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Decomposition of perturbed Chebyshev polynomials
, 2007
"... We characterize polynomial decomposition fn = r ◦ q with r, q ∈ C[x] of perturbed Chebyshev polynomials defined by the recurrence f0(x) = b, f1(x) = x − c, fn+1(x) = (x − d)fn(x) − afn−1(x), n ≥ 1, where a, b, c, d ∈ R and a> 0. These polynomials generalize the Chebyshev polynomials, which are o ..."
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Cited by 7 (2 self)
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We characterize polynomial decomposition fn = r ◦ q with r, q ∈ C[x] of perturbed Chebyshev polynomials defined by the recurrence f0(x) = b, f1(x) = x − c, fn+1(x) = (x − d)fn(x) − afn−1(x), n ≥ 1, where a, b, c, d ∈ R and a> 0. These polynomials generalize the Chebyshev polynomials, which are obtained by setting a = 1/4, c = d = 0 and b ∈ {1, 2}. At the core of the method, two algorithms for polynomial decomposition are provided, which allow to restrict the investigation to the resolution of six systems of polynomial equations in three variables. The final task is then carried out by the successful computation of reduced Gröbner bases with Maple 10. Some additional data for the calculations are available on the author’s web page.
Factoring Dickson polynomials over finite fields
 Finite Fields Appl
, 1999
"... We derive the factorizations of the Dickson polynomials Dn(X, a) and En(X, a), and of the bivariate Dickson polynomials Dn(X, a) − Dn(Y, a), over any finite field. Our proofs are significantly shorter and more elementary than those previously known. ..."
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Cited by 6 (3 self)
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We derive the factorizations of the Dickson polynomials Dn(X, a) and En(X, a), and of the bivariate Dickson polynomials Dn(X, a) − Dn(Y, a), over any finite field. Our proofs are significantly shorter and more elementary than those previously known.
Characterization Of Polynomial Prime Bidecompositions  A Simplified Proof
, 1995
"... Bidecompositions, i.e., solutions to r p = s q, play a central role in the study of uniqueness properties of complete decompositions with respect to functional composition. In [Rit22] all bidecompositions using polynomials over the complex number field have been characterized. Later the result was g ..."
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Cited by 4 (0 self)
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Bidecompositions, i.e., solutions to r p = s q, play a central role in the study of uniqueness properties of complete decompositions with respect to functional composition. In [Rit22] all bidecompositions using polynomials over the complex number field have been characterized. Later the result was generalized to more general fields. All proofs tend to be rather long and involved. The object of this paper is to develop a version that is simpler than the existing ones, while keeping completely elementary, thus making it accessible to a wider community.
Complete decomposition of Dicksontype recursive polynomials and a related Diophantine equation
, 2007
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A Note on the Discriminator
"... For f(X) ∈ Z[X], let Df (n) be the least positive integer k for which f(1),..., f(n) are distinct modulo k. Several results have been proven about the function Df in recent years, culminating in Moree’s characterization of Df (n) whenever f lies in a certain (large) subset of Z[X] and n is sufficie ..."
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Cited by 3 (2 self)
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For f(X) ∈ Z[X], let Df (n) be the least positive integer k for which f(1),..., f(n) are distinct modulo k. Several results have been proven about the function Df in recent years, culminating in Moree’s characterization of Df (n) whenever f lies in a certain (large) subset of Z[X] and n is sufficiently large. We give several improvements of Moree’s result, as well as further results on the function Df. 1
Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dicks ..."
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Cited by 2 (0 self)
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We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.
Complete decomposition of Dicksontype polynomials and related Diophantine equations
, 2007
"... We characterize decomposition over C of polynomials f (a,B) n (x) defined by the generalized Dicksontype recursive relation (n ≥ 1), f (a,B) 0 (x) = B, f (a,B) 1 (x) = x, f (a,B) n+1 (x) = xf (a,B) n (x) − af (a,B) n−1 (x), where B, a ∈ Q or R. As a direct application of the uniform decompositi ..."
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We characterize decomposition over C of polynomials f (a,B) n (x) defined by the generalized Dicksontype recursive relation (n ≥ 1), f (a,B) 0 (x) = B, f (a,B) 1 (x) = x, f (a,B) n+1 (x) = xf (a,B) n (x) − af (a,B) n−1 (x), where B, a ∈ Q or R. As a direct application of the uniform decomposition result, we fully settle the finiteness problem for the Diophantine equation f (a,B) n (x) = f (â, ˆ B) m This extends and completes work of Dujella/Tichy and Dujella/Gusić concerning Dickson polynomials of the second kind. The method of the proof involves a new sufficient criterion for indecomposability of polynomials with fixed degree of the right component. (y).
ON EXPONENTIAL SUMS, NOWTON IDENTITIES AND DICKSON POLYNOMIALS OVER FINITE FIELDS
"... Abstract. Let Fq be a finite field, Fqs be an extension of Fq, let f(x) ∈ Fq[x] be a polynomial of degree n with gcd(n, q) = 1. We present a recursive formula for evaluating the exponential sum ∑ c∈Fqs χ(s) (f(x)). Let a and b be two elements in Fq with a ̸ = 0, u be a positive integer. We obtain ..."
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Abstract. Let Fq be a finite field, Fqs be an extension of Fq, let f(x) ∈ Fq[x] be a polynomial of degree n with gcd(n, q) = 1. We present a recursive formula for evaluating the exponential sum ∑ c∈Fqs χ(s) (f(x)). Let a and b be two elements in Fq with a ̸ = 0, u be a positive integer. We obtain an estimate of the exponential sum ∑ c∈F ∗ qs χ(s) (acu + bc−1), where χ (s) is the lifting of an additive character χ of Fq. Some properties of the sequences constructed from these exponential sums are provided also. 1.