Abstract
We prove an effective form of Hilbert’s irreducibility theorem for polynomials over a global field K. More precisely, we give effective bounds for the number of specializations \(t \in {{\cal O}_K}\) that do not preserve the irreducibility or the Galois group of a given irreducible polynomial F(T, Y) ∈ K[T, Y]. The bounds are explicit in the height and degree of the polynomial F(T, Y), and are optimal in terms of the size of the parameter \(t \in {{\cal O}_K}\). Our proofs deal with the function field and number field cases in a unified way.
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Acknowledgments
M. Paredes was supported in part by a CONICET Postdoctoral Fellowship. R. Sasyk was supported in part through the grant PICT 2017-0883. We thank the anonymous referee for the careful reading of the manuscript and their many suggestions.
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Paredes, M., Sasyk, R. Effective Hilbert’s irreducibility theorem for global fields. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2604-7
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DOI: https://doi.org/10.1007/s11856-023-2604-7