Using the Aubry-Perret bound, the paper shows that image inclusion and preimage multiplicity conditions on rational functions over finite fields imply one is a composition of the other.
W eil, Sur les Courbes Alg´ ebriques et les Vari´ et´ es qui s’en D´ eduisent, Hermann et Cie., Paris
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An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields
Using the Aubry-Perret bound, the paper shows that image inclusion and preimage multiplicity conditions on rational functions over finite fields imply one is a composition of the other.