ON THE IRREDUCIBILITY OF SUM OF TWO RECIPROCAL POLYNOMIALS

Title & Authors
ON THE IRREDUCIBILITY OF SUM OF TWO RECIPROCAL POLYNOMIALS
Bang, Minsang; Kwon, DoYong;

Abstract
For a certain kind of reciprocal polynomials $\small{P(x),Q(x){\in}{\mathbb{Z}}[x}$$\small{]}$$\small{}$, their sums are considered. We demonstrate that the Mahler measure of polynomials plays a role to prove the irreducibility of the sums over the field of rationals.
Keywords
irreducible polynomial;reciprocal polynomial;Mahler measure;
Language
English
Cited by
References
1.
M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, Pisot and Salem numbers, Birkhauser Verlag, Basel, 1992.

2.
L. E. Dickson, Criteria for the irreducibility of a reciprocal equation, Bull. Amer. Math. Soc. 14 (1908), no. 9, 426-430.

3.
H. Kleiman, On irreducibility criteria of Dickson, J. London Math. Soc. (2) 7 (1974), 467-475.

4.
L. Kronecker, Zwei Satze uber Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.

5.
D. Y. Kwon, Minimal polynomials of some beta-numbers and Chebyshev polynomials, Acta Arith. 130 (2007), no. 4, 321-332.

6.
D. Y. Kwon, Reciprocal polynomials with all zeros on the unit circle, Acta Math. Hungar. 131 (2011), no. 3, 285-294.

7.
P. Lakatos, On zeros of reciprocal polynomials, Publ. Math. Debrecen 61 (2002), no. 3-4, 645-661.

8.
D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461-479.

9.
C. L. Siegel, Algebraic integers whose conjugates lie in the unit circle, Duke Math. J. 11 (1944), 597-602.

10.
C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175.