Displaying similar documents to “On a problem of Sidon for polynomials over finite fields”

On the Győry-Sárközy-Stewart conjecture in function fields

Igor E. Shparlinski (2018)

Czechoslovak Mathematical Journal

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We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product ( a b + 1 ) ( a c + 1 ) ( b c + 1 ) for distinct positive integers a , b and c . In particular, we show that, under some natural conditions on rational functions F , G , H ( X ) , the number of distinct zeros and poles of the shifted products F H + 1 and G H + 1 grows linearly with deg H if deg H max { deg F , deg G } . We also obtain a version of this result for rational functions over a finite field.

Polynomials and degrees of maps in real normed algebras

Takis Sakkalis (2020)

Communications in Mathematics

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Let 𝒜 be the algebra of quaternions or octonions 𝕆 . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f ( t ) 𝒜 [ t ] has a root in 𝒜 . As a consequence, the Jacobian determinant | J ( f ) | is always non-negative in 𝒜 . Moreover, using the idea of the topological degree we show that a regular polynomial g ( t ) over 𝒜 has also a root in 𝒜 . Finally, utilizing multiplication ( * ) in 𝒜 , we prove various results on the topological degree...

Robin functions and extremal functions

T. Bloom, N. Levenberg, S. Ma'u (2003)

Annales Polonici Mathematici

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Given a compact set K N , for each positive integer n, let V ( n ) ( z ) = V K ( n ) ( z ) := sup 1 / ( d e g p ) V p ( K ) ( p ( z ) ) : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions V K ( n ) are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, V K ( z ) := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, | | p | | K 1 ]. Our main result is that if K is regular, then all of the functions V K ( n ) are continuous; and their associated Robin functions ϱ V K ( n ) ( z ) : = l i m s u p | λ | [ V K ( n ) ( λ z ) - l o g ( | λ | ) ] increase to ϱ K : = ϱ V K for all z outside a pluripolar...

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...

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 .

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...

Thompson’s conjecture for the alternating group of degree 2 p and 2 p + 1

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group G denote by N ( G ) the set of conjugacy class sizes of G . In 1980s, J. G. Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N ( G ) = N ( L ) , then G L . We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z ( G ) = 1 and N ( G ) = N ( A i ) is necessarily isomorphic to A i , where i { 2 p , 2 p + 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 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...

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₊.

Variations on a question concerning the degrees of divisors of x n - 1

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial x n - 1 : “How often does x n - 1 have a divisor of every degree between 1 and n ?” In a previous paper, we considered the situation when x n - 1 is factored in [ x ] . In this paper, we replace [ x ] with 𝔽 p [ x ] , where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p .

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

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Let C be a hyperelliptic curve of genus g 1 over a number field K with good reduction outside a finite set of places S of K . We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g , S and K . In particular, we obtain that for any given number field K , finite set of places S of K and integer g 1 one can in principle determine the set of K -isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside...

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.

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 ) .