynomials of degree 3 with real coefficients (or ), j = 1,...,n and H’(x) is a nilpotent matrix for each . We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case , where . Note that the above inequality is not true when the coefficients...
Zhixiong Chen, Arne Winterhof (2015)
Acta Arithmetica
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For an odd prime p and an integer w ≥ 1, polynomial quotients are defined by with , u ≥ 0, which are generalizations of Fermat quotients . First, we estimate the number of elements for which for a given polynomial f(x) over the finite field . In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...