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

Karl Dilcher; Larry Ericksen

Communications in Mathematics (2014)

  • Volume: 22, Issue: 1, page 77-102
  • ISSN: 1804-1388

Abstract

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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 we state a few conjectures.

How to cite

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Dilcher, Karl, and Ericksen, Larry. "Reducibility and irreducibility of Stern $(0,1)$-polynomials." Communications in Mathematics 22.1 (2014): 77-102. <http://eudml.org/doc/261964>.

@article{Dilcher2014,
abstract = {The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials $a(n;x)$ defined by $a(0;x)=0$, $a(1;x)=1$, $a(2n;x)=a(n;x^2)$, and $a(2n+1;x)=x\,a(n;x^2)+a(n+1;x^2)$; 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 few conjectures.},
author = {Dilcher, Karl, Ericksen, Larry},
journal = {Communications in Mathematics},
keywords = {Stern sequence; Stern polynomials; reducibility; irreducibility; cyclotomic polynomials; discriminants; zeros; Stern sequence; Stern polynomials; reducibility; irreducibility; cyclotomic polynomials; discriminants; zeros},
language = {eng},
number = {1},
pages = {77-102},
publisher = {University of Ostrava},
title = {Reducibility and irreducibility of Stern $(0,1)$-polynomials},
url = {http://eudml.org/doc/261964},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Dilcher, Karl
AU - Ericksen, Larry
TI - Reducibility and irreducibility of Stern $(0,1)$-polynomials
JO - Communications in Mathematics
PY - 2014
PB - University of Ostrava
VL - 22
IS - 1
SP - 77
EP - 102
AB - The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials $a(n;x)$ defined by $a(0;x)=0$, $a(1;x)=1$, $a(2n;x)=a(n;x^2)$, and $a(2n+1;x)=x\,a(n;x^2)+a(n+1;x^2)$; 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 few conjectures.
LA - eng
KW - Stern sequence; Stern polynomials; reducibility; irreducibility; cyclotomic polynomials; discriminants; zeros; Stern sequence; Stern polynomials; reducibility; irreducibility; cyclotomic polynomials; discriminants; zeros
UR - http://eudml.org/doc/261964
ER -

References

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