Abstract
We investigateV f , the cardinality of the value set of a polynomialf of degreen over a finite field of cardinalityq. It has been shown that iff is not bijective, thenV f ≤q−(q−1)/n. Polynomials do exist which essentially achieve that bound. We do prove that if the degree off is prime to the characteristic andf is not bijective, then asymptoticallyV f ≤(5/6)q. We consider related problems for curves and higher dimensional varieties. This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well.
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Both authors partially supported by the NSF.
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Guralnick, R., Wan, D. Bounds for fixed point free elements in a transitive group and applications to curves over finite fields. Isr. J. Math. 101, 255–287 (1997). https://doi.org/10.1007/BF02760932
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DOI: https://doi.org/10.1007/BF02760932