Displaying similar documents to “On the proof of Erdős' inequality”

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 , . . . , f ) : n m the Łojasiewicz exponent ( F ) of F is attained on the set z n : f ( z ) · . . . · f ( z ) = 0 .

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 ) .

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...

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&amp;#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 | f | 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 - - 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 - - z - 1 for k 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.

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 = 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 c w 2 - w 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₊.

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

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 { P i } i = 1 generated by a three-term recurrence relation P i ( x ) + Q 1 ( x ) P i - 1 ( x ) + Q 2 ( x ) P i - 2 ( x ) = 0 with the standard initial conditions P 0 ( x ) = 1 , P - 1 ( x ) = 0 , where Q 1 ( x ) and Q 2 ( x ) are arbitrary real polynomials.

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 ) : There does not exist a sequence ( p k , q k ) ² of complex singular points of F such that the imaginary parts ( ( p k ) , ( q k ) ) tend to (0,0), the real parts ( ( p k ) , ( q k ) ) tend to ∞ and F ( ( p k ) , ( q k ) ) ) a ² . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition ( J ) and if, in addition, the restriction of F to every real level set P - 1 ( c ) is proper for values of |c| large enough.

Entire functions of exponential type not vanishing in the half-plane 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 1 , and let Q ( z ) : = z n P ( 1 / 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 ) | 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.

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 2 to all of 2 . The compact subsets of 2 on which every piecewise polynomial function is extensible to 2 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...

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 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 p [ x ] (n ≥ 0, p = c h a r q ) be the polynomial defined by the functional equation c 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 .

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

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Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of E with pole at infinity and M ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

Polynomials, sign patterns and Descartes' rule of signs

Vladimir Petrov Kostov (2019)

Mathematica Bohemica

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By Descartes’ rule of signs, a real degree d polynomial P with all nonvanishing coefficients with c sign changes and p sign preservations in the sequence of its coefficients ( c + p = d ) has pos c positive and ¬ p negative roots, where pos c ( mod 2 ) and ¬ p ( mod 2 ) . For 1 d 3 , for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair ( pos , neg ) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly ¬ negative roots (all of them simple). For d 4 ...

The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p 1/p , aj ∈ ℂ , such that ( x - 1 ) k divides P(x). For n ∈ ℕ and L > 0 let κ ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L m a x 1 j n | a j | , a j , such that ( x - 1 ) k divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that c 1 ( n / L ) - 1 κ ( n , L ) c 2 ( n / L ) for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space Λ p of polynomial p-forms in ℝⁿ we introduce some special inner product. Let H p be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that Λ p splits as the direct sum d * ( Λ p + 1 ) δ * ( Λ p - 1 ) H p , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

Sidon basis in polynomial rings over finite fields

Wentang Kuo, Shuntaro Yamagishi (2021)

Czechoslovak Mathematical Journal

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Let 𝔽 q [ t ] denote the polynomial ring over 𝔽 q , the finite field of q elements. Suppose the characteristic of 𝔽 q is not 2 or 3 . We prove that there exist infinitely many N such that the set { f 𝔽 q [ t ] : deg f < N } contains a Sidon set which is an additive basis of order 3 .

On monogenity of certain pure number fields of degrees 2 r · 3 k · 7 s

Hamid Ben Yakkou, Jalal Didi (2024)

Mathematica Bohemica

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Let K = ( α ) be a pure number field generated by a complex root α of a monic irreducible polynomial F ( x ) = x 2 r · 3 k · 7 s - m [ x ] , where r , k , s are three positive natural integers. The purpose of this paper is to study the monogenity of K . Our results are illustrated by some examples.

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

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Let f ( x ) be a cubic, monic and separable polynomial over a field of characteristic p 3 and let E be the elliptic curve given by y 2 = f ( x ) . In this paper we prove that the coefficient at x 1 2 p ( p - 1 ) in the p –th division polynomial of E equals the coefficient at x p - 1 in f ( x ) 1 2 ( p - 1 ) . For elliptic curves over a finite field of characteristic p , the first coefficient is zero if and only if E is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...