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

Karl Dilcher, Larry Ericksen (2014)

Communications in Mathematics

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 we state a...

Representations of multivariate polynomials by sums of univariate polynomials in linear forms

A. Białynicki-Birula, A. Schinzel (2008)

Colloquium Mathematicae

The paper is concentrated on two issues: presentation of a multivariate polynomial over a field K, not necessarily algebraically closed, as a sum of univariate polynomials in linear forms defined over K, and presentation of a form, in particular a zero form, as the sum of powers of linear forms projectively distinct defined over an algebraically closed field. An upper bound on the number of summands in presentations of all (not only generic) polynomials and forms of a given number of variables and...

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