#### 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 *B*_{1}(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 *B*_{1}(l) terms. Here, we give explicit estimates for this function or, more precisely, we prove that one may take for instance

*B*_{1}(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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