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arXiv · 2202.10420

Effective Hilbert's Irreducibility Theorem for global fields

Marcelo Paredes·Román Sasyk
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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 $t\in \mathcal{O}_K$ that do not preserve the irreducibility or the Galois group of a given irreducible polynomial $F(T,Y)\in K[T,Y]$. 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 $t\in \mathcal{O}_K$. Our proofs deal with the function field and number field cases in a unified way.

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Marcelo Paredes, Román Sasyk. 2022-02-21. Effective Hilbert's Irreducibility Theorem for global fields. https://arxiv.org/abs/2202.10420

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