Subgroups Generated by Rational Functions in Finite Fields

For a large prime $p$, a rational function $ψ\in F_p(X)$ over the finite field $F_p$ of $p$ elements, and integers $u$ and $H\ge 1$, we obtain a lower bound on the number consecutive values $ψ(x)$, $x = u+1, \ldots, u+H$ that belong to a given multiplicative subgroup of $F_p^*$.

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