We provide several characterizations of algebraic curves X : F ( x , y ) = 0 defined over ℚ ¯ that satisfy the following property: there exist a number field k and an infinite set S ⊂ k such that, for every y ∈ S , the roots of the polynomial F ( x , y ) belong to k .
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Fedor Pakovich (2026) studied this question.
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