Displaying similar documents to “Parametrization of integral values of polynomials”

Reducibility and irreducibility of Stern ( 0 , 1 ) -polynomials

Karl Dilcher, Larry Ericksen (2014)

Communications in Mathematics

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The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials a ( n ; x ) defined by a ( 0 ; x ) = 0 , a ( 1 ; x ) = 1 , a ( 2 n ; x ) = a ( n ; x 2 ) , and a ( 2 n + 1 ; x ) = x a ( n ; x 2 ) + a ( n + 1 ; x 2 ) ; these polynomials are Newman polynomials, i.e., they have only 0 and 1 as coefficients. In this paper we prove numerous reducibility and irreducibility properties of these polynomials, and we show that cyclotomic polynomials play an important role as factors. We also prove several related results, such as the fact that a ( n ; x ) can only have simple zeros, and...

On three questions concerning 0 , 1 -polynomials

Michael Filaseta, Carrie Finch, Charles Nicol (2006)

Journal de Théorie des Nombres de Bordeaux

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We answer three reducibility (or irreducibility) questions for 0 , 1 -polynomials, those polynomials which have every coefficient either 0 or 1 . The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible 0 , 1 -polynomial. The third is the analogous question for exponents of irreducible 0 , 1 -polynomials.

Generalized Kummer theory and its applications

Toru Komatsu (2009)

Annales mathématiques Blaise Pascal

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In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that ζ ∉ k and ω ∈ k where ζ is a primitive n -th root of unity and ω = ζ + ζ - 1 . In particular, this result with ζ ∈ k implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.