Abstract:
We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly $n$. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to $n=7$. Moreover, we obtain results regarding small values of polynomials and approximation to a real number by algebraic integers and units in small prescribed degree. The main ingredient are irreducibility criteria for integral linear combinations of coprime integer polynomials. Moreover, for cubic polynomials, these criteria improve results of Győry on a problem of Szegedy.
Key words and phrases:wirsing's problem, exponents of Diophantine approximation, irreducibility of integer polynomials.
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