Abstract
The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form \(f(x)-f(y)\).
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Notes
Another way to show that f(x) is a polynomial in t is to observe that x is integral over K[t], since x is a root of \((x+\beta /(\alpha -1))^d-t\) over K[t]. Since K[t] is integrally closed, the function f(x) is a polynomial in t.
Another way to see this is to observe that \({{\,\mathrm{\text {{LF}}}\,}}(P_1(x,y))\ldots {{\,\mathrm{\text {{LF}}}\,}}(P_s(x,y))={{\,\mathrm{\text {{LF}}}\,}}(f(x)-f(y))=x^n-y^n=\prod \nolimits _{i=0}^{n-1}(x-\zeta ^iy)\), which implies that no two of \({{\,\mathrm{\text {{LF}}}\,}}(P_k(x,y))\) are associates. Hence no two of \(P_1(x,y),\ldots , P_s(x,y)\) are associates.
See Problem 12138 in Amer. Math. Monthly, 126:8, 2019, p. 756.
References
Artin, E.: Algebra with Galois Theory, Courant lecture notes, vol. 15. AMS (2007)
Cassels, J.W.S.: Factorization of polynomials in several variables. In: Aubert K.E., Ljunggren W. (eds) Proceedings of the 15th Scandinavian Congress Oslo. Lecture Notes in Mathematics, vol. 118. Springer, Berlin (1968)
Davenport, H., Schinzel, A.: Two problems concerning polynomials. J. Reine Angew. Math. 214(5), 386–391 (1964)
Ozeki, K.: A certain property of polynomials. Aequ. Math. 25, 247–252 (1982)
Prasolov, V.V.: Polynomials. Springer, Berlin (2004)
Schinzel, A.: Polynomials with Special Regard to Reducibility. Cambridge University Press, Cambridge (2004)
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Bekker, B., Podkopaev, O. On the functional equation \(\varvec{f(\alpha x+\beta )=f(x)}\). Aequat. Math. 96, 349–360 (2022). https://doi.org/10.1007/s00010-021-00833-7
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DOI: https://doi.org/10.1007/s00010-021-00833-7