Glasnik Matematicki, Vol. 54, No. 1 (2019), 11-20.

EXPLICIT BOUNDS FOR COMPOSITE LACUNARY POLYNOMIALS

Christina Karolus

Department of Mathematics, University of Salzburg, 5020 Salzburg, Austria
e-mail: christina.karolus@sbg.ac.at

Abstract.   Let f, g, h ℂ [x] be non-constant complex polynomials satisfying f(x)=g(h(x)) and let f be lacunary in the sense that it has at most l non-constant terms. Zannier proved in [9] that there exists a function B1(l) on ℕ, depending only on l and with the property that h(x) can be written as the ratio of two polynomials having each at most B1(l) terms. Here, we give explicit estimates for this function or, more precisely, we prove that one may take for instance

B1(l)=(4l)(2l)(3l)l+1.
Moreover, in the case l=2, a better bound is obtained using the same strategy.

2010 Mathematics Subject Classification.   11C08, 11R09

Key words and phrases.   Decomposable polynomials, lacunary polynomials

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.1.02

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