Displaying similar documents to “The algebra of polynomials on the space of ultradifferentiable functions”

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

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 | 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 .

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 ∈...

On the lattice of polynomials with integer coefficients: the covering radius in L p ( 0 , 1 )

Wojciech Banaszczyk, Artur Lipnicki (2015)

Annales Polonici Mathematici

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The paper deals with the approximation by polynomials with integer coefficients in L p ( 0 , 1 ) , 1 ≤ p ≤ ∞. Let P n , r be the space of polynomials of degree ≤ n which are divisible by the polynomial x r ( 1 - x ) r , r ≥ 0, and let P n , r P n , r be the set of polynomials with integer coefficients. Let μ ( P n , r ; L p ) be the maximal distance of elements of P n , r from P n , r in L p ( 0 , 1 ) . We give rather precise quantitative estimates of μ ( P n , r ; L ) for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of μ ( P n , r ; L p ) for p ≠ 2. It follows that μ ( P n , r ; L p ) n - 2 r - 2 / p as n → ∞. The results...

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...

The factorization of f ( x ) x n + g ( x ) with f ( x ) monic and of degree 2 .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we investigate the factorization of the polynomials f ( x ) x n + g ( x ) [ x ] in the special case where f ( x ) is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that f ( x ) is monic and linear.

The norm of the polynomial truncation operator on the unit disk and on [-1,1]

Tamás Erdélyi (2001)

Colloquium Mathematicae

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Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. c ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials P c of the form P ( z ) : = j = 0 n a j z j , a j C , by S ( P ) ( z ) : = j = 0 n a ̃ j z j , a ̃ j : = a j | a j | m i n | a j | , 1 (here 0/0 is interpreted as 1). We define the norms of the truncation operators by S , D r e a l : = s u p P ( m a x z D | S ( P ) ( z ) | ) / ( m a x z D | P ( z ) | ) , S , D c o m p : = s u p P c ( m a x z D | S ( P ) ( z ) | ) / ( m a x z D | P ( z ) | . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

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...

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

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Let K be any subset of N . We define a pluricomplex Green’s function V K , θ for θ-incomplete polynomials. We establish properties of V K , θ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of N , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on K s u p p ( d d c V K , θ ) N . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute s u p p ( d d c V K , θ ) N when K is a compact...

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let · be the uniform norm in the unit disk. We study the quantities M n ( α ) : = inf ( z P ( z ) + α - α ) where the infimum is taken over all polynomials P of degree n - 1 with P ( z ) = 1 and α > 0 . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that inf α > 0 M n ( α ) = 1 / n . We find the exact values of M n ( α ) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

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₊.

On the Gauss-Lucas'lemma in positive characteristic

Umberto Bartocci, Maria Cristina Vipera (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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If f ( x ) is a polynomial with coefficients in the field of complex numbers, of positive degree n , then f ( x ) has at least one root a with the following property: if μ k n , where μ is the multiplicity of α , then f ( k ) ( α ) 0 (such a root is said to be a "free" root of f ( x ) ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree n ) with coefficients in a field of positive characteristic p > n (Sudbery's Conjecture). In this paper it...

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let 𝔻 denote the unit disk { z : | z | < 1 } in the complex plane . In this paper, we study a family of polynomials P with only one zero lying outside 𝔻 ¯ .  We establish  criteria for P to satisfy implying that each of P and P '   has exactly one critical point outside 𝔻 ¯ .

Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján

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The coefficients of the greatest common divisor of two polynomials f and g (GCD ( f , g ) ) can be obtained from the Sylvester subresultant matrix S j ( f , g ) transformed to lower triangular form, where 1 j d and d = deg(GCD ( f , g ) ) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of S j ( f , g ) for an arbitrary allowable j are in details described and an algorithm for the calculation of the GCD ( f , g ) is formulated. If inexact polynomials are given, then an approximate...

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...

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

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Let q [ t ] be the polynomial ring over the finite field q , and let N be the subset of q [ t ] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A N for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that D ( N ) q N ( l o g N ) 7 / N .

On the value set of small families of polynomials over a finite field, II

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in q [ T ] of degree d for which s consecutive coefficients a = ( a d - 1 , . . . , a d - s ) are fixed. Our estimate asserts that ( d , s , a ) = μ d q + ( q 1 / 2 ) , where μ d : = r = 1 d ( ( - 1 ) r - 1 ) / ( r ! ) . We also prove that ( d , s , a ) = μ ² d q ² + ( q 3 / 2 ) , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of q [ T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that ( d , 0 ) = μ ² d q ² + ( q ) , where ₂(d,0) denotes the average second moment for...

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.

On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)

Journal de Théorie des Nombres de Bordeaux

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We show that the discriminant of the generalized Laguerre polynomial L n ( α ) ( x ) is a non-zero square for some integer pair ( n , α ) , with n 1 , if and only if ( n , α ) belongs to one of 30 explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of L n ( α ) ( x ) over is the alternating group A n . For example, we establish that for all but finitely many positive integers n 2 ( mod 4 ) , the only α for which the Galois group of L n ( α ) ( x ) over is A n is...

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 .

Relative exactness modulo a polynomial map and algebraic ( p , + ) -actions

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

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Let F = ( f 1 , ... , f q ) be a polynomial dominating map from n to  q . We study the quotient 𝒯 1 ( F ) of polynomial 1-forms that are exact along the generic fibres of F , by 1-forms of type d R + a i d f i , where R , a 1 , ... , a q are polynomials. We prove that 𝒯 1 ( F ) is always a torsion [ t 1 , ... , t q ] -module. Then we determine under which conditions on F we have 𝒯 1 ( F ) = 0 . As an application, we study the behaviour of a class of algebraic ( p , + ) -actions on n , and determine in particular when these actions are trivial.

Linear maps preserving elements annihilated by the polynomial X Y - Y X

Jianlian Cui, Jinchuan Hou (2006)

Studia Mathematica

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Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by T the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies Φ ( A ) Φ ( B ) = Φ ( B ) Φ ( A ) for all A, B ∈ ℬ(H) with A B = B A if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that Φ ( A ) = c U A U for all A ∈ ℬ(H).