Displaying similar documents to “Chebyshev polynomials and Pell equations over finite fields”

On some properties of Chebyshev polynomials

Hacène Belbachir, Farid Bencherif (2008)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

Letting T n (resp. U n ) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ( X k T n - k ) k and ( X k U n - k ) k for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space n [ X ] formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also T n and U n admit remarkableness integer coordinates on each of the two basis.

Chebyshev Distance

Roland Coghetto (2016)

Formalized Mathematics

Similarity:

In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of [...] ℰTn T n and in [20] he has formalized that [...] ℰTn T n is second-countable, we build (in the topological sense defined in [23]) a denumerable base of [...] ℰTn T n . Then we introduce the n-dimensional intervals (interval in n-dimensional Euclidean space, pavé (borné) de ℝn [16], semi-intervalle (borné) de ℝn [22]). We conclude with the definition of Chebyshev distance [11]. ...

Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond, Wen-Bin Zhang (2013)

Acta Arithmetica

Similarity:

If the counting function N(x) of integers of a Beurling generalized number system satisfies both 1 x - 2 | N ( x ) - A x | d x < and x - 1 ( l o g x ) ( N ( x ) - A x ) = O ( 1 ) , then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that 1 | N ( x ) - A x | x - 2 d x < and x - 1 ( l o g x ) ( N ( x ) - A x ) = O ( f ( x ) ) do not imply the Chebyshev bound.

Explicit extension maps in intersections of non-quasi-analytic classes

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

Similarity:

We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ( ) ( [ - 1 , 1 ] r ) ; (b) there is no continuous linear extension map from Λ ( ) ( r ) into ( ) ( r ) ; (c) under some additional assumption on , there is an explicit extension map from ( ) ( [ - 1 , 1 ] r ) into ( ) ( [ - 2 , 2 ] r ) by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

The transfinite diameter of the real ball and simplex

T. Bloom, L. Bos, N. Levenberg (2012)

Annales Polonici Mathematici

Similarity:

We calculate the transfinite diameter for the real unit ball B d : = x d : | x | 1 and the real unit simplex T d : = x + d : j = 1 d x j 1 .

Discriminants of Chebyshev radical extensions

T. Alden Gassert (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let t be any integer and fix an odd prime . Let Φ ( x ) = T n ( x ) - t denote the n -fold composition of the Chebyshev polynomial of degree shifted by t . If this polynomial is irreducible, let K = ( θ ) , where θ is a root of Φ . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on t that ensure K is monogenic. For other values of t , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of K and compute an integral basis for the ring...

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

Similarity:

We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula π = n = 0 ( ( 5 n + 3 ) n ! ( 2 n ) ! ) / ( 2 n - 1 ( 3 n + 2 ) ! ) with convergence as 13 . 5 - n , in much the same way as the Euler transformation gives π = n = 0 ( 2 n + 1 n ! n ! ) / ( 2 n + 1 ) ! with convergence as 2 - n . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

Renormings of c 0 and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

The aim of this paper is to show that for every Banach space ( X , · ) containing asymptotically isometric copy of the space c 0 there is a bounded, closed and convex set C X with the Chebyshev radius r ( C ) = 1 such that for every k 1 there exists a k -contractive mapping T : C C with x - T x > 1 1 / k for any x C .

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

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

Similarity:

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...

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

Similarity:

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...

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

Wojciech Banaszczyk, Artur Lipnicki (2015)

Annales Polonici Mathematici

Similarity:

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...

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

Similarity:

We consider the space of ultradifferentiable functions with compact supports and the space of polynomials on . A description of the space ( ) of polynomial ultradistributions as a locally convex direct sum is given.

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

Similarity:

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...

Kolmogorov problem in W r H ω [ 0 , 1 ] and extremal Zolotarev ω-splines

Bagdasarov Sergey K.

Similarity:

AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) f ( m ) ( ξ ) s u p , f W r H ω [ ξ , b ] , | | f | | [ a , b ] B ,for all 0 < m ≤ r, ξ ≤ a or ξ = (a+b)/2, all B > 0 and concave moduli of continuity ω on ℝ₊. It is shown that any extremal function = B , r , m , ω , ξ of the problem (*) enjoys the following two characteristic properties. First, the function ( r ) ( · ) - ( r ) ( ξ ) is extremal for the problem(**) ξ b h ( t ) ψ ( t ) d t s u p , h H ω [ ξ , b ] , h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...

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

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

Similarity:

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 ∈...

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

Tamás Erdélyi (2001)

Colloquium Mathematicae

Similarity:

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...

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

(2016)

Acta Arithmetica

Similarity:

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...

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 Lincei. Matematica e Applicazioni

Similarity:

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...

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

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.

Unit vector fields on antipodally punctured spheres: big index, big volume

Fabiano G. B. Brito, Pablo M. Chacón, David L. Johnson (2008)

Bulletin de la Société Mathématique de France

Similarity:

We establish in this paper a lower bound for the volume of a unit vector field v defined on 𝐒 n { ± x } , n = 2 , 3 . This lower bound is related to the sum of the absolute values of the indices of v at x and - x .

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

Similarity:

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.

Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari (2018)

Czechoslovak Mathematical Journal

Similarity:

Let V be the complex vector space of homogeneous linear polynomials in the variables x 1 , ... , x m . Suppose G is a subgroup of S m , and χ is an irreducible character of G . Let H d ( G , χ ) be the symmetry class of polynomials of degree d with respect to G and χ . For any linear operator T acting on V , there is a (unique) induced operator K χ ( T ) End ( H d ( G , χ ) ) acting on symmetrized decomposable polynomials by K χ ( T ) ( f 1 * f 2 * ... * f d ) = T f 1 * T f 2 * ... * T f d . In this paper, we show that the representation T K χ ( T ) of the general linear group G L ( V ) is equivalent to the direct sum of χ ( 1 ) copies...

Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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...